Prime Order Isometries of Unimodular Lattices and Automorphisms of IHS Manifolds

Abstract

We characterize conjugacy classes of isometries of odd prime order in unimodular |${\mathbb {Z}}$|-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of the known deformation types of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.

1 Introduction

1.1 Isometries of unimodular lattices

A lattice consists of a finitely generated free |${\mathbb {Z}}$|-module |$L$| equipped with a nondegenerate symmetric bilinear form |$b\colon L \times L \rightarrow {\mathbb {Q}}$|⁠. It is called integral if |$b$| is integer valued. An integral lattice is even if for all |$x\in L$| we have |$b(x,x) \in 2{\mathbb {Z}}$|⁠, otherwise it is odd. An integral lattice is called unimodular if it is of determinant |$\pm 1$|⁠. An even unimodular lattice of signature |$(l_+, l_-)$| exists if and only if |$l_+ \equiv l_- \pmod {8}$|⁠, while in the odd case there is no restriction on the signature. If |$l_+$| and |$l_-$| are both nonzero, then the odd (even) unimodular lattice of signature |$(l_+, l_-)$| is unique up to isometry.

We treat the following problem: does |$L$| admit an isometry |$f \in O(L)$| of odd prime order |$p$|? If yes, classify their conjugacy classes.

In order to do so, we define the invariant and coinvariant lattices of |$f$| as

$$ \begin{align*} &L^f=\{x \in L \mid f(x)=x\} \quad \mbox{ and }\quad L_f=\big(L^f\big)^\perp.\end{align*}$$

If |$L$| is unimodular, then |$L^f$| and |$L_f$| are in fact |$p$|-elementary. In particular, |$|\det L_f|$| is a power of |$p$|⁠. Our 1st main result is the following theorem.

 

Theorem 1.1.

Let |$p$| be an odd prime number. There exists a unimodular lattice |$L$| of signature |$(l_+,l_-)$| admitting an isometry |$f \in O(L)$| of order |$p$| with coinvariant lattice |$L_f$| of signature |$(s_+,s_-)$| and determinant |$\det L_f = \left ((-1)^{s_-}\right ) p^n$| if and only if there exists |$m\in {\mathbb {Z}}_{\geq 0}$| such that

• (I)

  |$l_+ + l_- - s_+ - s_-> 0$| and |$L$| is odd or

• (II)

  |$l_+ \equiv l_- \pmod {8}$| and |$L$| is even;

• 1.

  |$s_+ + s_- = (n+2m)(p-1)>0$|⁠;

• 2.

  |$s_+, s_- \in 2{\mathbb {Z}}$|⁠;

• 3.

  |$s_+ \leq l_+$|⁠, |$ \quad s_- \leq l_-$|⁠;

• 4.

  |$s_+ + s_- + n \leq l_+ + l_-$|⁠;

• 5.

  if |$n = 0$| or |$n=l_+ +l_- - s_+ - s_-$|⁠, then |$s_+ \equiv s_- \pmod {8}$|⁠.

The invariants of the theorem determine the genus of the invariant and coinvariant lattice of |$f$|⁠. But in order to distinguish between conjugacy classes, we need additional invariants.

Let |$\zeta _p$| be a primitive |$p$|-th root of unity. The coinvariant lattice |$L_f$| has the structure of a |${\mathbb {Z}}[\zeta _p]$|-module via |$\zeta _p \cdot x = f(x)$|⁠. Moreover, it carries a nondegenerate |${\mathbb {Q}}[\zeta _p]$|-valued hermitian form |$h$| defined by

$$ \begin{align*} &h(x,y) = \sum_{i=0}^{p-1} b(x,f^i(y))\zeta_p^i, \qquad x,y \in L_f.\end{align*}$$

Let |$k = {\operatorname {rk}}_{{\mathbb {Z}}[\zeta _p]} L_f=n+2m$|⁠. Its determinant lattice|$\det (L_f,h_f)$| consists of the top exterior power |$\det L_f = \bigwedge ^k L_f$| equipped with the hermitian form |$\det h_f=\wedge ^k h_f$| defined by |$(\det h_f)(x,x)=\det (h_f(x_i,x_j))_{1\leq i,j\leq k}$| for |$x=x_1 \wedge \ldots \wedge x_k \in \det L_f$|⁠. Let |$E={\mathbb {Q}}[\zeta _p]$| and |$K= {\mathbb {Q}}[\zeta _p + \zeta _p^{-1}]$|⁠. Since |$\det L_f$| is a module of rank one, it is isomorphic to a fractional ideal |$I_f$| of |$E$|⁠. Seen as an element of the class group |${\operatorname {C}}(E)$|⁠, it is independent of the isomorphism and called the Steinitz class of |$L_f$|⁠. It measures the deviation of |$L_f$| from being a free |${\mathbb {Z}}[\zeta _p]$|-module. The Steinitz class |$[I_f]$| is contained in the relative class group|${\operatorname {C}}(E/K)$|⁠, which is defined as the kernel of the norm map |${\operatorname {C}}(E) \rightarrow {\operatorname {C}}(K)$|⁠. The order of |${\operatorname {C}}(E/K)$| is called the relative class number.

Finally, the signatures of |$(L,f)$| are given by the signatures |$(k_i^+,k_i^-)\in (2{\mathbb {N}})^2$| of the real quadratic spaces

$$ \begin{align}& K_i=\ker(f_{\mathbb{R}}+f_{\mathbb{R}}^{-1}- \zeta_p^i - \zeta_p^{-i}), \quad i \in \{1,\dots, (p-1)/2\}. \end{align}$$

(1)

They satisfy |$(s_+,s_-)=\sum _{i=1}^{(p-1)/2}(k_i^+,k_i^-)$| and |$k_i^+ + k_i^-$| is independent of |$i$|⁠. In fact, any collection of even nonnegative |$k_i^\pm $| satisfying these conditions occurs.

 

Theorem 1.2.

Let |$L$| be a unimodular lattice, |$f,g \in O(L)$| be isometries of odd prime order |$p$|⁠, |$E={\mathbb {Q}}[\zeta _p]$| and |$K= {\mathbb {Q}}[\zeta _p + \zeta _p^{-1}]$|⁠. Suppose that |$L_f$| is indefinite or of rank |$p-1$|⁠. Then |$f$| is conjugate to |$g$| if and only if

1. the invariant lattices |$L^f$| and |$L^g$| are isometric,

2. |$f$| and |$g$| have the same signatures,

3. the determinant lattices |$\det (L_f,h_f)$| and |$\det (L_g,h_g)$| are isometric.

The number of conjugacy classes with the same signature and invariant lattice as |$f$| is given by the relative class number |$\# {\operatorname {C}}(E/K)$|⁠.

If moreover the relative class number is odd, then the determinant lattices are isometric if and only if |$L_f$| and |$L_g$| have the same Steinitz invariant in |${\operatorname {C}}(E/K)$|⁠.

 

Remark 1.3.

For |$L_f$| or |$L^f$| definite, a classification of conjugacy classes includes an enumeration of isometry classes in the respective genera. This is typically of algorithmic nature and has been carried out for instance in [31, 44] where Kirschmer and Nebe classify extremal unimodular lattices of rank |$48$| admitting certain prime order isometries.

The analogous results for |$p=2$| are easily derived from [47, Thms. 3.6.2, 3.6.3, §16] due to Nikulin. They boil down to a classification of primitive |$2$|-elementary sublattices of unimodular lattices up to the action of the orthogonal group.

Quebbemann [51] classifies odd prime order automorphisms of unimodular lattices taking |$(L_f,f)$| and |$L^f$| as given. The addition of a classification for hermitian and |$p$|-elementary lattices allows us to reach the effective results given above.

In a recent breakthrough, Bayer-Fluckiger [5] classifies characteristic polynomials of isometries of unimodular lattices, under the condition that the polynomial has no linear factor, which prevents us from applying her results to our study.

1.2 Automorphisms of IHS manifolds

An irreducible holomorphic symplectic (IHS) manifold is a compact complex Kähler simply connected manifold |$X$| such that |$H^0(X, \Omega ^2_X) = {\mathbb {C}} \omega _X$|⁠, where |$\omega _X$| is an everywhere nondegenerate holomorphic form. Two-dimensional IHS manifolds are K3 surfaces and in higher dimension the known deformation types of IHS manifolds are: |$K3^{\left [n\right ]}$| (i.e., deformation equivalent to Hilbert schemes of |$n$| points on a K3 surface [7, Section 6]), |$\textrm {Kum}_n$| (i.e., deformation equivalent to the |$2n$|-dimensional generalized Kummer variety of an abelian surface [7, §7]), OG|$_6$| and OG|$_{10}$| (i.e., deformation equivalent, respectively, to the six-dimensional and ten-dimensional IHS manifolds first constructed by O’Grady [48, 49]). Whether or not there are more deformation types remains a wide open question.

The cohomology group |$H^2(X,{\mathbb {Z}})$| of an IHS manifold |$X$| admits an integral |${\mathbb {Z}}$|-lattice structure (by use of the Beauville–Bogomolov–Fujiki quadratic form [20, Thm. 4.7]). Let |${\operatorname {Mon}}^2(X)\subseteq O^+(H^2(X,{\mathbb {Z}}))$| denote the monodromy group of |$X$|⁠. The natural homomorphism

$$ \begin{align}& \rho_X \colon {\operatorname{Aut}}(X) \rightarrow {\operatorname{Mon}}^2(X),\quad \sigma\mapsto (\sigma^{-1})^* \end{align}$$

(2)

is the key tool in the study of automorphisms. Its image is computed by the strong Torelli theorem and it is injective for manifolds of type |$K3^{\left [n\right ]}$| and OG|$_{10}$|⁠, while it has a small and well understood kernel for |$\textrm {Kum}_n$| and OG|$_6$| (see Table 1). If |$\sigma ^*(\omega _X)=\omega _X$|⁠, then |$\sigma $| is called symplectic and otherwise non-smyplectic. We will be interested in studying non-symplectic automorphisms |$\sigma \in {\operatorname {Aut}}(X)$| whose action on cohomology |$\rho _X(\sigma )$| is an isometry of odd prime order.

Table 1.

Monodromy of the known deformation types of IHS manifolds

Type	|$\Lambda $|	|$\mathop {\textrm {Mon}}\nolimits ^2(\Lambda )$|	|$\ker \rho _X$|	|$M$|	|$V\leq M$|	|$S$|
K3	|$U^{\oplus 3}\oplus E_8^{\oplus 2}$|	|$ O^+(\Lambda )$|	|$1$|	|$\textrm {II}_{(3,19)}$|	-	-
K3|$^{[n]}$|	|$U^{\oplus 3}\oplus E_8^{\oplus 2} \oplus \langle 2-2n \rangle $|	|$ \Gamma ^+(\Lambda )$|	|$1$|	|$\textrm {II}_{(4,20)}$|	|$\langle 2n-2 \rangle $|	|$O$|
Kum|$_n$|	|$U^{\oplus 3}\oplus \langle -2n-2 \rangle $|	|$\ker (\det \cdot \chi )^+$|	|$C_{n+1}^4 \rtimes C_2$|	|$\textrm {II}_{(4,4)}$|	|$\langle 2n+2 \rangle $|	|$SO$|
OG|$_6$|	|$U^{\oplus 3}\oplus \langle -2\rangle ^{\oplus 2}$|	|$O^+(\Lambda )$|	|$C_2^8$|	|$_{(5,5)}$|	|$\langle 2 \rangle \oplus \langle 2 \rangle $|	|$O$|
OG|$_{10}$|	|$U^{\oplus 3}\oplus E_8^{\oplus 2} \oplus A_2$|	|$O^+(\Lambda )$|	|$1$|	|$_{(5,21)}$|	|$A_2(-1)$|	|$O$|

For K3 surfaces |$H^2(X,{\mathbb {Z}})$| is unimodular, |$\rho _X$| is injective and |${\operatorname {Mon}}^2(X) = O^+(H^2(X,{\mathbb {Z}}))$|⁠. Theorems 1.1 and 1.2 combined with the strong Torelli theorem provide a new and unified proof for the known classification of non-symplectic automorphisms of K3 surfaces up to deformation (Theorem 4.2).

In the case of higher-dimensional IHS manifolds, the classification of automorphisms is significantly more involved, for multiple reasons. Firstly, the second cohomology lattice is no longer unimodular. Secondly, the monodromy group depends on the deformation class of |$X$| and may be strictly contained in |$O^+(H^2(X,{\mathbb {Z}}))$|⁠. Thirdly, birational geometry enters the picture: we can have non-isomorphic birational models of the same pair |$(X, \sigma )$|⁠.

Previous works on non-symplectic automorphisms of IHS manifolds have mainly focused on describing the invariant and coinvariant lattices of the isometry of the second cohomology group induced via the map |$\rho _X$|⁠. For manifolds of type |$K3^{\left [n\right ]}$|⁠, a complete classification of the pairs of invariant and coinvariant lattices of non-symplectic automorphisms of odd prime order |$p$| is known in the case |$n=2$|⁠, by work of Boissière et al. [9, Thm. 7.1] and Boissière et al. [8, Thm. 6.1]. A similar classification has been achieved for |$n=3,4$| by Camere and the 2nd author [14, Thm. 1.2]. Moreover, for |$n = 2,3,4$| and |$p \neq 23$|⁠, examples of automorphisms for all possible pairs of lattices have been exhibited in [9] and [14]. For manifolds of type |$\textrm {Kum}_n$|⁠, Mongardi et al. [41] have classified the pairs of invariant and coinvariant lattices when |$n=2$| and presented examples of automorphisms for the cases where the rank of the coinvariant lattice is at most |$6$|⁠. For OG|$_6$| type IHS manifolds, Grossi has recently classified the invariant and coinvariant lattices [22] of non-symplectic automorphisms of prime order.

Our study moves beyond this case-by-case analysis to achieve a complete classification that applies to all known deformation types of IHS manifolds and all possible dimensions.

 

Theorem 1.4.

Let |$X^{\prime}$| be an IHS manifold of type |$K3^{\left [n\right ]}$|⁠, |$\textrm {Kum}_n$|⁠, OG|$_6$|⁠, or OG|$_{10}$|⁠. Let |$K$| be a primitive sublattice of |$\Lambda \cong H^2(X^{\prime},{\mathbb {Z}})$|⁠. Then there exist an IHS manifold |$X$| of the same deformation type of |$X^{\prime}$|⁠, a marking |$\eta : H^2(X,{\mathbb {Z}}) \rightarrow \Lambda $| and a non-symplectic automorphism |$\sigma \in {\operatorname {Aut}}(X)$| with |$\rho _X(\sigma ) \in O(H^2(X,{\mathbb {Z}}))$| of odd prime order |$p$| such that |$\eta (H^2(X,{\mathbb {Z}})_{\rho _X(\sigma )}) = K$| and |$\eta (H^2(X,{\mathbb {Z}})^{\rho _X(\sigma )}) = K^\perp $| if and only if |$K$| is |$p$|-elementary of discriminant |$p^a$| and signature |$(2, (a+2m)(p-1)-2)$| for some nonnegative integers |$a, m$|⁠.

 

Remark 1.5.

An analogous statement of Theorem 1.4 in the case of non-symplectic involutions of manifolds of type |$K3^{\left [n\right ]}$| has been obtained by Camere et al. [15, Thm. 2.3] building on work of Joumaah [28].

Beyond the case of K3 surfaces, the isometry classes of the invariant and coinvariant lattices are in principle no longer enough to reconstruct the automorphism up to deformation, not even if we allow for birational conjugation. In Section 3, we conduct a general study of deformations and conjugations by birational morphisms of pairs |$(X, G)$|⁠, with |$X$| an IHS manifold and |$G$| a group of automorphisms. We do so by giving a moduli theoretic interpretation for the results of [10, 28] on period spaces of |$(\rho , T)$|-polarized IHS manifolds of type |$K3^{\left [n\right ]}$| and by extending them to arbitrary IHS manifolds.

 

Theorem 1.6.

Let |$X$| be an IHS manifold and |$G \leq {\operatorname {Aut}}(X)$| a group of non-symplectic automorphisms with |$\ker \rho _X \leq G$| and |$\rho _X(G)$| of odd prime order |$p$|⁠. If |$X$| is of type OG|$_{10}$|⁠, then |$(X,G)$| is determined up to deformation and birational conjugation by the isomorphism classes of the lattices |$H^2(X,{\mathbb {Z}})^G$|⁠, |$H^2(X,{\mathbb {Z}})_G$| if and only if |$p\neq 23$|⁠. For manifolds of type |$K3^{\left [n\right ]}$| and |$\textrm {Kum}_n$|⁠, the lattices determine |$(X,G)$| up to deformation and birational conjugation except for |$p=23$| and the cases in Tables 3 and 4.

Table 2.

Orbit lengths of primitive vectors

|$L$|	|$G$|	|$\sum b(k) t^k$|
|$A_2(-1)$|	|$O$|	|$t + t^{3} + t^{7} + t^{13} + t^{19} + t^{21} + t^{31} + t^{37} + t^{39} + t^{43} + t^{49}\dots $|
|$A_2(-1)$|	|$SO$|	|$t + t^{3} + 2t^{7} + 2t^{13} + 2t^{19} + 2t^{21} + 2t^{31} + 2t^{37} + 2t^{39} + 2t^{43} \dots $|
|$K_7$|	|$SO$|	|$t + 2t^{2} + 2t^{4} + t^{7} + 2t^{8} + 2t^{11} + 2t^{14} + 2t^{16} + 4t^{22} + 2t^{23} \dots $|
|$F_{23a}$|	|$O$|	|$t + t^{6} + t^{8} + t^{12} + t^{18} + t^{23} + t^{26} + t^{27} + t^{36} + t^{39} + t^{48}\dots $|
|$F_{23b}$|	|$O$|	|$t^{2} + t^{3} + t^{4} + t^{6} + t^{9} + t^{12} + t^{13} + t^{16} + t^{18} + 2t^{24} + t^{26} + t^{29}\dots $|

Table 3.

Ambiguous automorphisms for |$K3^{\left [n\right ]}$|

|$(p,r,a)$|	|$\textit {div}(V)$|	|$n$|
|$(3,2,1)$|	−	|$92, 134, 218, 248, 260, 274, 302, 400, 404, 428, 470, 482, \dots $|
|$(5,4,1)$|	|$1$|	|$26, 101, 126, 151, 226, 276, 351, 401, 476, 501, 526, 601, \dots $|
|$(5,4,3)$|	|$5$|	|$26, 101, 126, 151, 226, 276, 351, 401, 476, 501, 526, 601, \dots $|
|$(23,2,1)$|	−	|$ 7, 13, 19, 25, 27, 37, 40, 49, 53, 55, 59, 63, 73, 79, 83,88 \dots $|

Table 4.

Ambiguous automorphisms for |$\textrm {Kum}_n$|

|$(p,r,a)$|	|$\textit {div}(V)$|	|$n$|
|$(3,2,1)$|	−	|$6, 12, 18, 20, 30, 36, 38, 42, 48, 56, 60, 66, 72, 78, 90, 92,96,\dots $|
|$(5,4,1)$|	|$1$|	|$24, 99, 124, 149, 224, 274, 349, 399, 474, 499, 524, 599, 624, \dots $|
|$(7,2,1)$|	−	|$3, 7, 10, 13, 15, 21, 22, 27, 28, 31, 36, 42, 43, 45, 52, 55, 57, 63 \dots $|

Since the kernel of |$\rho _X$| is well understood, the result provides an explicit classification of non-symplectic automorphisms of the known IHS manifolds with an action on cohomology of odd prime order up to deformation and birational conjugation. (See [22] for OG|$_6$|⁠.)

 

Remark 1.7.

The introduction of hermitian lattices to the study of automorphisms allows for an approach that is at large independent on the deformation type of the manifolds. Nevertheless, we have to restrict to the known types in the hypothesis of our theorems because we need an explicit description of the group of monodromy operators. Once the monodromy group is available the methods can be used for any IHS deformation type.

2 Prime Order Isometries of Unimodular Lattices

Our path towards a classification of prime order isometries starts from the invariant and coinvariant lattices. We already remarked that the coinvariant lattice |$L_f$| of an isometry |$f \in O(L)$| of odd prime order |$p$| can be seen as a hermitian lattice over |${\mathbb {Z}}[\zeta _p]$|⁠, whose classification is well understood. Assuming that |$L$| is unimodular gives us sufficient information to characterize |$L^f$| and |$L_f$|⁠. In a final step, we check when and how |$L^f$| and |$L_f$| glue to obtain a unimodular lattice.

2.1 |${\mathbb {Z}}$|-lattices

We assume that the reader is familiar with the basic theory of |${\mathbb {Z}}$|-lattices and refer to [17, 33, 47] for further details and definitions. If the bilinar form |$b$| is understood, we may abbreviate the lattice |$(L,b)$| to |$L$| and |$b(x,x)$| to |$x^2$|⁠. In this case, we write |$L(-1)$| for |$(L,-b)$|⁠. The genus of a lattice |$L$| is denoted by |$\mathfrak {g}(L)$|⁠. We use Conway–Sloane’s notation for genera of lattices (see [17, Ch. 15]). Our convention for the discriminant of |$L$| is to set it equal to |$|\det (L)|$|⁠, which coincides with the cardinality of its discriminant group.

For a prime number |$p$|⁠, a lattice is said to be |$p$|-elementary if its discriminant group is a sum of copies of |${\mathbb {Z}}/p{\mathbb {Z}}$|⁠. The following result gives a classification of |$p$|-elementary lattices, which we will need later on.

 

Theorem 2.1.

[17, Ch. 15, Thm. 13] Let |$l_{\pm },n \in {\mathbb {N}}$|⁠, |$\epsilon \in \{\pm 1\}$|⁠. Then the genus |$\textrm {II}_{(l_+,l_-)}p^{\epsilon n}$| exists if and only if |$\epsilon $| is given for |${\operatorname {rk}}\ L \neq n$| by

$$ \begin{align}& l_+ - l_- \equiv 2 \epsilon -2 -(p-1)n \pmod{8} \end{align}$$

(3)

while if |${\operatorname {rk}}\ L =n$|⁠, then |$\epsilon = \left(\frac {-1}{p}\right)^{l_-}$|⁠.

The genus |$\text{I}_{(l_+,l_-)}p^{\epsilon n}$| exists if and only if either |${\operatorname {rk}}\ L\! <\! n$| or |${\operatorname {rk}}\ L = n$| and |$\epsilon =\left(\frac {-1}{p}\right)^{l_-}$|⁠.

2.2 Hermitian lattices

In this section, we introduce hermitian lattices—our key tool to study the coinvariant lattice. We assume that the reader is familiar with basic concepts from algebraic number theory as developed in [45].

 

Definition 2.2.

Let |$K$| be a field of characteristic zero and let |$a \in K$|⁠. Set |$E = K[x]/(x^2-a)$| and denote by |$\sigma \colon E \rightarrow E, [x] \mapsto [-x]$| the canonical involution. A hermitian space|$(V,h)$| over |$(E,\sigma )$| is a finitely generated free |$E$|-module |$V$| equipped with a nondegenerate |$K$|-bilinear form

$$ \begin{align*} &h\colon V \times V \rightarrow E,\end{align*}$$

which is |$E$|-linear in the 1st argument and satisfies |$h(x,y) = h(y,x)^\sigma $| for all |$x,y \in V$|⁠. Let |${\mathbb {Z}}_E$| be the ring of integers of |$E$|⁠. A hermitian |${\mathbb {Z}}_E$|-lattice|$(L,h)$| consists of a finitely generated |${\mathbb {Z}}_E$|-module |$L\subseteq V$| of full rank, equipped with the hermitian form |$h$|⁠.

If |$E$| and |$h$| are understood, we drop them from notation and simply speak of a hermitian lattice |$L$|⁠. For |$a_1, \dots , a_n \in K$| we denote by |$\langle a_1, \cdots , a_n \rangle $| the free |${\mathbb {Z}}_E$|-module with the standard basis |$e_1, \dots , e_n$| equipped with the hermitian form defined by |$h(e_i,e_j)=\delta _{ij}a_i$|⁠.

Notation for cyclotomic fields. Let |$p$| be an odd prime and |$\zeta $| a fixed primitive |$p$|-th root of unity. See [59, Ch. 2] for details.

• |$\Phi _p(x)$| is the |$p$|-th cyclotomic polynomial.

• |$E = {\mathbb {Q}}[\zeta ]$| and |$K = {\mathbb {Q}}[\zeta + \zeta ^{-1}]$|⁠.

• |${\operatorname {C}}(E)$|⁠, |${\operatorname {C}}(K)$| are the respective class groups.

• |${\operatorname {C}}(E/K)={\operatorname {C}}(E)/{\operatorname {C}}(K)$| is the relative class group. Its order is the relative class number.

• |$\zeta ^{\sigma }=\zeta ^{-1}$|⁠, that is, the involution |$\sigma $| is complex conjugation.

• |${\mathbb {Z}}_E = {\mathbb {Z}}[\zeta ]$| and |${\mathbb {Z}}_K = {\mathbb {Z}}[\zeta + \zeta ^{-1}]$|⁠.

• |$\pi = (1-\zeta )$|⁠, |$\mathfrak {p}=\pi \pi ^\sigma {\mathbb {Z}}_K$|⁠, |$\mathfrak {P}=\pi {\mathbb {Z}}_E$|⁠.

• |$\mathfrak {A}= \mathfrak {D}^E_{\mathbb {Q}} = \pi ^ {p-2} {\mathbb {Z}}_E$| is the absolute different.

• |$\mathfrak {R}= \mathfrak {D}^E_K= \pi {\mathbb {Z}}_E$| is the relative different.

• |$N = N^E_K$| and |$T = {\operatorname {Tr}} ^E_K$| are the relative norm and trace.

• |$\Omega (K)$| is the set of places of |$K$|⁠.

• |$\mathbb {P}(K)$| is the set of non-zero prime ideals of |${\mathbb {Z}}_K$|⁠.

• |$E_\nu := E \otimes _{K} K_\nu $| is the completion of |$E$| at a place |$\nu \in \Omega (K)$|⁠.

Notation concerning hermitian lattices.

• |$(L,h)$| is a hermitian lattice.

• |$(L,b={\operatorname {Tr}}^E_{\mathbb {Q}} \circ h)$| is its trace lattice.

• |$L_\nu := L \otimes _{{\mathbb {Z}}_K} {\mathbb {Z}}_{K_\nu }$| is the completion of |$L$| at a place |$\nu \in \Omega (K)$|⁠.

• |$\mathfrak {s}(L)=h(L,L)$| is the scale of |$(L,h)$|⁠.

• |$\mathfrak {n}(L)=\sum _{x \in L} h(x,x){\mathbb {Z}}_K$| is the norm of |$(L,h)$|⁠.

2.3 The trace lattice

If |$(L,h)$| is a hermitian |${\mathbb {Z}}[\zeta ]$|-lattice, define |$b = {\operatorname {Tr}}^E_{\mathbb {Q}} \circ \; h$|⁠. Then |$(L,b)$| is a |${\mathbb {Z}}$|-lattice, called the trace lattice. Multiplication by |$\zeta $| induces an isometry |$f$| of |$(L,b)$| with minimal polynomial |$\Phi _p(x)$|⁠. Conversely, if |$(L,b)$| is a lattice and |$f$| an isometry with minimal polynomial |$\Phi _p(x)$|⁠, then |$\zeta \cdot x = f(x)$| defines a |${\mathbb {Z}}_E$|-module structure on |$L$| and

$$ \begin{align}& h(x,y) = \sum_{i=0}^{p-1} b(x,f^i(y)) \zeta^i \in E \end{align}$$

(4)

defines a hermitian form. We sum up this result in the following proposition. Note that it involves the choice of a fixed root of unity |$\zeta $|⁠.

 

Proposition 2.3.

The trace construction sets up an equivalence between the category of hermitian |${\mathbb {Z}}_E$|-lattices |$(L,h)$| and the category consisting of pairs |$((L,b),f)$|⁠, where |$f \in O(L,b)$| is an isometry of minimal polynomial |$\Phi _p(x)$| and morphisms are |$f$|-equivariant isometries.

We denote by

$$ \begin{align*} &L^\vee=(L,b)^\vee = \{ x \in L \otimes {\mathbb{Q}} \mid b(x,L) \subseteq {\mathbb{Z}} \} \end{align*}$$

and

$$ \begin{align*} &L^\#=(L,h)^\#= \{ x \in L \otimes {\mathbb{Q}} \mid h(x,L) \subseteq {\mathbb{Z}}_E \} \end{align*}$$

the respective dual lattices. They satisfy the following relation

$$ \begin{align}& (L,b)^\vee = \mathfrak{A}^{-1} (L,h)^\#. \end{align}$$

(5)

If |$\mathfrak {B}L^\#=L$| for some fractional ideal |$\mathfrak {B}$| of |$E$|⁠, we call |$(L,h)$||$\mathcal {B}$|-modular. A |${\mathbb {Z}}_E$|-modular hermitian lattice is called unimodular.

 

Lemma 2.4.

If |$(L,h)$| is a hermitian lattice over a prime cyclotomic field such that its trace lattice |$(L,b)$| is integral, then |$(L,b)$| is even.

 

Proof.

That |$(L,b)$| is integral is equivalent to |${\operatorname {{\mathfrak {s}}}}(L) \subseteq \mathfrak {A}^{-1}$|⁠. Let |$x\in L$|⁠. Then |$h(x,x) \in {\operatorname {{\mathfrak {n}}}}(L) \subseteq K \cap {\operatorname {{\mathfrak {s}}}}(L) \subseteq K \cap \mathfrak {A}^{-1} = (\mathfrak {D}^K_{\mathbb {Q}})^{-1}$|⁠. Thus, |${\operatorname {Tr}}^E_{\mathbb {Q}}(h(x,x))=2{\operatorname {Tr}}^K_{\mathbb {Q}}(h(x,x)) \in 2{\mathbb {Z}}$|⁠.

 

Example 2.5.

The root lattice |$A_{p-1}$| admits a fixed point free isometry of order |$p$|⁠. To see this, consider the extended Dynkin diagram |$\widetilde {A_{p-1}}$|⁠. It is a regular polygon with |$p$| vertices, with a rotational symmetry |$f$| of order |$p$|⁠, which fixes the sum of the |$p$| vertices. This sum is precisely the kernel of the bilinear form of |$\widetilde {A_{p-1}}$|⁠. Thus, |$f$| descends to a fixed point free isometry of the quotient. But the quotient is isomorphic to |$A_{p-1}$|⁠; therefore, it can be seen as a hermitian |${\mathbb {Z}}[\zeta ]$| lattice of rank one.

2.4 Genera of hermitian lattices

The classification of hermitian lattices largely parallels that of |${\mathbb {Z}}$|-lattices. In this section, we recall the parts of the classification needed for our purposes.

 

Definition 2.6.

Two hermitian lattices |$L$| and |$L^{\prime}$| are said to be in the same genus if the completions |$L_\nu $| and |$L^{\prime}_\nu $| are isomorphic for all |$\nu \in \Omega (K)$|⁠.

Let |$\nu \in \Omega (K)$| be a place of |$K$|⁠. If |$\nu $| is unramified in |$E/K$|⁠, then |$E_\nu $| is either isomorphic to |$K_\nu \times K_\nu $| or |$E_\nu /K_\nu $| is an unramified field extension of degree |$2$|⁠. Otherwise, |$\nu $| is ramified and then |$E_\nu /K_\nu $| is a ramified extension of degree |$2$|⁠. A hermitian lattice over |$E_\nu $| can be decomposed as an orthogonal direct sum of modular hermitian lattices. In the sequel, we recall the classification of modular hermitian lattices [27]. A unimodular hermitian lattice over |$E_\nu $| with |$\nu $| an unramified prime is unique up to isomorphism (cf. [29, Prop. 3.3.5]). In our case, the only ramified prime is |$\pi $|⁠. We denote by

$$H_i=\left ( \begin {matrix} 0 & \pi ^i \\ (\pi ^\sigma )^i & 0\\ \end {matrix} \right )$$

a hyperbolic plane of scale |$\mathfrak {P}^i$|⁠.

 

Proposition 2.7.

[29, Prop. 3.3.5] Let |$\pi $| be a ramified prime that is coprime to |$2$|⁠. Let |$L$| be a |$\pi ^i$|-modular lattice of rank |$r$| over |$E_\pi $|⁠. If |$i$| is even, then

$$ \begin{align*} &L \cong \langle (\pi \pi^\sigma)^{i/2},\dots, (\pi \pi^\sigma)^{i/2}, \det L (\pi \pi^\sigma)^{i(1-r)/2}\rangle.\end{align*}$$

If |$i$| is odd, then

$$ \begin{align*} &L \cong H_{i}^{\oplus r/2} = \left( \begin{matrix} 0 & \pi^i \\ (\pi^\sigma)^i & 0\\ \end{matrix} \right)^{\oplus r/2}.\end{align*}$$

Let |$v_1, \dots , v_s$| be the real places of |$K$|⁠. Denote by |$n_i$| the number of negative entries in a diagonal Gram matrix of |$(L_{v_i},h_{v_i})$|⁠. They are called the signatures of |$(L,h)$|⁠. Let |$(s_+,s_-)$| be the signature of the trace lattice |$(L,b)$| and |$k^\pm _i$| the signatures of |$(L,b,f)$| as defined in the introduction. For a place |$q$| of |${\mathbb {Q}}$|⁠, we obtain the orthogonal splitting

$$ \begin{align*} &(L,h) \otimes {\mathbb{Z}}_q \cong \bigoplus_{\nu \mid q} (L,h)_\nu.\end{align*}$$

For |$q=-1$| the infinite place, we obtain the orthogonal splitting of Equation (1). By carrying out the trace construction for a hermitian lattice over |${\mathbb {C}}/{\mathbb {R}}$| of rank one, we get that |$k_i^-=2n_i$| and thus

$$ \begin{align}& s_- = 2 \sum_{i=1}^s n_i. \end{align}$$

(6)

The genus of a hermitian lattice |$(L,h)$| is uniquely determined by its signatures |$n_i$| and its modular decompositions at all primes dividing its discriminant.

 

Proposition 2.8.

[29, 3.4.2 (3) and 3.5.6] Given hermitian lattices |$(L_\nu ,h_\nu )$| at each place |$\nu \in \Omega (K)$|⁠, all but finitely many of which are unimodular, there is a global hermitian lattice |$(L,h)$| with |$(L,h)_\nu \cong (L_\nu ,h_\nu )$| at all places if and only if the set |$S=\{ \nu \in \Omega (K) | \det (L_\nu ,h_\nu ) \not \in N(E_\nu ^\times ) \}$| is of even cardinality.

In order to analyze the classes in a genus more closely, we need the following facts and notation:

• |$\mathcal {I}$| the group of fractional ideals of |$E$|⁠,

• |$J = \{\mathfrak {A} \in \mathcal {I} \mid \mathfrak {A}\mathfrak {A}^\sigma = {\mathbb {Z}}_E\}$|⁠,

• |$J_0 = \{e {\mathbb {Z}}_E \mid e \in E^\times \mbox { with } ee^\sigma =1\}$|⁠,

• |$C={\operatorname {C}}(E)$| the class group of |$E$|⁠,

• |$C_0=\{ [\mathfrak {A}] \in C \mid \mathfrak {A}=\mathfrak {A}^\sigma \}$| the subgroup of |$C$| generated by the image of |${\operatorname {C}}(K)$| and the prime ideals of |${\mathbb {Z}}_E$| ramified in |$E/K$|⁠,

• |$C/C_0 \rightarrow J/J_0$| induced by |$\mathfrak {A} \mapsto \mathfrak {A}/\mathfrak {A}^\sigma $| is an isomorphism.

In our case, only a single prime ideal of |${\mathbb {Z}}_E$| ramifies in |$E/K$|⁠, namely the principal ideal |$\mathfrak {P}=(1-\zeta )$|⁠. By [59, Thm. 4.14], the natural homomorphism |${\operatorname {C}}(K) \rightarrow {\operatorname {C}}(E)$| is injective. This yields that |$\#(C/C_0)=\#{\operatorname {C}}(E)/\#{\operatorname {C}}(K)=\#{\operatorname {C}}(E/K)$| is the relative class number.

 

Proposition 2.9.

Let |$(L_1,h_1)$| and |$(L_2,h_2)$| be hermitian lattices of rank one in the same genus. Write |$L_i=\mathfrak {I}_i x_i$| and |$d_i=h_i(x_i,x_i)$| for |$x_i \in L_i$|⁠, |$\mathfrak {I}_i \in \mathcal {I}$| and |$i=1,2$|⁠. Let |$n \in E$| with |$n n^\sigma = d_1/d_2$|⁠. Then |$(L_1,h_1)$| is isometric to |$(L_2,h_2)$| if and only if

$$ \begin{align*} &\mathfrak{I}_1/\mathfrak{I}_2 \cdot n \in J_0.\end{align*}$$

Furthermore, |$\# \mathfrak {g}(L,h) = \# {\operatorname {C}}(E/K)$|⁠.

 

Remark 2.10.

In terms of the isomorphism |$J/J_0 \cong C/C_0$|⁠, this means that we find a fractional ideal |$\mathfrak {I}$| of |$E$| with |$\mathfrak {I}_1/\mathfrak {I}_2 \cdot n =\mathfrak {I}/\mathfrak {I}^\sigma $|⁠. Then |$\mathfrak {I}_1/\mathfrak {I}_2 \cdot n \in J_0$| if and only if |$[\mathfrak {I}] \in C_0$|⁠.

 

Proof.

Since |$(L_1,h_1)$| and |$(L_2,h_2)$| are in the same genus, the corresponding quadratic spaces |$(V_i,h_i)$| are isomorphic. In particular, |$d_1/d_2$| is a norm. Hence, we find |$n$| as in the proposition that gives the isometry |$g\colon (V_1,h_1) \rightarrow (V_2,h_2)$|⁠, |$x_1 \mapsto n x_2$|⁠. The image of |$L_1$| is |$n \mathfrak {I}_1 x_2$|⁠. Now |$g(L_1)$| and |$L_2$| are isomorphic if and only if we find an isometry |$e \in O(V_2,h_2)\cong J_0$| with |$e n \mathfrak {I}_1 x_2 = \mathfrak {I}_2 x_2$|⁠, i.e. |$\mathfrak {I}_1/\mathfrak {I}_2 \cdot n \in J_0$|⁠. Since the lattices in |$\mathfrak {g}(L_2,h_2)$| are all of the form |$(\mathfrak {A}L_2,h_2)$|⁠, |$\mathfrak {A} \in J$|⁠, the number of classes in the genus is |$\# J / J_0$|⁠.

Next we use the case of rank one to settle the indefinite case of rank at least two. The crucial ingredient is the strong approximation theorem for unitary groups by Shimura [55].

 

Definition 2.11.

Let |$(L,h)$| be a hermitian lattice of rank |$r$|⁠. Its determinant lattice |$\det (L,h)$| consists of the module |$\det L = \bigwedge ^r L$| equipped with the hermitian form |$\wedge ^{r}h$|⁠. Explicitly, if |$L$| is given by the pseudo basis |$L = \sum _{i=1}^r \mathfrak {A}_i x_i$| for fractional ideals |$\mathfrak {A}_i$| of |$E$| and |$x_i \in L$|⁠, then |$\det L = \prod _{i=1}^r \mathfrak {A}_i x$| where |$x=x_1 \wedge \dots \wedge x_r$|⁠. Furthermore, |$(\det h)(x,x) = \det (h(x_i,x_j))_{1\leq i,j \leq r}$|⁠.

Let |$L,L^{\prime}$| be |${\mathbb {Z}}_E$| lattices in the |$E$|-vector space |$V$|⁠. The index ideal is |$[L: L^{\prime}]_{{\mathbb {Z}}_E}=\{\det \sigma | \sigma \in {\operatorname {Hom}}_{{\mathbb {Z}}_E}(L,L^{\prime})\}$|⁠. More concretely, let |$L,L^{\prime}\subseteq V=E^n$| with standard basis |$e_1,\dots , e_n$| be given by pseudo bases |$[\mathfrak {A}_i,x_i]$| and |$[\mathfrak {B}_i,y_i]$|⁠. Then |$[L:L^{\prime}] = \prod \mathfrak {A}_i / \mathfrak {B}_i \cdot (\det X / \det Y)$| where |$X$| is the matrix with rows |$x_i \in E^n$| and |$Y$| the matrix with rows |$y_i$|⁠. We compute |$\det L= \prod \mathfrak {A}_i \det (X) e_1 \wedge \dots \wedge e_n$| and |$\det L^{\prime}= \prod \mathfrak {B}_i \det (Y) e_1 \wedge \dots \wedge e_n$|⁠. Clearly,

$$ \begin{align*} &[L: L^{\prime}]_{{\mathbb{Z}}_E}=[\det L^{\prime}: \det L]_{{\mathbb{Z}}_E}.\end{align*}$$

The following is inspired by Bayer–Fluckiger’s use of determinant lattices in the classification of unimodular hermitian lattices [4].

 

Proposition 2.12.

Let |$E = {\mathbb {Q}}[\zeta ]$| and |$K={\mathbb {Q}}[\zeta + \zeta ^{-1}]$| where |$\zeta $| is a primitive |$p$|-th root of unity and |$p$| an odd prime. Let |$(L,h)$| be a hermitian lattice over |$E/K$|⁠. Suppose that |$(L,h)$| is of rank at least two and indefinite. Then the number of classes in the genus |$\mathfrak {g}(L,h)$| is the relative class number |$\#C(E/K)$|⁠. Two lattices in |$\mathfrak {g}(L,h)$| are isometric if and only if they have isometric determinant lattices.

 

Proof.

Let |$(L,h)$| be indefinite of rank two with ambient hermitian space denoted by |$(V,h)$| and |$\mathfrak {q} \in \mathbb {P}(K)$| be a prime. Recall the following definitions and facts from [30]:

• |$\mathcal {E}_0^{\mathfrak {q}}=\{u \in {\mathbb {Z}}_{E_{\mathfrak {q}}}^* \mid uu^\sigma =1\}$|⁠,

• |$\mathcal {E}_1^{\mathfrak {q}}=\{u/u^{\sigma } \mid u \in {\mathbb {Z}}_{E_{\mathfrak {q}}}^*\} \subseteq \mathcal {E}_0^{\mathfrak {q}}$|⁠,

• |$\mathcal {E}(L_{\mathfrak {q}})=\{\det (g) \mid g \in O(L_{\mathfrak {q}},h_{\mathfrak {q}}) \}\subseteq \mathcal {E}_0^{\mathfrak {q}}$|⁠,

• |$P(L)=\{\mathfrak {q} \in \mathbb {P}(E) \mid \mathcal {E}(L_{\mathfrak {q}}) \neq \mathcal {E}_0^{\mathfrak {q}}\}$| consists only of primes ramified in |$E/K$|⁠,

• |$\mathcal {E}(L) = \prod _{\mathfrak {q} \in \mathbb {P}(L)} \mathcal {E}_0^{\mathfrak {q}}/\mathcal {E}(L_{\mathfrak {q}})$|⁠,

• |$R(L)=\{(e \mathcal {E}(L_{\mathfrak {q}}))_{\mathfrak {q}\in P(L)} \in \mathcal {E}(L) \mid e \in {\mathbb {Z}}_E \mbox { and } ee^\sigma =1\}$|⁠.

The number of special genera in the genus of |$L$| is |$[C:C_0] [\mathcal {E}(L):R(L)]$| (cf. [55, 5.28]). Since the genus is indefinite, each special genus consists of a single isometry class. By definition, two lattices in |$(V,h)$| that are isometric lie in the same special genus. Thus, the number above is actually the number of classes in the genus. As |$\mathfrak {p}$| is the unique prime ramified in |$E/K$|⁠, |$\mathcal {E}(L)=\mathcal {E}_0^{\mathfrak {p}} /\mathcal {E}(L_{\mathfrak {p}})$|⁠. By [30, Thm. 3.7], |$\mathcal {E}_1^{\mathfrak {p}} \subseteq \mathcal {E}(L_{\mathfrak {p}})$| and further the quotient |$\mathcal {E}_0^{\mathfrak {p}}/ \mathcal {E}_1^{\mathfrak {p}}$| is of order |$2$| generated by |$-1 \cdot \mathcal {E}_1^{\mathfrak {p}}$| (cf. [30, Lem. 3.5]). Since |$(-1\cdot \mathcal {E}_1^{\mathfrak {p}})$| is clearly in |$R(L)$|⁠, the index |$[\mathcal {E}(L):R(L)]=1$|⁠. This implies the following: if |$L^{\prime} \subseteq V$| is a lattice in the genus of |$L$|⁠, then it is isometric to |$(L,h)$| if and only if the index |$[L: L^{\prime}]_{{\mathbb {Z}}_E}=[\det L: \det L^{\prime}]_{{\mathbb {Z}}_E}$| lies in |$J_0$|⁠, that is, if and only if |$\det (L,h)$| is isometric to |$\det (L^{\prime},h)$|⁠.

 

Lemma 2.13.

Suppose that the relative class number of |$C(E/K)$| is odd. Then

$$ \begin{align*} &\Xi\colon J/J_0 \rightarrow C(E/K),\quad \mathfrak{A}J_0 \mapsto [\mathfrak{A}]\end{align*}$$

is an isomorphism.

 

Proof.

Since in our setting |$\# J/J_0 = \#{\operatorname {C}}(E/K)$|⁠, it suffices to show that |$\Xi $| is injective. Let |$({\mathbb {Z}}_K^\times )^+$| be the set of totally positive units of |${\mathbb {Z}}_K$|⁠. Suppose that the relative class number is odd, then |$({\mathbb {Z}}_E^\times )^+= N({\mathbb {Z}}_E^\times )$| by [56, Prop. A.2]. Let |$\mathfrak {A}=a{\mathbb {Z}}_E \in J$| be principal. By the definition of |$J$|⁠, |$aa^\sigma {\mathbb {Z}}_E = \mathfrak {A}\mathfrak {A}^\sigma = {\mathbb {Z}}_E$|⁠. Moreover, |$aa^\sigma $| lies in |$K$| and is totally positive since it is a norm. Thus |$aa^\sigma $| lies in |$ ({\mathbb {Z}}_K^\times )^+ = N({\mathbb {Z}}_E^\times )$| and we can find |$b \in {\mathbb {Z}}_E^\times $| with |$bb^\sigma = aa^\sigma $|⁠. Hence |$\mathfrak {A} = a {\mathbb {Z}}_E = a/b {\mathbb {Z}}_E$| lies in |$J_0$|⁠.

Let |$(L,h)$| be a hermitian lattice. The volume of |$(L,h)$| is the fractional ideal |$\mathfrak {v}(L)=[L^\#: L]_{{\mathbb {Z}}_E}$|⁠. Let |$[\mathfrak {A}_i,e_i]$| be a pseudo basis for |$L$|⁠. Set |$\mathfrak {I} = \prod _{i=1}^n \mathfrak {A}_i$| and |$a = \det (h(e_i,e_j))_{1\leq ij \leq r}$|⁠. Then the volume of |$(L,h)$| is |$\mathfrak {v}(L)= a \mathfrak {I} \mathfrak {I}^\sigma $|⁠. Note that the class of |$\mathfrak {I}$| in |$C(E)$| is the Steinitz invariant of |$L$|⁠.

 

Lemma 2.14.

Suppose that the relative class number of |$C(E/K)$| is odd. Let |$(L,h)$| be a hermitian lattice that is indefinite or of rank one. Suppose that the volume |$\mathfrak {v}(L)=\mathfrak {p}^l$| for some |$l$| and let |$L^{\prime} \in \mathfrak {g}(L,h)$|⁠. Then |$L$| is isomorphic to |$L^{\prime}$| if and only if they have the same Steinitz class in |$C(E/K)$|⁠.

 

Proof.

Let |$\mathfrak {I}$| be as above. Since |$\mathfrak {p}^l = a \mathfrak {I} \mathfrak {I}^\sigma $| is principal, the Steinitz class |$[\mathfrak {I}]$| lies in the kernel of the norm map, that is, it is an element of the relative class group. Let |$(L^{\prime},h)$| be in the genus of |$(L,h)$|⁠. Its Steinitz invariant is given by |$[\mathfrak {I}] [L:L^{\prime}] \in C(E/K)$|⁠.

As |$\Xi $| is an isomorphism, mapping |$L^{\prime}$| to its Steinitz class gives a bijection between the isometry classes in the genus of |$(L,h)$| and the relative class group.

2.5 Fixed point free isometries of prime order

Let |$L$| be a unimodular lattice and |$f \in O(L)$| of prime order |$p$|⁠. Since |$L$| is unimodular, there is an |$f$|-equivariant isomorphism of the discriminant groups |$(L^f)^\vee / L^f \cong L_f^\vee / L_f$|⁠. In particular, we see that |$f$| acts trivially on the discriminant group of |$L_f$|⁠. This observation sets the topic of this section.

 

Proposition 2.15.

Let |$p$| be an odd prime number and |$(L,b)$| an integral lattice of signature |$(s_+,s_-)$|⁠. Then some lattice in the genus of |$(L,b)$| admits a fixed point free isometry |$f$| of order |$p$| acting trivially on the discriminant group if and only if there are nonnegative integers |$n,m \in {\mathbb {Z}}$| such that

• (i)

  |$L$| is even, |$p$|-elementary of discriminant |$p^n$|⁠,

• (ii)

  |$s_+ + s_- = (n + 2m)(p-1)$|⁠,

• (iii)

  |$s_+ \in 2{\mathbb {Z}}$|⁠.

Conversely, if the integers |$s_+,s_-,n,m \in {\mathbb {Z}}_{\geq 0}$| satisfy (ii), (iii), and

• (iv)

  if |$n=0$|⁠, then |$s_+ \equiv s_- \pmod {8}$|⁠,

then a triple |$(L,b,f)$| with |$L$| satisfying (i) exists.

Suppose that |$(L,b)$| is indefinite or of rank |$(p-1)$|⁠. Let |$f$|⁠, |$f^{\prime}$| be fixed point free isometries of order |$p$| of |$L$| with associated hermitian lattices |$(L,h)$|⁠, |$(L,h^{\prime})$|⁠. Then |$f$| is conjugate to |$f^{\prime}$| if and only if they have the same signature and |$\det (L,h)$| is isometric to |$\det (L,h^{\prime})$|⁠. The number of conjugacy classes with a given signature is either |$0$| or |$\#C(E/K)$|⁠. If moreover the relative class number is odd, then |$\det (L,h)$| is isometric to |$\det (L,h^{\prime})$| if and only if |$(L,f)$| and |$(L,f^{\prime})$| (seen as |${\mathbb {Z}}[\zeta _p]$|-modules) have the same Steinitz class in |${\operatorname {C}}(E/K)$|⁠.

 

Proof.

(i) Since |$f=\zeta $| acts as the identity on the discriminant group and |$(L,b)$| is integral, we have

$$ \begin{align}& (1-\zeta) (L,b)^\vee \subseteq (L,b)\subseteq (L,b)^\vee. \end{align}$$

(7)

Thus, |$(L,b)^\vee / (L,b)$| is isomorphic to |$({\mathbb {Z}}[\zeta ]/ (1-\zeta ))^n$| for some |$n \leq \dim _E L\otimes {\mathbb {Q}}$|⁠. Since |${\mathbb {Z}}[\zeta ]/ (1-\zeta ) \cong {\mathbb {F}}_p$| as abelian groups, we see that |$(L,b)$| is |$p$|-elementary. Moreover, it is even by Lemma 2.4.

(ii) Recall that |$\pi = (1-\zeta )$|⁠. Using Equation (5), we translate Equation (7) to

$$ \begin{align}& \pi^{3-p}(L,h)^\# \subseteq (L,h) \subseteq \pi^{2-p} (L,h)^\#. \end{align}$$

(8)

Thus, |$(L,h)$| is unimodular at all primes |$\mathfrak {q}$| of |${\mathbb {Z}}_K$| except |$\mathfrak {p}=\pi \pi ^\sigma {\mathbb {Z}}_K$|⁠. Since the primes |$\mathfrak {q}$| are unramified, a unimodular hermitian lattice over |${\mathbb {Z}}_{K_{\mathfrak {q}}}$| is uniquely determined by its rank. By Equation (8), the modular decomposition of |$L$| at |$\mathfrak {p}$| may only have |$\pi ^i$| modular blocks for |$2-p \leq i \leq 3-p$|⁠. From the classification of |$\pi ^i$|-modular hermitian lattices in Proposition 2.7, we extract that

$$ \begin{align*} &(L,h)_{\mathfrak{p}} \cong M \oplus H_{2-p}^{\oplus m},\end{align*}$$

where |$M$| is |$\pi ^{3-p}$|-modular of rank |$k$| and determinant |$\epsilon \in K_{\mathfrak {p}}^\times / N(E_\pi ^\times )$|⁠. Since |$\mathfrak {A}^{-1}H_{2-p}^\#$| equals |$H_{2-p}$|⁠, we obtain

$$ \begin{align*} &L^\vee/L=\mathfrak{A}^{-1} L^\# / L \cong \mathfrak{A}^{-1}M^\#/M \cong {\mathbb{Z}}_E/\mathfrak{P}^k\cong {\mathbb{F}}_p^k\end{align*}$$

giving |$\det (L,b) = p^k$| and thus |$k=n$|⁠.

(iii) This follows from (ii) and the fact that |$s_- = 2 \sum _{i=1}^s n_i$| is even.

(iv) Note that |$n=0$| if and only if |$(L,b)$| is unimodular; hence, |$(iv)$| is necessary.

Recall from Proposition 2.8 that a collection |$(L_{v},h_v)_{v\in \Omega (K)}$| of local hermitian lattices, all but finitely many of which are unimodular, comes from a single global hermitian lattice |$(L,h)$| if and only if the set |$S=\{v \in \Omega (K) | \det L_v \not \in N(E_v) \}$| is finite of even cardinality. An infinite place |$v$| lies in |$S$| if and only if |$n_v$| is odd. Thus, we obtain

$$ \begin{align}& \# S \equiv \sum_{i=1}^s n_i + \begin{cases} 0 \mbox{ if } \epsilon (-1)^m \in N(E^\times_\pi)\\ 1 \mbox{ if } \epsilon (-1)^m \not\in N(E^\times_\pi) \end{cases} \pmod{2} \end{align}$$

(9)

where |$\epsilon = 1$| if |$n=0$|⁠. We see that for |$n\neq 0$| the condition |$\# S \equiv 0 \mod 2$| uniquely determines the norm class of |$\epsilon $|⁠. It remains to show that for |$n=0$| condition (iv) implies that |$\#S$| is even. To this end, note that |$-1$| is a local norm at |$\pi $| if and only if |$p \equiv 1 \pmod {4}$|⁠. Thus we can rewrite Equation (9) with |$\epsilon =1$| as

$$ \begin{eqnarray*} 2 \#S &\equiv& 2 \sum_{i=1}^s n_i + m(p-1) \pmod{4}\\ &\equiv& s_- + m(p-1) \pmod{4}. \end{eqnarray*}$$

Multiplying by |$2$| we arrive at |$4\#S + 2s_- \equiv 2m(p-1) \pmod {8}$|⁠. Now (ii) and (iv) yield that |$\#S$| is even.

The lattice |$(L,b)$| determines |$n$| and |$m$|⁠. By the previous considerations, this determines the isomorphism class of |$(L,h)_\nu $| at all finite places |$\nu \in \mathbb {P}(K)$|⁠. The signatures |$n_i$| determine it at the infinite places. Thus, they give the genus of |$(L,h)$|⁠. By Propositions 2.9 and 2.12, the classes in the genus of |$(L,h)$| are determined by the determinant lattice, and for odd relative class number by the Steinitz class (cf. Lemma 2.14).

 

Remark 2.16.

The relative class number |$h^{-}({\mathbb {Q}}[\zeta _p])$| is |$1$| for all primes |$p\leq 19$|⁠. For |$p=23$|⁠, |$29$|⁠, |$31$|⁠, |$37$|⁠, |$41$| it equals |$3$|⁠, |$2^3$|⁠, |$3^2$|⁠, |$37$|⁠, |$11^2$|⁠, respectively. The next prime with even relative class number is |$p=113$| with |$h^-=2^3 \cdot 17 \cdot 11853470598257$| (cf. [59, Tables §3]).

2.6 The action on the discriminant group

For an integral |${\mathbb {Z}}$|-lattice |$(L,b)$| and |$f \in O(L,b)$| fixed point free of order |$p$|⁠, we denote by |$O(L,b,f)$| the centralizer of |$f$| in |$O(L,b)$|⁠. Note that |$O(L,b,f)= O(L,h)$|⁠, where |$h$| is defined as in Equation (4). Let |$\bar f$| be the isometry of |$L^{\vee } \! / L $| induced by |$f$|⁠. In what follows, we want to compute the image of the natural morphism

$$ \begin{align}& O(L,h) \rightarrow O(L^{\vee} \! / L ) \end{align}$$

(10)

in the case |$\bar f = {\operatorname {id}}_{L^{\vee } \! / L }$|⁠.

Let |$x \in L^\vee $| and |$[x]=x + L$|⁠. If |$b(x,x)\not \equiv 0 \mod {\mathbb {Z}}$|⁠, we have the reflection

$$ \begin{align*} &\tau_{[x]}\colon L^\vee/L \rightarrow L^\vee/L, \quad y \mapsto y -2\frac{b(x,y)}{b(x,x)}y.\end{align*}$$

This can be adapted for hermitian lattices. For any |$\delta \in \mathcal {E}_0=\{\delta \in {\mathbb {Z}}_E \mid \delta \delta ^\sigma =1\}$| and |$x \in V=L\otimes E$| with |$h(x,x) \neq 0$|⁠, we obtain the quasi reflection |$\tau _{x,\delta } \in O(V,h)$| defined by

$$ \begin{align*} &\tau_{x,\delta}: V \rightarrow V, \quad y\mapsto y + (\delta-1)\frac{h(y,x)}{h(x,x)}x.\end{align*}$$

It maps |$x$| to |$\delta x$| and it is trivial on |$x^\perp $|⁠.

 

Lemma 2.17.

Let |$x\in L^\vee \otimes {\mathbb {Z}}_p=L^\vee _{\mathfrak {p}}$| with |$b(x,x) \not \equiv 0 \mod {\mathbb {Z}}_p$|⁠. Suppose that |$\mathfrak {P} L_{\mathfrak {p}}^\vee \subseteq L_{\mathfrak {p}}$|⁠. Then |$\tau _{x,-1}$| acts as the reflection |$\tau _{[x]}$| on the discriminant group |$L^\vee /L$|⁠.

 

Proof.

As |${\operatorname {Tr}}^{E_{\mathfrak {p}}}_{{\mathbb {Q}}_p}(h(x,x)) = b(x,x) \not \equiv 0 \mod {\mathbb {Z}}_p$|⁠, we obtain that |$h(x,x) \not \in \mathfrak {A}^{-1}$|⁠. On the other hand, |$h(x,x) \in \mathfrak {s}(L^\vee ,h)_{\mathfrak {p}}= \mathfrak {P}^{-1} \mathfrak {A}^{-1}$|⁠. Thus, |$h(x,x){\mathbb {Z}}_{E_{\mathfrak {p}}} = \mathfrak {s}(L^\vee )$| and we can write |$L^\vee = {\mathbb {Z}}_{E_{\mathfrak {p}}} x \oplus x^\perp $|⁠. This induces a compatible splitting on |$L^{\vee } \! / L $|⁠. Since |$\pi x \in L$|⁠, we have |$x{\mathbb {Z}}_{E_{\mathfrak {p}}}+L_{\mathfrak {p}} = x {\mathbb {Z}}_p+L_{\mathfrak {p}}$| and this splitting is |$[x]{\mathbb {Z}}_p \oplus [x]^\perp $|⁠. We can now conclude by comparing the actions of |$\tau _{x,-1}$| and |$\tau _{[x]}$| on |$[x]$| and |$[x]^\perp $|⁠.

 

Lemma 2.18.

The morphism

$$ \begin{align*} &O(L_{\mathfrak{p}},h_{\mathfrak{p}}) \rightarrow O(L^{\vee} \! / L )\end{align*}$$

is surjective.

 

Proof.

Note that in our case |$O(L^{\vee } \! / L )$| is just an orthogonal group over a field. Then the Cartan–Dieudonné theorem [54, Ch. 1, Thm. 5.4] says that it is generated by reflections. But these reflections lie in the image.

 

Corollary 2.19.

The special orthogonal group |$SO(L^{\vee } \! / L )$| is contained in the image of |$O(L,h) \rightarrow O(L^{\vee } \! / L )$|⁠.

 

Proof.

The special orthogonal group is generated by pairs |$\tau _{[x]}\tau _{[y]}$|⁠. Then the corresponding isometry |$t=\tau _{x,-1}\tau _{y,-1}$| has determinant one. By the strong approximation theorem, for any |$k$| we can find |$g \in O(L)$| with |$(g-t)(L) \subseteq \mathfrak {p}^kL$|⁠. For us |$k=1$| is enough.

Since |$[O(L^{\vee } \! / L ):SO(L^{\vee } \! / L )]$| is at most |$2$|⁠, we need only one more generator to get full surjectivity of the map in Equation (10).

 

Proposition 2.20.

Let |$(L,h)$| be a hermitian |${\mathbb {Z}}_E$|-lattice with |$\mathfrak {P} L^\vee \subseteq L$|⁠. If |$(L,h)$| is indefinite or of rank one, then the morphism

$$ \begin{align*} &O(L,h) \rightarrow O(L^{\vee} \! / L )\end{align*}$$

is surjective.

 

Proof.

Set |$V = L\otimes E$|⁠. If |$(L,h)$| is of rank one, then |$O(L^\vee /L)\subseteq \{\pm 1\}$| and the natural map is surjective. If |$(L,b)$| is unimodular, the proposition is certainly true. Otherwise, by the proof of Proposition 2.15, |${\operatorname {{\mathfrak {s}}}}(L) = {\operatorname {{\mathfrak {n}}}}(L){\mathbb {Z}}_{E}$|⁠. Let |$x \in L$| be a local norm generator at |$\mathfrak {p}$|⁠, that is, |$h(x,x){\mathbb {Z}}_{K_{\mathfrak {p}}} = {\operatorname {{\mathfrak {n}}}}(L)_{\mathfrak {p}}$|⁠. Since |${\operatorname {{\mathfrak {n}}}}(L_{\mathfrak {p}}){\mathbb {Z}}_{E_{\mathfrak {p}}}={\operatorname {{\mathfrak {s}}}}(L_{\mathfrak {p}})$|⁠, the reflection |$\tau =\tau _{x,-1} \in O(V,h)$| satisfies |$\tau (L_{\mathfrak {p}}) = L_{\mathfrak {p}}$|⁠. However, it has denominators at the finite set of primes

$$ \begin{align*} &Q = \{\mathfrak{q} \in \mathbb{P}(K) \mid \tau(L_{\mathfrak{q}}) \neq L_{\mathfrak{q}}\}.\end{align*}$$

We shall use the strong approximation theorem ([55, 5.12], [32]) in the formulation of [29, Thm. 5.1.3] to compensate the denominators. Take |$S = \mathbb {P}(K)$|⁠. Since |$V$| is indefinite |$\Omega (K) \setminus S$| contains an isotropic place. We set |$T = Q \cup \{ \mathfrak {p}\}$|⁠, and define |$\sigma _{\mathfrak {q}} = \tau ^{-1} \circ \phi _{\mathfrak {q}} $| for |$\mathfrak {q} \in Q$| where |$\phi _{\mathfrak {q}}\in O(L_{\mathfrak {q}})$| is of determinant |$-1$| (which is possible by [30, Cor. 3.6]). Finally, set |$\sigma _{\mathfrak {p}} = {\operatorname {id}}_{L_{\mathfrak {p}}}$|⁠. By the strong approximation theorem for any |$k \in {\mathbb {N}}$|⁠, we can find |$\sigma \in O(V)$| with

• |$\sigma (L_{\mathfrak {r}}) = L_{\mathfrak {r}}$| for |$\mathfrak {r} \in S \setminus T$| and

• |$(\sigma - \sigma _{q})(L_{\mathfrak {q}})\subseteq \mathfrak {q}^kL_{\mathfrak {q}}$| for |$\mathfrak {q} \in T$|⁠.

Choose |$k\geq 1$| large enough such that |$\mathfrak {q}^k \tau ( L_{\mathfrak {q}}) \subseteq L_{\mathfrak {q}}$| for all |$\mathfrak {q} \in Q$|⁠. Then

$$ \begin{eqnarray*} \tau \circ \sigma (L_{\mathfrak{q}}) &\subseteq& \tau \circ \big(\sigma - \sigma_{\mathfrak{q}}\big)(L_{\mathfrak{q}}) + \tau \circ \sigma_{\mathfrak{q}}(L_{\mathfrak{q}})\\ &\subseteq& \tau(\mathfrak{q}^k L_{\mathfrak{q}}) + \tau\circ \sigma_{\mathfrak{q}}(L_{\mathfrak{q}})\\ &\subseteq& \mathfrak{q}^k \tau(L_{\mathfrak{q}}) + \phi_{\mathfrak{q}}(L_{\mathfrak{q}})\\ &\subseteq& L_{\mathfrak{q}}. \end{eqnarray*}$$

Hence, |$\tau \circ \sigma $| preserves |$L_{\mathfrak {q}}$| for all |$\mathfrak {q} \in \mathbb {P}(K)$|⁠. As it is moreover an element of |$O(V,h)$|⁠, it must be in |$O(L,h)$|⁠. Since |$k\geq 1$|⁠, both |$\tau $| and |$\tau \circ \sigma $| induce the reflection |$\tau _{[x]}$| on the discriminant group. This reflection generates |$O(L^{\vee } \! / L )/SO(L^{\vee } \! / L )$|⁠.

For later use, we prove the following lemma.

 

Lemma 2.21.

If |$(L,h)$| is a hermitian |${\mathbb {Z}}_E$|-lattice with trace lattice |$(L,b,f)$|⁠, then |$O(L,b,f)=SO(L,b,f)$|⁠.

 

Proof.

Let |$g \in O(L,b,f)=O(L,h)$|⁠. When we view |$g$| as an |$E$|-linear map its determinant |$d=\det _E(g) \in E$| satisfies |$dd^\sigma =1$|⁠. Viewed as a |${\mathbb {Q}}$|-linear map one obtains |$\det _{\mathbb {Q}}(g)=N^E_{\mathbb {Q}}(\det _E(g))=N^K_{\mathbb {Q}}\circ N^E_K(\det _E(g))=N^K_{\mathbb {Q}}(1)=1$|⁠.

2.7 The classification

We now use the results of the previous section to obtain existence and uniqueness results on prime order isometries of unimodular lattices. To make notation lighter, in the following we will denote by |$A_L = L^\vee /L$| the discriminant group of an even |${\mathbb {Z}}$|-lattice |$(L,b)$| and by |$q_L$| its discriminant quadratic form. The length|$l(A_L)$| is defined as the minimal number of generators of the group |$A_L$|⁠.

 

Proof of Theorem 1.1.

If |$L$| is unimodular, there exists an isometry of order |$p$| if and only if there exists a primitive sublattice |$S \subset L$| and a fixed point free isometry |$f \in O(S)$| of order |$p$| which acts trivially on the discriminant group of |$S$|⁠. Indeed, such an isometry |$f \in O(S)$| glues with |${\operatorname {id}}_{S^\perp }$| (see [47, Cor. 1.5.2]) to give an isometry of |$L$| whose coinvariant lattice is |$S$|⁠. By Proposition 2.15, |$S$| is |$p$|-elementary of signature |$(s_+,s_-)$|⁠, rank |$s_+ + s_- = (n+2m)(p-1)$| and discriminant |$p^n$|⁠, for some nonnegative integers |$s_+, s_-, n,m$|⁠, and |$s_+$| even. This gives conditions (1), (2), and condition (5) for |$n=0$|⁠. By [47, Thm. 1.12.2 (resp. 1.16.5)], such a lattice |$S$| embeds primitively into some even (resp. odd) unimodular lattice |$L$| of signature |$(l_+,l_-)$| if and only if (II) (resp. (I)), (3), (4) hold (note that |$l(A_S)=n$|⁠) and further

$$ \begin{align*} &\text{if } n = l_+ + l_- - (n+2m)(p-1) \text{, then } (-1)^{l_+ - s_+}p^n \equiv \text{disc}(K(q_S)) \mod \big( {\mathbb{Z}}_p^*\big)^2\end{align*}$$

where |$K(q_S)$| is the unique |$p$|-adic lattice of rank |$n$| and discriminant form |$q_S$|⁠, that is, the |$p$|-modular Jordan component of |$S \otimes {\mathbb {Z}}_p$|⁠. Its determinant is obtained in Theorem 2.1 via |$s_+ - s_- \equiv 2\epsilon -2 - (p-1)n \pmod {8}$|⁠, where |$\epsilon \in \{\pm 1\}$| indicates the unit-square class of the determinant. The left side is computed by the Legendre symbol |$\left (\frac {-1}{p}\right )^{l_+ - s_+} \equiv (p-1)(l_+-s_+) +1 \pmod {4}$|⁠. Inserting this for |$\epsilon $|⁠, we arrive at

$$ \begin{align*} &s_+ - s_- \equiv (p-1)(2l_+ - 2s_+ -n) \equiv (p-1)(l_+ - l_- - s_+ + s_-) \equiv (1-p)(s_+ - s_-)\pmod{8}\end{align*}$$

which gives (5). (Note that in the odd case (5) implies condition (1) of [47, Thm 1.16.5]).

 

Proof of Theorem 1.2.

Let |$L$| be a unimodular lattice and |$f,g\in O(L)$| prime order isometries with |$L_f, L_g$| indefinite or of rank |$p-1$|⁠. Suppose that |$L^f\cong L^g$|⁠, that the signatures agree, and that the determinant lattices of |$L_f$| and |$L_g$| are isometric. We prove that |$f$| and |$g$| are conjugated as isometries of |$L$|⁠. By assumption, there is an isometry |$u\colon L^f \rightarrow L^g$|⁠, and by Proposition 2.15 an isometry |$v\colon (L_f,f|_{L_f})\rightarrow (L_g,g|_{L_g})$|⁠. Let |$\epsilon _f\colon A_{L_f} \rightarrow A_{L^f}$| and |$\epsilon _g\colon A_{L_g} \rightarrow A_{L^g}$| be the standard isomorphisms between the discriminant groups of orthogonal primitive sublattices inside a unimodular lattice. By Proposition 2.20, there exists an isometry |$w \in O(L_f)$| centralizing |$f\vert _{L_f}$| whose action on the discriminant group |$A_{L_f}$| is |$\bar {w} = \bar {v}^{-1} \circ \epsilon _g^{-1} \circ \bar {u} \circ \epsilon _f$|⁠. It follows from [47, Cor. 1.5.2] that |$u \oplus (w \circ v)\colon L^f \oplus L_f \rightarrow L^g \oplus L_g$| extends to an isometry of |$O(L)$| conjugating |$f$| and |$g$|⁠.

By using Theorem 1.1, for a given genus |$\textrm {II}_{(l_+,l_-)}$|⁠, we can list all triples |$(p,n,m)$| such that there exists a unimodular lattice |$L \in \textrm {II}_{(l_+,l_-)}$| and an isometry |$f \in O(L)$| of odd prime order |$p$| with |$L_f$| of signature |$(2, (n+2m)(p-1)-2)$| and discriminant |$p^n$|⁠. Notice that, while the lattice |$L_f$| is uniquely determined (up to isometries) by the triple |$(p,n,m)$|⁠, a priori there can be several distinct isometry classes in the genus of |$L^f$|⁠. For geometric applications, we study the uniqueness of the invariant lattice for some selected unimodular genera.

 

Lemma 2.22.

Let |$L$| be a unimodular lattice in one of the genera |$\textrm {II}_{(4,4)}$|⁠, |$\textrm {II}_{(5,5)}, \textrm {II}_{(3,19)}, \textrm {II}_{(4,20)}, \textrm {II}_{(5,21)}$|⁠. Let |$f \in O(L)$| be an isometry of odd prime order |$p$| with |$L_f$| of signature |$(2, (n+2m)(p-1)-2)$| and discriminant |$p^n$|⁠, for some |$n,m \in {\mathbb {Z}}_{\geq 0}$|⁠. Then the lattice |$L^f$| is unique in its genus unless |$L \in \textrm {II}_{(4,20)}$| and |$(p,n,m) = (23,1,0)$|⁠, where |$L^f$| is isometric to either

$$F_{23a} = \left (\begin {smallmatrix} 2 & 1 \\ 1 & 12 \end {smallmatrix}\right )$$

or

$$F_{23b} = \left (\begin {smallmatrix} 4 & 1 \\ 1 & 6 \end {smallmatrix}\right )$$

⁠.

 

Proof.

Since |$L$| is unimodular, |$q_{L^f} \cong -q_{L_f}$|⁠; hence, |$L^f$| is also |$p$|-elementary. Denote by |$(l_+, l_-)$| the signature of |$L$|⁠. If |$L^f$| is indefinite and |${\operatorname {rk}}\ L^f \geq 3$|⁠, then |$L^f$| is unique in its genus by [17, Ch. 15, Thm. 14]. For |$L$| as in the statement, the only cases where one of these two conditions fails are the following: |$(l_+,l_-,p,n,m) = (3,3,3,0,1)$|⁠, |$(3,3,3,2,0)$|⁠, |$(3,3,5,1,0)$|⁠, |$(4,4,3,1,1)$|⁠, |$(4,4,7,1,0)$|⁠, |$(4,20,3,1,5)$|⁠, |$(4,20,23,1,0)$|⁠. The genera of |$L^f$| are, respectively, |$\textrm {II}_{(1,1)}$|⁠, |$\textrm {II}_{(1,1)}3^{-2}$|⁠, |$\textrm {II}_{(1,1)}5^{-1}$|⁠, |$\textrm {II}_{(2,0)}3^{-1}$|⁠, |$\textrm {II}_{(2,0)}7^1$|⁠, |$\textrm {II}_{(2,0)}3^{-1}$|⁠, |$\textrm {II}_{(2,0)}23^1$|⁠. By using [17, Ch. 15, Tables 15.1 and 15.2a] we check that there is only one isometry class for each of these genera, except for |$\textrm {II}_{(2,0)}23^1$|⁠, which contains the two distinct isometry classes given in the statement.

3 Automorphisms of IHS Manifolds

In this section, we apply our results on isometries of unimodular lattices to obtain a classification of automorphisms of irreducible holomorphic symplectic manifolds with an action of odd prime order on the 2nd cohomology lattice.

As mentioned in the introduction, if |$X$| is IHS then |$H^2(X,{\mathbb {Z}})$| is torsion-free and equipped with a nondegenerate symmetric bilinear form of topological origin, which gives it the structure of an integral lattice of signature |$(3,b_2(X)-3)$| (see [20, Thm. 4.7]). For all known examples of IHS manifolds, this lattice has been computed explicitly: it is even and only depends on the deformation type of the manifold. Let |$H^2(X,{\mathbb {Z}}) \cong \Lambda $| for some fixed lattice |$\Lambda $|⁠. The choice of an isometry |$\eta : H^2(X,{\mathbb {Z}}) \rightarrow \Lambda $| is called a marking of |$X$| and two marked IHS manifolds |$(X, \eta )$|⁠, |$(X^{\prime}, \eta ^{\prime})$| are equivalent if there is an isomorphism |$f: X \rightarrow X^{\prime}$| such that |$\eta ^{\prime} = \eta \circ f^*$|⁠.

There exists a coarse moduli space |$\mathcal {M}_\Lambda $| which parametrizes (equivalence classes of) |$\Lambda $|-marked IHS manifolds |$(X, \eta )$| for |$X$| of a fixed deformation type (see [26]). Two points |$(X, \eta )$| and |$(X^{\prime}, \eta ^{\prime})$| belong to the same connected component of |$\mathcal {M}_\Lambda $| if and only if |$\eta ^{\prime} \circ \eta ^{-1}$| is a parallel transport operator (see [36, §1.1] for the definition). Denote by |${\operatorname {Mon}}^2(X) \subset O(H^2(X,{\mathbb {Z}}))$| the monodromy group, which is the group of monodromy operators of |$X$|⁠, that is, parallel transport operators |$\gamma \colon H^2(X,{\mathbb {Z}}) \rightarrow H^2(X,{\mathbb {Z}})$|⁠. Let

$$ \begin{align*} & \Omega_\Lambda:= \left\{ {\mathbb{C}}\omega \in \mathbb{P}(\Lambda \otimes {\mathbb{C}}) \mid (\omega, \omega) = 0, (\omega, \overline{\omega})> 0\right\}\end{align*}$$

be the period domain and define the period map

$$ \begin{align*} & \mathcal{P}: \mathcal{M}_\Lambda \rightarrow \Omega_\Lambda, \quad (X,\eta) \mapsto \eta(H^{2,0}(X)).\end{align*}$$

By [7, Thm. 5] and [26, Thm. 8.1], |$\mathcal {P}$| is a local homeomorphism and, for any connected component |$\mathcal {M}_\Lambda ^0 \!\subset\! \mathcal {M}_\Lambda $|⁠, its restriction |$\mathcal {P}_0\!:\! \mathcal {M}_\Lambda ^0\! \rightarrow\! \Omega _\Lambda $| is surjective. Moreover, for any |$p \!\in\! \Omega _\Lambda $| the fiber |$\mathcal {P}_0^{-1}(p)$| consists of pairwise inseparable points and if |$(X, \eta ), (X^{\prime}, \eta ^{\prime}) \in \mathcal {M}_\Lambda $| are inseparable, |$X$| and |$X^{\prime}$| are bimeromorphic (global Torelli theorem; see [36, Theorem 2.2]).

The Kähler cone|${\mathcal {K}}_X \subseteq H^{1,1}(X,{\mathbb {R}})$| consists of the classes of Kähler metrics. The positive cone|${\mathcal {C}}_X$| is the connected component of |$\{x \in H^{1,1}(X,{\mathbb {R}}) \mid x^2>0\}$| containing a Kähler class. The positive cone admits two important wall and chamber decompositions. We denote by |$\Delta (X) \subset H^{1,1}(X,{\mathbb {R}}) \cap H^2(X,{\mathbb {Z}})$| the set of primitive integral monodromy birationally minimal (MBM) classes (see [1, Def. 1.13]) and by |${\mathcal {B}} \Delta (X)\subseteq \Delta (X)$| the set of stably prime exceptional divisors of |$X$|⁠. Then the chambers of

$$ \begin{align*} &{\mathcal{C}}_X \setminus \bigcup_{\delta \in \Delta(X)} \delta^\perp\end{align*}$$

are called the Kähler type chambers. One of them is the Kähler cone. The chambers of

$$ \begin{align*} &{\mathcal{C}}_X \setminus \bigcup_{\delta \in {\mathcal{B}} \Delta(X)} \delta^\perp\end{align*}$$

are called the exceptional chambers. The exceptional chamber containing the Kähler cone is the fundamental exceptional chamber|$\mathcal {F}\mathcal {E}_X$|⁠. Its closure equals the closure of the birational Kähler cone. An element |$\delta $| of |${\mathcal {B}} \Delta (X)$| defines the reflection |$\tau _\delta (x):= x - 2\frac {(x,\delta )}{(\delta ,\delta )} \delta $|⁠, which turns out to be a monodromy (see [36, §6.2]). The group |$W_{Exc}(X) \leq {\operatorname {Mon}}^2(X)$| generated by these reflections is called the Weyl-group. It acts transitively on the set of exceptional chambers and the fundamental exceptional chamber is a fundamental domain for the action of |$W_{Exc}(X)$| on |${\mathcal {C}}_X$|⁠.

Fixing a connected component|${\mathcal {M}}^\circ _\Lambda $| of |${\mathcal {M}}_\Lambda $| one can transport these objects to |$\Lambda $|⁠, so that we may speak of |$\Delta (\Lambda )$|⁠, |${\mathcal {B}}\Delta (\Lambda )$|⁠, |${\mathcal {C}}_\Lambda $|⁠, |${\operatorname {Mon}}^2(\Lambda )$|⁠, and so on. For instance, given |$(X,\eta ) \in {\mathcal {M}}_\Lambda ^\circ $| we have |$\Delta (X) = \eta ^{-1}(\Delta (\Lambda )) \cap H^{1,1}(X,{\mathbb {R}})$|⁠.

The importance of theses cones for our work lies in Verbitsky’s strong Torelli theorem [58] for IHS manifolds as formulated by Markman.

 

Theorem 3.1.

[36, Thms. 1.2, 1.6, Cor. 5.7] Let |$X_1$|⁠, |$X_2$| be irreducible holomorphic symplectic manifolds and |$\phi \colon H^2(X_1,{\mathbb {Z}}) \rightarrow H^2(X_2,{\mathbb {Z}})$| a parallel transport operator, which is a Hodge isometry.

• There exists an isomorphism |$f\colon X_2 \rightarrow X_1$| with |$\phi =f^\ast $| if and only if |$\phi ({\mathcal {K}}_{X_1})={\mathcal {K}}_{X_2}$|⁠.

• If |$X_1$|⁠, |$X_2$| are projective, there exists a birational map |$f\colon X_2 \dashrightarrow X_1$| with |$\phi =f^\ast $| if and only if |$\phi (\mathcal {F}\mathcal {E}_{X_1})=\mathcal {F}\mathcal {E}_{X_2}$|⁠.

Thus, birational models of a projective IHS manifold |$X$| are given by the Kähler type chambers contained in the fundamental exceptional chamber.

By [36, Thm. 3.1], if |$\sigma \in {\operatorname {Aut}}(X)$| then |$\sigma ^* \in O(H^2(X, {\mathbb {Z}}))$| is a monodromy operator. Theorem 3.1 computes the exact image of the homomorphism |$\rho _X \colon {\operatorname {Aut}}(X) \rightarrow {\operatorname {Mon}}^2(X)$| defined in Equation (2). Its kernel is finite [26, Prop. 9.1] and, as a group, a deformation invariant of |$X$| [23, Thm. 2.1]. We briefly present its structure for the known deformation types of IHS manifolds and summarize the results in Table 1. For K3 surfaces and manifolds of type |$K3^{\left [n\right ]}$|⁠, the injectivity of |$\rho _X$| follows after deforming to manifolds where |${\operatorname {Aut}}(X)$| is trivial. This is the argument used by Beauville [6, Prop. 10] who proved injectivity for the Hilbert scheme of |$n$| points on a K3 surface after observing that |${\operatorname {Aut}}(S^{[n]}) = 0$| if |$S$| is a K3 with trivial Picard group and no non-trivial automorphisms.

In the case of manifolds of type |$\textrm {Kum}_n$|⁠, the map |$\rho _X$| is no longer injective. Let |$A$| be a complex |$2$|-torus and |$K_n(A) \subset A^{[n+1]}$| its |$2n$|-dimensional generalized Kummer variety. The translation |$t_a \in {\operatorname {Aut}}(A)$| by a point |$a \in A$| induces an automorphism of |$A^{[n+1]}$|⁠, which leaves the fiber |$K_n(A)$| invariant if and only if |$a$| has order |$n+1$|⁠. Since |$H^2(K_n(A), {\mathbb {Z}})$| is torsion-free, the resulting |$\tilde {t}_a \in {\operatorname {Aut}}(K_n(A))$| acts trivially on it. As shown in [11, Cor. 5], the generators of |$\ker (\rho _{K_n(A)})$| are the automorphisms |$\tilde {t}_a$|⁠, for |$a \in A$| of order |$n+1$|⁠, and the automorphism induced by the group homomorphism |$-{\operatorname {id}} \in {\operatorname {Aut}}(A)$|⁠. Thus, |$\ker (\rho _X) \cong \left (C_{n+1}\right )^4 \rtimes C_2$| for any manifold |$X$| of type |$\textrm {Kum}_n$|⁠.

The description of |$\ker (\rho _X)$| for manifolds of OG|$_6$| and OG|$_{10}$| type requires significantly more work. It is obtained in [43] by looking at desingularizations of specific moduli spaces of sheaves on abelian surfaces and K3 surfaces, respectively. We omit the details here as we will not need them for our work and just recall the results. For manifolds of type OG|$_{10}$|⁠, |$\rho _X$| is injective. In the OG|$_6$| case |$\ker (\rho _X) \cong (C_2)^8$|⁠: the elements in the kernel can be interpreted as automorphisms induced by translations by points of order |$2$| on |$A \times A^\vee $|⁠, where |$A$| is the abelian surface underlying the moduli space and |$A^\vee $| its dual.

Let |$\sigma \in {\operatorname {Aut}}(X)$| be a (biholomorphic) automorphism of an IHS manifold |$X$|⁠. It can be readily checked (cf. [12, §5]) that the coinvariant lattice |$H^2(X,{\mathbb {Z}})_{\rho _X(\sigma )}$| of a symplectic automorphism |$\sigma \in {\operatorname {Aut}}(X)$| is negative definite and contained in |${\operatorname {NS}}(X)$|⁠. On the other hand, if |$\sigma $| is non-symplectic, then |$H^2(X,{\mathbb {Z}})_{\rho _X(\sigma )}$| has signature |$(2,\ast )$| and it contains the transcendental lattice of |$X$|⁠.

3.1 Deformations

In this section, we apply results by Horikawa to show the existence of a universal deformation for pairs |$(X,f)$| consisting of an IHS manifold and an automorphism. The result is surely known to the experts but for lack of a reference we give a detailed proof.

 

Definition 3.2.

Let |$X$| be an IHS manifold and |$f\in {\operatorname {Aut}}(X)$|⁠. A deformation of the pair |$(X,f)$| consists of a smooth proper holomorphic map |$p \colon {\mathcal {X}} \rightarrow B$|⁠, |$0\in B$| with a distinguished fiber |${\mathcal {X}}_0 := p^{-1}(0)=X$| and an automorphism |$F \in {\operatorname {Aut}}({\mathcal {X}}/B)$| such that |$F|_{{\mathcal {X}}_0}=f$|⁠.

From a morphism |$s \colon B^{\prime} \rightarrow B$|⁠, we obtain a family of deformations |${\mathcal {X}}^{\prime}={\mathcal {X}} \times _B B^{\prime} \!\rightarrow\! B^{\prime}$| and |$F^{\prime} = F \times {\operatorname {id}}_{B^{\prime}}$|⁠. We call it the family induced from |$({\mathcal {X}},B,p,F)$| via |$s$|⁠.

In the following, only the germ at |$0 \in B$| of a deformation is of relevance and all statements are to be read in this sense.

 

Definition 3.3.

A deformation |$\mathcal {B}=({\mathcal {X}},B,p,F)$| of |$(X,f)$| is called versal, if for every deformation |$\mathcal {B}^{\prime}=({\mathcal {X}}^{\prime},B^{\prime},p^{\prime},F^{\prime})$| of |$(X,F)$| there exists a morphism |$s\colon B^{\prime} \rightarrow B$| such that |$\mathcal {B}^{\prime}$| is induced by |$\mathcal {B}$| via |$s$|⁠. If moreover |$s$| is unique, then the deformation is called universal.

In [9, §4], the authors construct a family of deformations of the pair |$(X,f)$|⁠. This family will later turn out to be universal. We review the construction. Let |$p \colon \mathcal {X} \rightarrow D=\mbox {Def}(X)$| be the universal family of deformations |$X$|⁠. By [25, Thm 8.1], we obtain (the germ of) a family |$\mathcal {X}^{\prime} \rightarrow D$| of deformations of |$X$| and a holomorphic map |$ \mathcal {X}\rightarrow \mathcal {X}^{\prime}$| whose restriction to |$X$| coincides with |$f$|⁠. By the universality of |$p,$| we obtain |$\mathcal {X}^{\prime}$| as a pullback of |$\mathcal {X}$| under a unique map |$\gamma \colon D \rightarrow D$|⁠.

By composition, we have a map |$\Phi \colon \mathcal {X} \rightarrow \mathcal {X}$|⁠. Then the restriction of |$\Phi $| to |$D^\gamma = \{d \in D \mid \gamma (d)=d\}$| gives a family of deformations of |$(X,f)$|⁠. Note that |$D^{\gamma }$| is smooth.

 

Proposition 3.4.

The restriction |$F \colon \mathcal {X}|_{D^\gamma } \rightarrow {\mathcal {X}}|_{D^\gamma }$| of |$\Phi $| is the universal deformation of |$(X,f)$|⁠.

To prove the versality, we will use the results and language of [24]. For this purpose, we set |$\Phi _0 = \Phi |_X \colon X \rightarrow {\mathcal {X}}$| and |$\hat {\Phi }=(\Phi ,p)\colon {\mathcal {X}} \rightarrow {\mathcal {X}} \times D, x\mapsto (\Phi (x),p(x))$|⁠. Then |$({\mathcal {X}},\hat {\Phi },p,D)$| is a deformation of |$\Phi _0$| in the sense of [24, Def. 1.1].

 

Lemma 3.5.

The deformation |$({\mathcal {X}},\hat {\Phi },p,D)$| is a versal deformation of |$\Phi _0$| in the sense of [24, Def. 1.2].

 

Proof.

This follows from [24, Thm. 2.1]. It applies if we check that the so-called characteristic map of |$(\Phi ,p)$| is surjective. For this purpose, let |$i\colon X \rightarrow {\mathcal {X}}$| be the inclusion and consider the exact sequences

where |${\mathcal {L}}$| is defined as the cokernel of the lower part. Note that the vertical arrows in the middle are isomorphisms. We take the corresponding long exact sequence in cohomology and obtain a commutative diagram

where |$\rho $| is the Kodaira–Spencer map of |$p\colon {\mathcal {X}} \rightarrow D$|⁠. Since |$p$| is the universal deformation, |$\rho $| is an isomorphism, and |$\tau $| is an isomorphism by definition. By [24, Prop. 1.4], |$\tau = \rho \circ \delta ^{-1}$| is the characteristic map.

 

Proof of Proposition 3.4.

Let |${\mathcal {X}}^{\prime} \rightarrow B, F^{\prime}\in {\operatorname {Aut}}({\mathcal {X}}^{\prime}/B)$| be a family of deformations of |$(X,f)$|⁠. By the universality of |$p$|⁠, we obtain a diagram

and the family |$\hat {\Phi }^{\prime}= (p_{\mathcal {X}} \circ F^{\prime},p^{\prime})\colon {\mathcal {X}}^{\prime} \rightarrow {\mathcal {X}} \times B$| of deformations of |$\Phi _0$|⁠. By the versality of |$\hat {\Phi }$|⁠, we obtain that |$\hat {\Phi }^{\prime} = \hat {\Phi } \times {\operatorname {id}}_B \colon {\mathcal {X}}^{\prime} \rightarrow ({\mathcal {X}} \times D) \times _D B = {\mathcal {X}} \times B$|⁠. Note that in fact |$\hat {\Phi }^{\prime}$| factors through |${\mathcal {X}} \times _D B ={\mathcal {X}}^{\prime}$|⁠. This is summarized in the following diagram:

Looking at various subdiagrams, one can check that the diagram is actually commutative. In particular, |$\gamma \circ s = s$|⁠. Hence, |$s(B) \subseteq D^\gamma $|⁠. Thus, it makes sense to restrict everything to the fibers over |$D^\gamma $|⁠, and |${\mathcal {X}}^{\prime} = {\mathcal {X}}|_{D^\gamma } \times _{D^\gamma } B$|⁠. We obtain the diagram

where the inner quadrangle must commute since the outer one does. We infer that |$F^{\prime}=F \times {\operatorname {id}}_B$| is the pullback of |$F$|⁠.

If, instead of a single automorphism |$f$|⁠, we have a group |$G=\langle f_1,\dots f_n \rangle $|⁠, then we obtain an action of |$G$| on |$D$| via |$\gamma _1, \dots , \gamma _n$|⁠. Now the universal family arises by restricting to the fixed locus |$D^G$|⁠.

3.2 Moduli spaces

In this section, we use universal deformations to construct a moduli space for pairs |$(X,G)$|⁠. In a 2nd step, we take a closer look at the period map, to determine where it is injective. This study is modeled on previous work in the case of K3 surfaces by Dolgachev, Kondō [18] and for IHS manifolds of type |$K3^{\left [n\right ]}$| by Joumaah [28] and Boissière et al. [10].

 

Definition 3.6.

Let |$X,X^{\prime}$| be IHS manifolds and |$G \leq {\operatorname {Aut}}(X)$|⁠, |$G^{\prime} \leq {\operatorname {Aut}}(X^{\prime})$|⁠. Then we call |$(X,G)$| and |$(X^{\prime},G^{\prime})$|bimeromorphically conjugate if there exists a bimeromorphic map |$\phi \colon X \dashrightarrow X^{\prime}$| such that |$\phi G \phi ^{-1} = G^{\prime}$|⁠. They are called deformation equivalent, if there exists a connected family |$\psi \colon \mathcal {X} \rightarrow B$| of IHS manifolds, a group of automorphisms |$\mathcal {G}$| of |$\mathcal {X}/B$| and two points |$b,b^{\prime}$| such that the restriction of |$(\mathcal {X},\mathcal {G})$| to the fibers above |$b$| and |$b^{\prime}$| give |$(X,G)$| and |$(X^{\prime},G^{\prime})$|⁠.

Fix a connected component |$\mathcal {M}_\Lambda ^\circ $| of the moduli space of |$\Lambda $|-marked IHS manifolds of a given deformation type. Let |$(X,\eta )$| be a marked pair belonging to this component. We define |${\operatorname {Mon}}^2_\circ (\Lambda ) = \eta {\operatorname {Mon}}^2(X)\eta ^{-1}$|⁠. This choice is independent of |$(X,\eta )$| as long as they stay in the same connected component. A different connected component results in a conjugate subgroup of |$O(\Lambda )$|⁠.

 

Definition 3.7.

Let |$H \leq O(\Lambda )$| be a subgroup. An |$H$|-marked IHS manifold is a triple |$(X,\eta ,G)$| such that |$(X,\eta )$| is a marked pair, |$\ker \rho _X \leq G \leq {\operatorname {Aut}}(X)$| and |$\eta \rho _X(G)\eta ^{-1}=H$|⁠. Two |$H$|-marked IHS manifolds |$(X_1,\eta _1,G_1)$| and |$(X_2,\eta _2,G_2)$| are called isomorphic (respectively, bimeromorphic) if there exists an isomorphism (respectively, a bimeromorphic map) |$f\colon X_1 \rightarrow X_2$| such that |$\eta _1 \circ f^* = \eta _2$|⁠. In particular, |$f G_1 f^{-1} = G_2$|⁠. We call |$H$|effective if there exists at least one |$H$|-marked IHS manifold.

Let |$(X,\eta ,G)$| be an |$H$|-marked IHS manifold. The action of |$G$| on |$H^{2,0}(X)$| induces via the marking a character |$\chi \colon H \rightarrow {\mathbb {C}}^*$|⁠. We can construct a coarse moduli space parametrizing isomorphism classes of |$H$|-marked IHS manifolds |$(X,\eta ,G)$| with |$\chi (\eta \rho _X(g)\eta ^{-1}) \cdot \omega _X = (g^*)^{-1} \omega _X$| for all |$g \in G$|⁠, by gluing the base spaces of the universal deformations of |$(X,\eta ,G)$| constructed in Section 3.1. We denote the resulting moduli space by |$\mathcal {M}^\chi _H$|⁠.

 

Proposition 3.8.

The forgetful map |$\phi \colon \mathcal {M}^\chi _H \rightarrow \mathcal {M}_\Lambda , (X,\eta ,G) \mapsto (X,\eta )$| is a closed embedding.

 

Proof.

Let |$(X,\eta ,G)$| be an |$H$|-marked IHS manifold. The universal deformation |${\mathcal {X}} \rightarrow D$| of |$X$| with marking induced by |$\eta $| provides the neighborhood |$D$| of the point |$(X,\eta )$| in the moduli. Then |$D^G= {\operatorname {im}} \phi \cap D$| is closed by its definition as a fixed point set of a finitely generated group of automorphisms. It remains to show that the map is injective. Indeed, since |$\ker \rho _X \leq G$|⁠, we can reconstruct the group as |$G = \rho _X^{-1}(\eta ^{-1} H \eta )$|⁠.

 

Remark 3.9.

Let |$G$| be a group and |$\rho \colon G \rightarrow O(\Lambda )$| a representation. Then one can define a |$\rho $|-marked IHS manifold as a triple |$(X,\rho ^{\prime},\eta )$| where |$\rho ^{\prime}\colon G \rightarrow {\operatorname {Aut}}(X)$| is a monomorphism such that |$\rho _X(\rho ^{\prime}(g)) = \eta ^{-1} \rho (g) \eta $| for all |$g \in G$|⁠. The difference to |$H$|-marked IHS manifolds is that |$\rho $| may contain information on cohomologically trivial automorphisms whereas |$H$| does not. Indeed, the forgetful map |$(X,\rho ^{\prime},\eta )\mapsto (X,\eta )$| is in general not injective. Therefore, we chose to work with |$H$|-markings.

The corresponding period domain is

$$ \begin{align*} &\Omega_\Lambda^\chi=\{{\mathbb{C}} \omega \in \Omega_\Lambda \mid h(\omega)=\chi(h)\cdot \omega \mbox{ for all } h \in H\}.\end{align*}$$

This results in a period map |${\mathcal {P}}\colon {\mathcal {M}}^\chi _H \rightarrow \Omega _\Lambda ^\chi $|⁠, which is a local isomorphism by Proposition 3.8. Following [10, 28], we exhibit a bijective restriction of the period map. We set |$\mathcal {M}_H^{\circ ,\chi } = \phi ^{-1}(\mathcal {M}_\Lambda ^\circ $|⁠). Let

$$ \begin{align*} & \Lambda^\chi_{\mathbb{C}} := \{x \in \Lambda_{\mathbb{C}} \mid h(x)=\chi(h)\cdot x \mbox{ for all } h \in H\}\end{align*}$$

and |$N = (\Lambda _{\mathbb {C}}^\chi )^\perp \cap \Lambda $|⁠. For |$H$|-marked IHS manifolds, we have |$N\subseteq \eta ({\operatorname {NS}}(X))$| with equality for a very general subset of IHS manifolds. Suppose that |$\chi $| is nontrivial. Then |$N$| is of signature |$(1,t)$| for some |$t$|⁠. In this case |$H$|-marked IHS manifolds are projective, and we will speak of birational instead of bimeromorphic maps to emphasize this.

Let |${\mathcal {C}}_N = {\mathcal {C}}_\Lambda \cap N_{\mathbb {R}}$|⁠. We say that |$(X,\eta ,H)$| is |$(H,N)$|-polarized, if |${\mathcal {C}}_N \cap \eta (\mathcal {K}_X) \neq \emptyset $|⁠. Indeed, |$(X,\eta ^{-1}|_N)$| is an |$N$|-polarized IHS manifold in the sense of [13].

Recall that |$\Delta (\Lambda )$| denotes the set of MBM-classes of |$\Lambda $|⁠. Set |$M = N^\perp $| and |$\Delta (M)=\Delta (\Lambda ) \cap M$|⁠. Let |$\Delta ^{\prime}(M)$| consist of those |$v \in \Delta (\Lambda )$| such that |$v = v_N + v_M$|⁠, |$v_N \in N_{\mathbb {Q}}$|⁠, |$v_M \in M_{\mathbb {Q}}$| with |$q(v_N)<0$|⁠.

 

Definition 3.10.

Suppose that |$\chi $| is nontrivial. Let |$K(N)$| be a Kähler-type chamber of |${\mathcal {C}}_N$| preserved by |$H$|⁠. An |$(H,N)$|-polarized IHS manifold |$(X,\eta ,G)$| is called |$K(N)$|-general, if |$\eta (\mathcal {K}_X) \cap N_{\mathbb {R}} = K(N)$|⁠. It is called |$H$|-general, if it is |$K(N)$|-general for some |$K(N)$|⁠.

Note that |$(X,\eta ,G)$| is |$H$|-general if and only if |$\Delta ^{\prime}(M) \cap \eta ({\operatorname {NS}}(X))=\emptyset $|⁠. Let |$\mathcal {M}^{\circ ,\chi }_{K(N)}$| be the subset of |$\mathcal {M}_H^{\circ ,\chi }$| consisting of |$K(N)$|-general |$(H,N)$|-polarized manifolds. For |$\delta \in \Delta (\Lambda )$|⁠, let |$H_\delta \subset {\mathbb {P}}(\Lambda _{\mathbb {C}})$| be the hyperplane orthogonal to |$\delta $|⁠. We denote by |$\Delta (K(N)) \subset \Delta ^{\prime}(M)$| the subset of elements |$\delta $| such that |$H_\delta \cap K(N) \neq \emptyset $|⁠.

 

Proposition 3.11.

Suppose that |$H\leq Mon^2(\Lambda )$| is finite and |$\chi $| nontrivial. Set |$\Delta = \bigcup _{\delta \in \Delta (M)} H_\delta $| and |$\Delta ^{\prime} = \bigcup _{\delta \in \Delta (K(N))} H_\delta $|⁠.

1. The period map

   $$ \begin{align*} &\mathcal{P} \colon \mathcal{M}^{\circ,\chi}_{H} \rightarrow \Omega^\chi_\Lambda \setminus \Delta\end{align*}$$

   is surjective.
2. Its restriction

   $$ \begin{align*} &{\mathcal{P}}_{K(N)}\colon\mathcal{M}^{\circ,\chi}_{K(N)} \rightarrow \Omega^\chi_\Lambda \setminus \left(\Delta \cup \Delta^{\prime}\right)\end{align*}$$

   is bijective.

Suppose in addition that |$H$| is of prime order.

• (3)

  Let |$w \in \Omega ^\chi _\Lambda \setminus (\Delta \cup \Delta ^{\prime})$| and |$(X_i,G_i,\eta _i)$| for |$i=1,2$| be two |$H$|-general elements of the fiber |${\mathcal {P}}^{-1}(w)$|⁠. Then |$(X_1,G_1)$| is birationally conjugate to |$(X_2,G_2)$|⁠.

 

Proof of (1) and (2).

We adapt the proofs of [10, Thms. 4.5, 5.6] to our situation by using Proposition 3.8.

Let |$(X,\eta ,G)$| be an |$H$|-marked IHS manifold and |$\delta \in \Delta (\Lambda )$|⁠. If |${\mathcal {P}}(X,\eta ,G) \in H_\delta $|⁠, then |$\eta ^{-1}(\delta ) \in {\operatorname {NS}}(X)$| is an MBM class of |$X$|⁠, hence |$\delta ^\perp \cap \eta ({\mathcal {K}}_X) = \emptyset $|⁠. Since |$\eta ({\mathcal {K}}_X) \cap N_{\mathbb {R}}$| is nonempty, we obtain that |$\delta \not \in N^\perp = M$|⁠. Thus, the 1st map is well defined.

Suppose moreover that |$(X,\eta ,G)$| is |$K(N)$|-general, that is, |$\eta ({\mathcal {K}}_X) \cap N_{\mathbb {R}}=K(N)$|⁠. Let |$\delta \in \Delta ^{\prime}(M)$|⁠. If |${\mathcal {P}}(X,\eta ,G) \in H_\delta $|⁠, then |$\delta \in \eta (NS(X))$|⁠, which is a contradiction. Thus, the 2nd map is well defined. Let |$\omega \in \Omega ^\chi _\Lambda \setminus \Delta $|⁠. By surjectivity of the period map, we can find a marked IHS manifold |$(X,\eta )$| in |${\mathcal {M}}_\Lambda ^\circ $| with |${\mathcal {P}}(X,\eta ) = \omega $|⁠. Since |$\Lambda ^H$| is hyperbolic and |$w \not \in \Delta $|⁠, we can find a Kähler-type chamber of |$\eta ({\operatorname {NS}}(X))$|⁠, which contains an |$H$|-invariant vector |$\kappa $|⁠. Hence, we have a birational map |$f\colon X \dashrightarrow X^{\prime}$| and a monodromy operator |$\tau \in {\operatorname {Mon}}^2(X)$| preserving the Hodge structure of |$X$| such that |$\eta ^{\prime -1}(\kappa ) \in {\mathcal {K}}_{X^{\prime}}$|⁠, where |$\eta ^{\prime} = \eta \circ \tau \circ f^*$|⁠. By construction |$\eta ^{\prime -1}H \eta ^{\prime}$| is a group of monodromies that preserves |$\eta ^{\prime -1}(\kappa )$|⁠. Set |$G^{\prime}=\rho _{X^{\prime}}^{-1}(\eta ^{\prime -1}H \eta ^{\prime})$|⁠. Then |$(X^{\prime},\eta ^{\prime},G^{\prime})$| is an |$H$|-marked IHS manifold with period |$w$|⁠.

To see the injectivity of the 2nd map, let |$(X_i,\eta _i,G_i)$| for |$i = 1,2$| be |$K(N)$|-general IHS manifolds with the same period. Then |$\eta _1^{-1} \circ \eta _2$| is a monodromy operator compatible with the Kähler cones and Hodge structures. It is therefore induced by an isomorphism |$f\colon X_1 \rightarrow X_2$|⁠, so that |$(X_1,\eta _1)$| and |$(X_2,\eta _2)$| are isomorphic. Since |$\phi \colon {\mathcal {M}}^\chi _H \rightarrow {\mathcal {M}}_\Lambda $| is injective, the preimages of |$(X_i,\eta _i)$| under |$\phi $| are isomorphic as well.

Recall that |${\mathcal {B}}\Delta (X) \leq \Delta (X):= {\operatorname {NS}}(X) \cap \eta ^{-1}(\Delta (\Lambda ))$| consist of the classes of stably prime exceptional divisors of |$X$|⁠. Let |$H$| be of prime order and |${\mathcal {C}}_X^{G}= {\mathcal {C}}_X \cap H^2(X,{\mathbb {R}})^G$| be the connected component of the |$G$|-fixed part of the positive cone which contains Kähler classes. We call the connected components of |${\mathcal {C}}_X^{G}\setminus \bigcup \{v^\perp : v \in {\mathcal {B}} \Delta (X)\}$| the |$G$|-stable exceptional chambers. The |$G$|-invariant fundamental exceptional chamber is |$\mathcal {F}\mathcal {E}_X^G=\mathcal {F}\mathcal {E}_X \cap H^2(X,{\mathbb {R}})^G$|⁠.

 

Lemma 3.12.

If |$(X,\eta ,G)$| is |$H$|-general, then the |$G$|-invariant fundamental chamber is a |$G$|-stable exceptional chamber.

 

Proof.

Write |${\mathcal {B}}\Delta (X)$| as the disjoint union of |${\mathcal {B}}\Delta ^G(X)$|⁠, |${\mathcal {B}}\Delta _G(X) := {\mathcal {B}}\Delta (X) \cap H^2(X,{\mathbb {Z}})_G$|⁠, |${\mathcal {B}}\Delta ^{\prime}(X):= {\mathcal {B}}\Delta (X) \cap \eta ^{-1}(\Delta ^{\prime}(M))$| and |${\mathcal {B}}\Delta ^{\prime\prime}$| for the remaining |${\mathcal {B}}\Delta ^{\prime\prime}\subseteq {\mathcal {B}}\Delta (X)$|⁠. In |${\mathcal {C}}^G_X \setminus \bigcup \{v^\perp : v \in {\mathcal {B}} \Delta (X)\}$| the hyperplanes coming from |${\mathcal {B}}\Delta ^{\prime\prime}$| do not meet |${\mathcal {C}}_X^{G}$| and can be omitted. Moreover, by the generality assumption, |${\mathcal {B}}\Delta _G(X)$| and |${\mathcal {B}}\Delta ^{\prime}(X)$| are empty.

Recall that |$W_{Exc}(X)\leq {\operatorname {Mon}}^2(X)$| is the subgroup generated by the reflections in stable prime exceptional divisors. We define |$W^G_{Exc}(X)$| as the subgroup generated by reflections in |$G$|-invariant stable prime exceptional divisors.

 

Proof of Proposition 3.11 (3).

Let |$(X_1,\eta _1, G_1)$| and |$(X_2,\eta _2, G_2)$| be in the fiber of |${\mathcal {P}}$| above |$w\in \Omega ^\chi _\Lambda \setminus (\Delta \cap \Delta ^{\prime})$|⁠. Since |$(X_1,\eta _1)$| and |$(X_2,\eta _2)$| lie in the same connected component |$\mathcal {M}_\Lambda ^\circ $|⁠, the composition |$\psi =\eta _1^{-1} \circ \eta _2$| is a parallel transport operator; hence, |$\psi ({\mathcal {F}}{\mathcal {E}}_{X_2})$| is an exceptional chamber.

Both |$(X_i,\eta _i,G_i)$| are |$H$|-general; hence, the invariant fundamental exceptional chambers |${\mathcal {F}}{\mathcal {E}}^{G_2}_{X_2}$| and |$\psi ^{-1}({\mathcal {F}}{\mathcal {E}}_{X_1}^{G_1})$| are |$G_2$|-stable chambers. Thus, there exists an element |$w \in W^{G_2}_{Exc}(X_2)$| with |$\psi \circ w ({\mathcal {F}}{\mathcal {E}}_{X_2}^{G_2})= {\mathcal {F}}{\mathcal {E}}_{X_1}^{G_1}$|⁠. We obtain |$\psi \circ w ({\mathcal {F}}{\mathcal {E}}_{X_2})= {\mathcal {F}}{\mathcal {E}}_{X_1}$| from the fact that two exceptional chambers are either disjoint or equal. By Theorem 3.1, |$\psi \circ w$| is induced by a birational map |$f\colon X_1 \dashrightarrow X_2$|⁠. Recall that |$G_i = \rho _{X_i}^{-1}(\eta _i^{-1}H\eta _i)$| for |$i\in \{1,2\}$|⁠. Since |$w$| is a reflection in a |$G_2$|-invariant class, |$w \rho _{X_2}(G_2)w =\rho _{X_2}(G_2)$|⁠. Using this one obtains |$\eta _2\rho _{X_2}(fG_1f^{-1})\eta _2^{-1} = H$|⁠, which implies |$f G_1 f^{-1}=G_2$| as desired.

Note that for |$H$| a cyclic group, giving |$\chi \colon H \rightarrow {\mathbb {C}}^*$| is the same as fixing a generator |$h$| of |$H$| acting by a fixed primitive root of unity |$\zeta $|⁠.

 

Theorem 3.13.

Fix a deformation type of IHS manifolds and a connected component |${\mathcal {M}}_\Lambda ^\circ $| of the moduli of |$\Lambda $|-marked IHS manifolds of the given deformation type. Let |$h_1, \dots , h_n$| be a complete set of representatives of the conjugacy classes of monodromies |${\operatorname {Mon}}^2_\circ (\Lambda )$| of odd prime order |$p$| with |$\ker (h_i + h_i^{-1} - \zeta _p - \zeta _p^{-1})\leq \Lambda _{\mathbb {R}}$| of signature |$(2,*)$|⁠. Let |$H_i = \langle h_i \rangle \leq {\operatorname {Mon}}^2_\circ (\Lambda )$| and |$\chi _i\colon H_i \rightarrow {\mathbb {C}}^*$| the character defined by |$\chi _i(h_i):= \zeta _p$|⁠. For each |$H_i$| choose |$(X_i,G_i,\eta _i) \in {\mathcal {M}}^{\circ ,\chi _i}_{H_i}$|⁠. Then |$(X_1, G_1), \dots , (X_n, G_n)$| is a complete set of representatives of pairs |$(X,G)$| up to deformation and birational conjugation where |$X$| is an IHS manifold of the given deformation type, |$G\leq {\operatorname {Aut}}(X)$| is non-symplectic and |$\ker \rho _X \leq G$|⁠, |$|\rho _X(G)|=p$|⁠.

 

Proof.

Let |$(X,G)$| be a pair as above. The choice of a marking |$\eta $| with |$(X,\eta ) \in {\mathcal {M}}_\Lambda ^\circ $| determines |$H=\eta \rho _X(G)\eta ^{-1}$| up to conjugation in |${\operatorname {Mon}}^2_\circ (\Lambda )$|⁠. Deformations of |$(X,G,\eta )$| leave |$H$| invariant. Similarly, birational modifications of |$X$| are induced by parallel transport operators. In particular, they conjugate |$H$| by a monodromy. Hence, the conjugacy class of |$H$| does not change under birational conjugation and deformation of |$(X,G)$|⁠. We see that the |$(X_i,G_i)$|⁠, |$i=1,\dots , n$|⁠, are pairwise not equivalent. It remains to show that |$(X,G)$| is equivalent to |$(X_i,G_i)$| for some |$i \in \{ 1, \dots , n\}$|⁠. Let |$g\in G$| with |$(g^*)^{-1} \omega _X = \zeta _p\omega _X$|⁠. Then |$h = \eta \rho _X(g) \eta ^{-1} \in H$| is independent of the choice of |$g$|⁠. By assumption, there exists |$i \in \{1,\dots n\}$| and |$f \in {\operatorname {Mon}}^2_\circ (\Lambda )$| with |$h = f h_i f^{-1}$|⁠. Then |$(X,G,f\circ \eta )$| gives an element of |${\mathcal {M}}^{\circ ,\chi _i}_{H_i}$|⁠. After a small deformation of |$(X_i,G_i,\eta _i)$| and |$(X,G,\eta )$|⁠, we may and will assume that they are |$H$|-general. Then each belongs to some moduli space |${\mathcal {M}}^{\circ , \chi _i}_{K_i(N)}$|⁠, respectively, |${\mathcal {M}}^{\circ , \chi _i}_{K(N)}$| for suitable Kähler-type chambers |$K_i(N)$| and |$K(N)$|⁠. By Proposition 3.11, both moduli spaces are isomorphic to the period domain |$\Omega ^{\chi _i}_\Lambda \setminus (\Delta \cup \Delta ^{\prime})$|⁠. Since |$p$| is odd, this period domain is connected. Hence, after a deformation, we may assume that their periods agree. We conclude with (3) of Proposition 3.11.

4 Monodromies

We have seen that a classification of non-symplectic automorphisms of IHS manifolds with an action of odd prime order on the 2nd cohomology lattice is given, up to birational conjugation and deformation, by the corresponding conjugacy classes in the monodromy group. We obtain the conjugacy classes by reduction to the unimodular case which allows to handle all known deformation types in a uniform way.

Let |$(L,b)$| be a lattice. The real spinor norm|$\vartheta =\mbox {spin}_{\mathbb {R}}: O(L) \rightarrow {\mathbb {R}}^*/\left ({\mathbb {R}}^*\right )^2 \cong \left \{ \pm 1\right \}$| is defined as

$$ \begin{align*} & \vartheta(g) = \left( -\frac{b(v_1, v_1)}{2}\right) \ldots \left( -\frac{b(v_r, v_r)}{2}\right) \mod \left({\mathbb{R}}^*\right)^2\end{align*}$$

if |$g_{\mathbb {R}} \in O(L)_{\mathbb {R}}$| factors as a product of reflections |$g_{\mathbb {R}} = \tau _{v_1} \circ \ldots \circ \tau _{v_r}$| with respect to elements |$v_i \in (L)_{\mathbb {R}}$| (in particular, |$r \leq {\operatorname {rk}}\ L$| by the Cartan–Dieudonné theorem). Note that this is the real spinor norm corresponding conventionally to the quadratic form |$-b$|⁠.

 

Example 4.1.

Let |$(L,b)$| be a lattice of signature |$(l_+,l_-)$|⁠. We can diagonalize |$b\otimes {\mathbb {R}}$|⁠, that is, we can find |$e_1,\dots e_n \in L \otimes {\mathbb {R}}$| giving a diagonal Gram matrix with |$l_+$| ones and |$l_-$| minus ones on the diagonal. Then |$-{\operatorname {id}} = \tau _{e_1} \circ \dots \circ \tau _{e_n}$|⁠. We see that the spinor norm of |$-{\operatorname {id}}$| is |$(-1)^{l_+}$|⁠.

We set |$O^+(L) = \ker \vartheta $| and for |$G\leq O(L)$| set |$G^+=G \cap O^+(L)$|⁠. The importance of this group lies in the fact that |${\operatorname {Mon}}^2(\Lambda ) \subseteq O^+(\Lambda )$|⁠. Here and in the following sections, root lattices associated to Dynkin diagrams are always assumed to be negative definite.

4.1 K3 surfaces

For any K3 surface |$S,$| we have |$H^2(S, {\mathbb {Z}}) \cong \Lambda = U^{\oplus 3} \oplus E_8^{\oplus 2}$|⁠, which is the unique lattice in the genus |$\textrm {II}_{(3,19)}$| up to isometry. Its monodromy group is given by |${\operatorname {Mon}}^2(\Lambda )=O^+(\Lambda )$|⁠.

 

Theorem 4.2.

Let |$p$| be an odd prime number and |$r,a$| nonnegative integers. There exists a K3 surface |$S$| and a non-symplectic automorphism |$\sigma \in {\operatorname {Aut}}(S)$| of odd prime order |$p$| with invariant lattice |$H^2(S,{\mathbb {Z}})^\sigma $||$p$|-elementary hyperbolic of rank |$r \geq 1$| and discriminant |$p^a$| if and only if

1. |$p \leq 19$| and |$22-r \equiv 0 \pmod {p-1}$|⁠;

2. |$0 \leq a \leq \min \{r, \frac {22-r}{p-1}\}$| and |$a \equiv \frac {22-r}{p-1} \pmod {2}$|⁠;

3. if |$a=0$| or |$a=r$|⁠, then |$r \equiv 2 \pmod {8}$|⁠.

Moreover, the triple |$(p,r,a)$| determines the pair |$(S, \langle \sigma \rangle )$| up to deformation.

 

Proof.

Since |$\vartheta (-{\operatorname {id}}_\Lambda )=-1$| and |$-{\operatorname {id}}_\Lambda $| lies in the center of |$O(\Lambda )$|⁠, conjugacy classes in |$O(\Lambda )$| and |$O^+(\Lambda )$| coincide. From Theorem 1.1, we obtain the conditions for the existence of an isometry |$f \in O(\Lambda )$| of odd prime order |$p$| when |$l_+ = 3$|⁠, |$l_- = 19$|⁠, |$s_+ = 2$|⁠. Theorem 1.2 implies that the action of |$f$| is determined (up to conjugacy on |$\Lambda $|⁠) by the isometry class of |$\Lambda ^f$|⁠, which is unique by Lemma 2.22. We conclude with Theorem 3.13.

 

Remark 4.3.

A study of non-symplectic automorphisms of odd prime order on K3 surfaces was already conducted by Artebani et al. [2] and [3], where all possible isometry classes of the invariant and coinvariant lattices inside the 2nd cohomology lattice are listed and related to the topology of the fixed locus of the automorphism. In a 2nd step, the authors prove that a K3 surface |$S$| with a non-symplectic prime order automorphism |$\sigma $| of given invariant lattice belongs to an explicit irreducible family. Since the action on the cohomology lattice does not vary within the family, this shows that the action is unique up to conjugacy. We can follow the converse approach: since the fixed locus is a deformation invariant of the pair |$(S,\sigma )$|⁠, Theorem 4.2 gives another explanation why the fixed locus is determined by the invariant lattice.

 

Remark 4.4.

If the action of the group |$G$| is symplectic, then |$\Lambda _G$| is negative definite and we have the additional condition that |$\Lambda _G$| has maximum norm at most |$-4$| [46, Thm. 4.3]. Thus, our results do not apply to recover the classification of symplectic automorphisms, which is best understood in terms of the Niemeier lattices. See also [34] for a survey.

4.2 K3|$^{[n]}$|⁠, Kum|$_n$|⁠, OG|$_6$|⁠, and OG|$_{10}$|

In order to treat the four deformation classes in a uniform way, we embed |$\Lambda \cong H^2(X,{\mathbb {Z}})$| primitively into a unimodular lattice |$M$|⁠. The orthogonal complement |$\Lambda ^\perp \subset M$| will be denoted by |$V$|⁠. These lattices are given in Table 1 (see [7], [52], [53], [36, Lem. 9.2], [37, Cor. 4.8], [39, Thm. 4.3], [40], [50], [6, Prop. 10], [11, Cor. 5], [43]).

Let |$L$| be a lattice and |$\gamma : O(L) \rightarrow O(A_L)$| the natural map. We define |$\Gamma (L)=\gamma ^{-1}(\{\pm {\operatorname {id}}_{A_L}\})$|⁠. Denote by |$\chi = \gamma |_{\Gamma (L)}\colon \Gamma (L) \rightarrow \{\pm 1\}$| the corresponding character. For |$H\leq G$| a group and |$\alpha \colon G \to {\mathbb {C}}^*$| a character, we set |$H^\alpha =G^\alpha \cap H$|⁠.

 

Definition 4.5.

For |$K3^{\left [n\right ]}$| and |$\textrm {Kum}_n$| set |$h_V=-{\operatorname {id}}_V$|⁠, while for OG|$_6$| and OG|$_{10}$| let |$h_V \in O(V)$| be represented by the matrix

$$\left (\begin {smallmatrix} 0 & 1\\ 1 & 0 \end {smallmatrix}\right )$$

with respect to the obvious basis. Fix |$v \in V$| primitive with |$h_V(v)=v$|⁠. Let |$O(M,V,v)$| be the joint stabilizer in |$O(M)$| of |$V$| and |$v$|⁠. For all deformation types the image of |$O(V) \rightarrow O(A_V)$| is generated by |$\bar h_V$|⁠, which has order two. This defines in a natural way a character |$\chi _V\colon O(M,V,v) \rightarrow \{\pm 1\}$|⁠. We have the restriction map |$O(M,V,v) \rightarrow O(\Lambda )$| and a section

$$ \begin{align*} &\gamma \colon {\operatorname{Mon}}^2(\Lambda) \hookrightarrow O(M,V,v), \quad f \mapsto \hat{f}:= \hat{\chi}(f) \oplus f\end{align*}$$

where |$\hat {\chi }(g)= {\operatorname {id}}_V$| if |$\bar g = {\operatorname {id}}_{A_\Lambda }$| and |$\hat {\chi }(g) = h_V$| else. In the following, let |$S(M) \in \left \{O(M), SO(M) \right \}$| be as defined in Table 1 for each deformation type and |$S^{\vartheta \cdot \chi _V}(M)$| be the subgroup where the product of the real spinor norm |$\vartheta $| and the character |$\chi _V$| is trivial.

 

Lemma 4.6.

The image of |$\gamma $| is given by

$$ \begin{align*} &G =S^{\vartheta \cdot \chi_V}(M,V,v)=\begin{cases} O^{\vartheta \cdot \chi_V}(M,V,0) & \mbox{for } K3^{\left[n\right]},\\ SO^{\vartheta \cdot \chi_V}(M,V,0) & \mbox{for } \textrm{Kum}_n,\\ O^{\vartheta \cdot \chi_V}(M,V,v) & \mbox{for } \textrm{OG}_6,\\ O^{\vartheta \cdot \chi_V}(M,V,v) & \mbox{for } \textrm{OG}_{10}. \end{cases}\end{align*}$$

 

Proof.

Let |$f \in {\operatorname {Mon}}^2(\Lambda )\leq O^+(\Lambda )$|⁠. If |$\bar f = {\operatorname {id}}$|⁠, then |$\hat {f}= {\operatorname {id}}_V \oplus f$| and

$$ \begin{align*} &(\vartheta \cdot \chi_V)(\hat{f})=\vartheta(\hat{f})=\vartheta(f)=1.\end{align*}$$

Otherwise, |$\hat {f} = h_V \oplus f$| and

$$ \begin{align*} &(\vartheta \cdot \chi_V)(\hat{f})=-\vartheta(\hat{f})=-\vartheta(h_V)\vartheta(f)=-\vartheta(h_V)=1.\end{align*}$$

By construction, |$\hat {f}$| preserves |$V$| and |$v$|⁠. In the |$\textrm {Kum}_n$|-case, we have in addition |$\det (\hat {f})=\det (\hat {\chi }(f))\det (f) = \chi (f) \det (f)=1$|⁠. We conclude that |$G$| contains the image of |$\gamma $|⁠. To show the converse inclusion, let |$g \in G$| and set |$f=g|_\Lambda $|⁠. Since |$M$| is unimodular, |$\chi _V(g)=1$| if and only if |$\bar f = {\operatorname {id}}$|⁠. Moreover, |$g\vert _V \in O(V,v)=\langle h_V \rangle $|⁠, which is of order two. Thus, |$\hat \chi (f) = g\vert _V$|⁠, which implies |$\hat f = g$|⁠. It is immediate to check that |$f \in {\operatorname {Mon}}^2(\Lambda )$|⁠.

Let |$A$| be a group and |$g,h \in A$| elements. We denote conjugation by |$^{h}{}{g}=hgh^{-1}$| and conjugacy classes by |$^ {A}{}{g}$|⁠. The set of conjugacy classes of elements of |$A$| is |${\operatorname {cl}}(A)$|⁠. For a subgroup |$B \leq A$|⁠, we have a natural map |${\operatorname {cl}}(B) \rightarrow {\operatorname {cl}}(A)$|⁠.

 

Theorem 4.7.

Let |$\psi \colon {\operatorname {cl}}(Mon^2(\Lambda ))\cong {\operatorname {cl}}(G) \rightarrow {\operatorname {cl}}(S(M))$| be the natural map. Let |$g \in S(M)$| be of odd prime order with |$M_g$| of signature |$(2,\ast )$| and |$V \subseteq M^g$|⁠. Then the map

$$ \begin{align*} & \begin{array}{rccl} \varphi \colon &\psi^{-1}\left(^{S(M)}{}{g}\right) &\longrightarrow &S(M^g) \setminus \left\{W \subseteq M^g \text{ primitive} \mid W \cong V \right\}\\[2pt] &^{G}{}{h} &\longmapsto&S(M^g)fV \mbox{ where } g=\ ^{f}{h}, f\in S(M) \end{array} \end{align*}$$

is a bijection.

 

Proof.

Note that |$gfV=fhf^{-1}fV=fhV = fV$| and similarly |$gfv=fv$|⁠. Since |$p$| is odd and |${\operatorname {rk}} V \leq 2$|⁠, this implies |$fV \leq M^g$|⁠. Let |$s \in G$| and |$f^{\prime} \in S(M)$| such that |$h^{\prime} =\, ^{s}{h}$| and |$g=\, ^ {f^{\prime}}{h^{\prime}}$|⁠. We have to show that the left cosets |$S(M^G)fV$| and |$S(M^g)f^{\prime}V$| coincide. With |$t = f^{\prime}sf^{-1}$|⁠, we have |$g=\ ^{t}{g}$|⁠, thus |$t$| preserves |$M^g$| and its restriction is in |$O(M^g)$|⁠. We have |$tfV=f^{\prime}sV=f^{\prime}V$|⁠. For |$\textrm {Kum}_n$|⁠, we have to show in addition that |$t|_{M^g} \in SO(M^g)$|⁠. Since |$gt=tg$|⁠, Lemma 2.21 applies and gives |$\det t|_{M_g}=1$|⁠. Thus, |$1 = \det t = \det t|_{M^g} \cdot \det t|_{M_g}= \det t|_{M^g}$|⁠. This shows that |$\varphi $| is well defined.

Let |$W \leq M^g$| be a primitive sublattice with |$W \cong V$|⁠. By [47, Prop. 1.6.1], we can find |$f \in O(M)$| with |$W=fV$| and since |$V^\perp $| contains a hyperbolic plane, we may even assume |$f \in S(M)$|⁠. Set |$h=\, ^{f^{-1}}{g}$|⁠. It satisfies |$h|_V={\operatorname {id}}_V$| and it is of odd order, hence |$h \in G=S^{\vartheta \cdot \chi _V}(M,V,v)$|⁠. By definition, |$g =\ ^{f}{h}$| and |$\varphi (^ {G}{}{h})=S(M^g)W$|⁠. Thus, |$\varphi $| is surjective.

Consider |$h=\, ^{f^{-1}}{g}$| and |$h^{\prime}=\, ^{f^{-1}}{g}$| with |$f,f^{\prime} \in S(M)$|⁠. Their images under |$\varphi $| are represented by |$fV$| and |$f^{\prime}V$|⁠. To prove injectivity, suppose that there is |$t \in S(M^g)$| satisfying |$tfV=f^{\prime}V$|⁠. As |$\Gamma (V)\subseteq {\operatorname {im}}(S(M^g,V)\rightarrow O(V))$| acts transitively on the vectors of length |$v^2$|⁠, we may even assume |$tfv=f^{\prime}v$|⁠. The natural map

$$ \begin{align*} &SO(M_g,g)=O(M_g,g) \rightarrow O(M_g^{\vee} \! / M_g )\end{align*}$$

is surjective (cf. Proposition 2.20) and |$M$| is unimodular; hence, we can extend |$t$| to an isometry |$\widetilde {t} \in S(M)$| commuting with |$g$|⁠. Then |$u=(f^{\prime})^{-1} \circ \widetilde {t} \circ f$| conjugates |$h$| to |$h^{\prime}$| and preserves |$V$| and |$v$|⁠. If |$(\vartheta \cdot \chi _V) (u) \neq 1$|⁠, we can multiply |$u$| by |$\hat \chi (-{\operatorname {id}}_\Lambda )\oplus (-{\operatorname {id}}_\Lambda )$| to enforce |$u \in G$|⁠.

Together Theorems 1.1, 1.2, 3.13, and Theorem 4.7 pave the way for classifying non-symplectic monodromies of odd prime order up to conjugation. It remains to compute the orbits of primitive sublattices in the invariant lattice |$M^f$|⁠. This is done in Sections 5 and 6. Before we continue with the monodromy classification, we give the proof of Theorem 1.4, which characterizes the invariant and coinvariant lattices of non-symplectic automorphisms with an action of odd prime order on the 2nd cohomology lattice, for IHS manifolds of type |$K3^{\left [n\right ]}$|⁠, |$\textrm {Kum}_n$|⁠, OG|$_6$|⁠, and OG|$_{10}$|⁠.

 

Proof of Theorem 1.4.

In order for |$K$| to be the coinvariant lattice of a non-symplectic automorphism, its signature needs to be |$(2,{\operatorname {rk}} K -2)$|⁠. Any |$p$|-elementary lattice of discriminant |$p$| and signature |$(2,(a+2m)(p-1)-2)$| is unique in its genus, since it is either indefinite or of signature |$(2,0)$| (and therefore isomorphic to |$A_2(-1)$|⁠; see [17, Table 15.1]). Hence, by Proposition 2.15, the lattice |$K$| admits a fixed-point free isometry |$f$| of order |$p$| with |$\bar f = {\operatorname {id}}_{A_K}$| if and only if it is |$p$|-elementary and |${\operatorname {rk}} K = (l(A_K) + 2m)(p-1)$| for some |$m \in {\mathbb {Z}}_{\geq 0}$|⁠. Let |$T:= K^\perp \subset \Lambda $|⁠. We may assume that |$(K,f)$| has signature |$(k_1^+,k_1^-)=(2,\ast )$|⁠. By [47, Cor. 1.5.2], we have that |${\operatorname {id}}_T \oplus f \in O(T \oplus K)$| extends to an isometry |$\Phi \in O(\Lambda )$| such that |$\Lambda ^\Phi = T$| and |$\Lambda _\Phi = K$|⁠. Since |$p$| is odd this isometry is a monodromy. Theorem 3.13 provides |$(X,\eta ,\sigma )$|⁠.

For an IHS manifold |$X$| of type |$K3^{\left [n\right ]}$|⁠, a group of automorphisms |$G \subset {\operatorname {Aut}}(X)$| is called induced if there exists a projective |$K3$| surface |$S$|⁠, a group |$F \subset {\operatorname {Aut}}(S)$|⁠, a Mukai vector |$v \in H^*(S, {\mathbb {Z}})^F$|⁠, and a |$v$|-generic stability condition |$\tau $| such that |$(X, G)$| is isomorphic to |$(M_\tau (S,v), \tilde {F})$|⁠, where |$\tilde {F}$| is the induced action of |$F$| on the moduli space. For type |$\textrm {Kum}_n$|⁠, the same definition holds with |$S$| an abelian surface and the induced action of |$F$| on the fiber |$K_\tau (S,v)$| over |$(0,0)$| of the Albanese map |$M_\tau (S,v) \rightarrow S \times \hat {S}$|⁠. Combining Theorems 1.1, 3.13, and 4.7 with the results of [42, §3] we obtain the following (see also [41, §2] for |$\textrm {Kum}_n$|⁠).

 

Proposition 4.8.

For manifolds of type |$K3^{\left [n\right ]}$| or |$\textrm {Kum}_n$|⁠, let |$g \in O(M)$| be an isometry of odd prime order |$p$| with |$M^g$| of signature |$(2, r-2)$| and discriminant |$p^a$|⁠, for some |$r = {\operatorname {rk}} M^g \geq 2$| and |$a \geq 0$|⁠. Then the monodromy classes in the fiber |$\psi ^{-1}(^{S(M)}g)$| admit a geometric realization as actions of induced automorphisms if and only if |$M^g$| contains |$U$| as a direct summand. In particular, this happens if and only if |$(p,r,a) \neq (3,2,1), (3,4,4), (3,6,5), (3,8,6), (5,4,3), (23,2,1)$| for type |$K3^{\left [n\right ]}$|⁠; |$(p,r,a) \neq (3,2,1), (7,2,1)$| for type |$\textrm {Kum}_n$|⁠.

Notice that, if there is a primitive embedding |$U \hookrightarrow M^g$|⁠, then the fiber |$\psi ^{-1}(^ {S(M)}{}g)$| is nonempty for all values |$n \geq 2$| (we can take a primitive vector of square |$2(n-1)$| or |$2(n+1)$| inside the copy of |$U$|⁠). We also remark that, for manifolds of type |$K3^{\left [n\right ]}$|⁠, the isometries of |$M$| with |$(p,r,a) = (3,4,4), (3,6,5), (3,8,6), (5,4,3)$| can be realized as extended actions of induced automorphisms on moduli spaces of twisted sheaves on |$K3$| surface, since |$M_g$| contains a primitive copy of |$U(p)$| (see [16, Thm. 3.4]). Here however, for a fixed |$n$|⁠, one has to check whether there exists an embedding |$V \hookrightarrow M^g$|⁠. The complete lists of triples |$(p,r,a)$| for odd prime order isometries of |$M$|⁠, which extend non-symplectic monodromies of manifolds of type |$K3^{\left [n\right ]}$| and |$\textrm {Kum}_n$| are given in Figures 1 to 3.

Fig. 1.

Isometries of |$\textrm {II}_{(4,20)}$| and |$K3^{\left [n\right ]}$|⁠.

Fig. 2.

Isometries of |$\textrm {II}_{(4,20)}$| of order at least |$7$| and |$K3^{\left [n\right ]}$|⁠. Cases |$23a$| and |$23b$| are as in Lemma 2.22.

Fig. 3.

Isometries of |$\textrm {II}_{(4,4)}$| and automorphisms of type |$\textrm {Kum}_n$|⁠.

IHS manifolds of type OG|$_6$| and OG|$_{10}$| can be obtained as symplectic desingularizations of singular moduli spaces of stable objects on abelian surfaces and K3 surfaces, respectively; hence, a definition of induced automorphisms can be given for these deformation types too (see [42, §5]). For a lattice-theoretical characterization of induced automorphisms of manifolds of type OG|$_6$|⁠, we refer to [22].

5 Primitive Vectors in |$p$|-Elementary Lattices and |$K3^{\left [n\right ]}$|⁠, |$\textrm {Kum}_n$|

In this section, we classify orbits of primitive vectors in |$p$|-elementary lattices. The classification comes in two flavors: if the lattice is positive definite, the theta series, a modular form, governs the representation behavior; if it is indefinite, then strong approximation for the spin group allows to handle orbits in terms of congruences and spinor exceptions. This boils down to linear algebra over |${\mathbb {F}}_2$|⁠.

Let |$p$| be an odd prime and |$L \in \textrm {II}_{(l_+,l_-)}p^{\epsilon n}$|⁠, |$\epsilon \in \left \{\pm 1\right \}$|⁠, |$l_+, l_- \geq 1$|⁠, |$l_+ + l_- \geq 3,$| and |$0<k \in 2{\mathbb {Z}}$|⁠. Set |$R=L(-1)\oplus U$|⁠. Primitive sublattices |${\mathbb {Z}} x=\langle k \rangle \subseteq L$| are determined, up to the action of |$O(L)$|⁠, by a primitive extension |$\langle k \rangle \oplus R \subseteq \overline {\langle k \rangle \oplus R}$| and a primitive embedding of |$\overline {\langle k \rangle \oplus R}$| into the unimodular overlattice |$\overline {L \oplus R}$| of |$L \oplus R$| [47, Proposition 1.15.1]. For later use, we define the following torsion quadratic forms

$$ \begin{align*} & \begin{array}{lrcc} u_{p,k}\colon &({\mathbb{Z}}/p^k{\mathbb{Z}})^2 \longrightarrow {\mathbb{Q}}/2{\mathbb{Z}},& (x,y) \mapsto 2xy p^{-k} + 2{\mathbb{Z}} ;&\\ w_{p,k}^{\epsilon}\colon & {\mathbb{Z}}/p^k{\mathbb{Z}} \longrightarrow {\mathbb{Q}}/2{\mathbb{Z}},& x \mapsto ux^2p^{-k} +2{\mathbb{Z}}, &u\in {\mathbb{Z}}, \left(\frac{u}{p}\right)=\epsilon. \end{array} \end{align*}$$

5.1 Existence

We begin by investigating the primitive extension.

 

Lemma 5.1.

Let |$p$| be an odd prime, |$A$|⁠, |$B$| be nondegenerate torsion quadratic modules such that |$A = w_{p,a}^\epsilon $| for some |$a\geq 1$|⁠, |$\epsilon \in \{\pm 1\}$|⁠, and the underlying module of |$B$| is isomorphic to |$({\mathbb {Z}}/p{\mathbb {Z}})^n$|⁠. Denote by |$d$| a generator of |$A \cong {\mathbb {Z}}/p^a {\mathbb {Z}}$|⁠. Suppose that there is an embedding |$\phi \colon {\mathbb {Z}} p^{a-1}d \rightarrow B(-1)$|⁠, and let |$H$| be its graph in |$A \oplus B$|⁠.

1. Let |$a=1$| and write |$B=A(-1)\oplus C$|⁠. Then |$H^\perp /H \cong C$|⁠.

2. Let |$a\geq 2$| and write |$B = u_{p,1} \oplus C$|⁠. Then |$H^\perp /H \cong A \oplus C$|⁠.

In either case, the map |$\phi $| is unique up to the action of |$O(B)$|⁠. Moreover, |$l(H^\perp /H) = l(B)-1$|⁠.

 

Proof.

Case (1) is clear. For the uniqueness, use Witt’s theorem [54, Ch. 1, Thm. 5.3]. In case (2), let |$e,f \in u_{p,1}$| be generators such that |$b(e,f)=1/p$|⁠. Let |$d^2 = \epsilon /p^a$|⁠. Then |$p^{a-1}d$| and its image are isotropic. By Witt’s theorem, an isotropic vector in a regular quadratic space is unique up to isometries. So we may assume that |$\phi (p^{a-1}d)=e$|⁠. Then |$H$| is generated by |$p^{a-1}d+e$|⁠, while |$H^\perp $| is generated by |$e$|⁠, |$d- \epsilon f$|⁠, and |$C$|⁠. Thus, |$H^\perp /H \cong {\mathbb {Z}}/p^a {\mathbb {Z}} \oplus C$|⁠. Since |$(d- \epsilon f)^2=d^2$|⁠, we see that |${\mathbb {Z}}(d- \epsilon f)$| is isomorphic to |$A$|⁠.

We now relate the index of the primitive extension |$\langle k \rangle \oplus R \subseteq \overline {\langle k \rangle \oplus R}$| to the divisibility |$\textit {div}(x)$| of the image of |$x$| in |$(L,b)$|⁠, i.e. the positive generator of the ideal |$b(x, L)$|⁠. A similar computation (for |$L$| not necessarily |$p$|-elementary) can be found in [21, Prop. 6.1]. We denote by |$\nu _p(k)$| the valuation of |$k \in {\mathbb {Z}}$| at the prime |$p$|⁠.

 

Lemma 5.2.

We have |$\textit {div}(x)=1$| if and only if |$\overline {\langle k \rangle \oplus R}=\langle k \rangle \oplus R$|⁠. Otherwise, |$\textit {div}(x)=p$|⁠. Moreover |$q_{\langle k \rangle ^\perp } \cong -q \oplus r$| where

1. |$q=q_{\langle k \rangle }$|⁠, |$r = q_L$| if |$\textit {div}(x) =1$|⁠;

2. |$q\oplus w_{p,1}^\epsilon =q_{\langle k \rangle }$|⁠, |$r\oplus w_{p,1}^\epsilon = q_L$| if |$\textit {div}(x) =p$| and |$\nu _p(k) =1$|⁠;

3. |$q=q_{\langle k \rangle }$|⁠, |$r \oplus u_{p,1} = q_L$| if if |$\textit {div}(x) =p$| and |$\nu _p(k) \geq 2$|⁠.

 

Proof.

Set |$K=\langle k \rangle $| and let |$\pi _{\langle k \rangle }\colon L \rightarrow {\langle k \rangle }^\vee $| be the orthogonal projection. Set |$H_{\langle k \rangle } = \pi _{\langle k \rangle }(L)/{\langle k \rangle }$| and |$h=|H_{\langle k \rangle }|= [L:{\langle k \rangle }\oplus {\langle k \rangle }^\perp ]$|⁠. Then the divisibility of |$x$| is given by the ideal

$$ \begin{align*} &\langle x, L \rangle=\langle x,\pi_{\langle k \rangle}(L) \rangle= \langle x,x/h\rangle={\mathbb{Z}} (k/h).\end{align*}$$

We compute |$h^2=[L:{\langle k \rangle }\oplus {\langle k \rangle }^\perp ]^2 = \det {\langle k \rangle } \det {\langle k \rangle }^\perp /\det L $|⁠. Using |$k =\det {\langle k \rangle }$| and |$\det {\langle k \rangle }^\perp =\det \left (\overline {{\langle k \rangle }\oplus R}\right ) = \det {\langle k \rangle } \det L /\left [\overline {({\langle k \rangle } \oplus R)}:({\langle k \rangle } \oplus R)\right ]^2$| we arrive at |$k/h = \left [\overline {({\langle k \rangle } \oplus R)}:({\langle k \rangle } \oplus R)\right ]$|⁠. Since |$R$| is |$p$|-elementary and |${\langle k \rangle }$| of rank one, this index is either |$p$| or |$1$|⁠. The computation of |$q_{{\langle k \rangle }^\perp }$| follows from Lemma 5.1.

Before we compute when such a sublattice exists, we introduce some notation. Let |$s=s^{\prime}p^{\nu _p(s)} \in {\mathbb {Q}}_p$|⁠. We set |$\chi _p(s)=\left(\frac {s^{\prime}}{p}\right)$|⁠. For a torsion quadratic form |$r$|⁠, we denote |$\chi _p(r) = \chi _p(\det K_p(r))$| where |$K_p(r)$| is the unique |$p$|-adic lattice of rank |$l(r)$| with |$q_{K_p(r)}=r$|⁠.

 

Theorem 5.3.

Let |$k$| be an even positive number and |$p$| be an odd prime number. Let |$L\in \textrm {II}_{(l_+,l_-)}p^{\epsilon n}$| with |$l_+, l_- \geq 1$| and |$l_+ + l_-\geq 3$|⁠. Write |$a = \nu _p(k)$| and |$k = k^{\prime}p^a$|⁠. Then there exists a primitive vector |$x \in L$| with |$x^2 = k$| and |$\textit {div}(x) =1$| if and only if one of the following conditions is satisfied:

• (I-1)

  |$a=0$|⁠, |$n < {\operatorname {rk}}\ L - 1$|⁠;

• (II-1)

  |$a=0$|⁠, |$n = {\operatorname {rk}}\ L - 1$| and |$\left(\frac {-1}{p}\right)^{l_-}\left(\frac {k^{\prime}}{p}\right)=\epsilon $|⁠;

• (III-1)

  |$a>0$|⁠, |$n < {\operatorname {rk}}\ L - 2 $|⁠;

• (IV-1)

  |$a>0$|⁠, |$n = {\operatorname {rk}}\ L - 2$| and |$l_+ - l_- \equiv 0 \pmod 8$|⁠.

There exists a primitive vector |$x \in L$| with |$x^2 = k$| and |$\textit {div}(x) =p$| if and only if one of the following conditions is satisfied:

• (I-p)

  |$a = 1$|⁠, |$n> 1$|⁠;

• (II-p)

  |$a = 1$|⁠, |$n = 1$| and |$\left(\frac {k^{\prime}}{p}\right)=\epsilon $|⁠;

• (III-p)

  |$a> 1$|⁠, |$n> 2$|⁠;

• (IV-p)

  |$a> 1$|⁠, |$n = 2$| and |$l_+ - l_- \equiv 0 \pmod 8$|⁠.

If |$l_+=1$| or |$l_-=0$|⁠, the given conditions are still necessary but possibly not sufficient.

 

Proof.

Suppose that |$\textit {div}(x) =1$|⁠. By [47, Thm. 1.12.2], an embedding exists if and only if |$l(q \oplus -r) < {\operatorname {rk}}\ L - 1$| or |$l(q \oplus -r) = {\operatorname {rk}}\ L - 1$| and

$$ \begin{align}& \left(\frac{(-1)^{l_-}|q \oplus -r|}{p}\right) = \left(\frac{(-1)^{l_-} k^{\prime}}{p}\right)= \chi_p(-q \oplus r). \end{align}$$

(11)

For |$a=0$|⁠, |$l(q\oplus -r)=\max \{1,n\}$| while for |$a>0$|⁠, |$l(q\oplus -r)=n+1$|⁠. This yields (I-1) and (III-1). In case (II-1), we have |$\chi _p(-q)\chi _p(r)=1\cdot \epsilon $|⁠. In case (IV-1), we have |$\chi _p(-q) =\left( \frac{-k^{\prime}}{p}\right)$| and |$\chi _p(r)=\epsilon $|⁠. Then Equation (11) gives |$1+(p-1)(l_-+1) \equiv \left( \frac {-1}{p}\right)^{l_-+1}\equiv \epsilon \pmod 4$|⁠. Inserting into Equation (3) we obtain |$l_+ \equiv l_- \pmod 8$|⁠.

Suppose that |$\textit {div}(x) = p$|⁠. For an embedding to exist, we have to be able to write |$q_C= -q \oplus r$| as in Lemma 5.2. Suppose that |$a=1$|⁠. Then we need |$q_{\langle k \rangle } = q \oplus w_{p,1}^{\delta }$|⁠, |$q_L = r \oplus w_{p,1}^{\delta }$| for |$\delta =\left( \frac {k^{\prime}}{p}\right)$|⁠. This is possible if |$n> 1$| (I-p) or |$n=1$| and |$\epsilon =\delta $| (II-p).

Now, suppose that |$a>1$|⁠. Then we can write |$q=q_{\langle k \rangle }$| and |$r \oplus u_{p,1} = q_L$| if and only if |$n>2$| (III-p) or |$n=2$| and |$q_L=u_{p,1}$|⁠, that is, |$\epsilon = \left( \frac {-1}{p}\right)\equiv 1 + (p-1) \pmod 4$|⁠. Inserting into Equation (3) gives |$l_+ - l_- \equiv 0$| (IV-p).

In both cases, one computes |$\chi _p(-q \oplus r)=\left( \frac {k^{\prime}}{p}\right)\epsilon $|⁠. If |${\operatorname {rk}}\ L - 1 = l(-q \oplus r)= n-1$|⁠, we additionally have to check Equation (11). By Theorem 2.1, we have |$\epsilon = \left( \frac {-1}{p}\right)^{l_-}$|⁠. So the condition is satisfied.

 

Remark 5.4.

As already noted in [14], Theorems 5.3 and 4.7 imply that the classification of non-symplectic automorphisms with a cohomology action of odd prime order |$p$| on IHS manifolds of type |$K3^{\left [n\right ]}$| (resp. |$\textrm {Kum}_n$|⁠) is usually richer when |$p$| divides |$2(n-1)$| (resp. |$2(n+1)$|⁠), since in this case the Mukai vector inside the invariant lattice of the extended action on the Mukai lattice might be chosen of divisibility |$p$| other than |$1$|⁠.

5.2 Spinor exceptions

At this point, we can effectively determine when a primitive sublattice |$\langle k \rangle \hookrightarrow L$| exists. Further, we know that |$O(L)$|-orbits of primitive sublattices are locally determined by their divisibility. However, globally this need not be true, due to spinor exceptions. In this subsection, we quantify the exceptional cases.

Setting. Let |$p$| be an odd prime and |$L \in \textrm {II}_{(l_+,l_-)}p^{\epsilon n}$|⁠, |$\epsilon \in \left \{\pm 1\right \}$|⁠, |$l_+ \geq 2$|⁠, |$l_-\geq 1$|⁠, and |$l_+ + l_- \geq 4$|⁠. For |$0<k \in 2{\mathbb {Z}}$| let |${\mathbb {Z}} x=\langle k \rangle \subseteq L$| be a primitive sublattice and |$C = \langle k \rangle ^\perp $|⁠. We write |$q_C = -q \oplus r$| as in Lemma 5.2. We set |$p^m = |r|$| and have |$j=|q|=j^{\prime} p^b$| with |$b=\nu _p(j)$|⁠. Note that our conditions on |$l_\pm $| imply that |$C$| is indefinite of rank at least |$3$|⁠. We define

$$ \begin{align*} &\mathcal{L}_1 = \{\ell \in {\mathbb{P}}\setminus \{2,p\} \mid \nu_\ell(j) \equiv_2 1\} \cup \{2 \mid 0 <\nu_2(j) \equiv_2 0 \}.\end{align*}$$

Write |$k= k^{\prime}p^{\nu _p(k)}$| and denote by |$\dagger $| the following set of conditions:

$$ \begin{align*}& \begin{cases} {\operatorname{rk}}\ L = 4,\\ \left| \det L \right| /\textit{div}(x)^2=p,\\ \nu_p(k)\geq 2,\\ \left(\frac{-2k^{\prime}}{p}\right)=1,\\ \text{either } p \equiv 1 \pmod 4, \textrm{or}\ p \equiv 3 \pmod 4\ \textrm{and}\ \exists \, \ell\in {\mathcal{L}_1} \text{ with}\ \left(\frac{\ell}{p}\right)=-1. \end{cases} \qquad{\dagger} \end{align*}$$

With the above notation, the main result of the section is the following.

 

Theorem 5.5.

Let |$p$| be an odd prime number, |$k= k^{\prime}p^{\nu _p(k)}>0$| even and |$L$| a |$p$|-elementary lattice of signature |$(l_+,l_-)$| with |$l_+\geq 2$|⁠, |$l_-\geq 1$|⁠, and |$l_+ + l_- \geq 4$|⁠. If condition |$\dagger $| holds, then the number of |$O(L)$|-orbits of primitive vectors |$x \in L$| with |$x^2=k$| and given divisibility |$\textit {div}(x)$| is two. Otherwise, the number of orbits is either zero or one.

Notation. We rely on results of Miranda and Morrison [38]. We recall the necessary notation.

• |$|\mathfrak {g}(C)|$| the number of isomorphism classes of lattices in the genus of |$C$|⁠,

• |${\mathbb {P}}$| the set of primes,

• |${\mathbb {A}}$| the finite adele ring of |${\mathbb {Q}}$|⁠,

• |$S$| the finite set of square free integers dividing |$2\det C$|⁠,

• |${\mathbb {U}}_\ell = {\mathbb {Q}}_\ell ^\times / \left ({\mathbb {Q}}_\ell ^\times \right )^2$|

• |${\mathbb {E}}_\ell = \{s \in {\mathbb {U}}_\ell \mid \nu _\ell (s) \equiv 0 \pmod 2\}$|⁠,

• |$\Gamma _{\ell }=\{ \pm 1 \} \times {\mathbb {U}}_\ell $|⁠,

• |$\Gamma _{\ell ,0}=\{\pm 1\} \times {\mathbb {E}}_\ell $|⁠,

• |$\Gamma _{\mathbb {A}}=\left \{(d_\ell ,s_\ell )_{\ell \in {\mathbb {P}}} \in \prod _{\ell \in {\mathbb {P}}}\Gamma _{\ell } \mid (d_\ell ,s_\ell ) \in \Gamma _{\ell ,0} \mbox { for all but finitely many } \ell \right \}$|⁠,

• |$\Gamma _{\mathbb {Q}}= \{\pm 1\} \times {\mathbb {Q}}^\times / \left ({\mathbb {Q}}^\times \right )^2\leq \Gamma _{\mathbb {A}}$|⁠,

• |$\Gamma _S=\{(d,s) \in \Gamma _{\mathbb {Q}} \mid s \in S\}$|⁠,

• |$O(L\otimes \mathbb {A})=\left \{f \in \prod _{\ell \in {\mathbb {P}}} O(L_\ell \otimes {\mathbb {Q}}_\ell ) \mid f_\ell \in O(L_\ell ) \mbox { for all but finitely many }\ell \right \}$|⁠,

• |${\operatorname {spin}}(\tau _x):= b(x,x)/2$| for |$x \in L$| anisotropic and |$\tau _x$| the reflection in |$x$|⁠,

• |$\sigma = (\det ,{\operatorname {spin}})\colon O(L\otimes \mathbb {A}) \rightarrow \Gamma _{\mathbb {A}}$|⁠,

• |$\Sigma (L_\ell ) = {\operatorname {im}}(\sigma _\ell \colon O(L_\ell ) \rightarrow \Gamma _\ell )$|⁠,

• |$\Sigma ^\#(L_p) = {\operatorname {im}}(\sigma _\ell \colon O^\#(L_\ell ) \rightarrow \Gamma _\ell )$|⁠,

• |$\Sigma (L) = \prod _{\ell \in {\mathbb {P}}}\Sigma (L_\ell ) \leq \Gamma _{\mathbb {A}}$|⁠,

• |$\Sigma ^\#(L) = \prod _{\ell \in {\mathbb {P}}}\Sigma ^\#(L_\ell )$|⁠.

Note that we view |$\Gamma _\ell $| as a subgroup of |$\Gamma _{\mathbb {A}}$|⁠.

 

Lemma 5.6

(Key lemma).

Write |$q_C = -q \oplus r$| as in Lemma 5.2 and set

$$ \begin{align*} &\Delta(C):=\left(\Gamma_S \cap \Sigma(C)\right)\cdot \Sigma^\#(C) \cdot \sigma\left(O(r)\times \{\pm {\operatorname{id}}_{-q}\}\right). \end{align*}$$

Then the number of primitive sublattices |$\langle k \rangle \subseteq L$| with |$q_{\langle k \rangle ^\perp } \cong q_C$| up to the action of |$O(L)$| is given by |$|\mathfrak {g}(C)|\cdot [\Sigma (C):\Delta (C)]$|⁠.

 

Proof.

Fix (the isomorphism class of) the orthogonal complement |$C$| and let |$\phi $| be the obvious anti-isometry

$$ \begin{align*} &\phi \colon q_{\overline{\langle{k}\rangle \oplus R}}= q \oplus -r \longrightarrow -q \oplus r=q_C.\end{align*}$$

Let |$H= \overline {\langle k \rangle \oplus R}/(\langle k \rangle \oplus R)$| be its graph and |$T = \mbox {Stab}(H,O(\langle k\rangle ) \times O(R))$| the corresponding stabilizer. Then, following [47, Prop. 1.15.1], the number of primitive sublattices |$\langle k \rangle $| with orthogonal complement isomorphic to |$C$| is given by the double coset

$$ \begin{align*} & \overline{O(C)}\backslash O(q_C)/ \phi \overline{T} \phi^{-1}, \end{align*}$$

where the bar indicates that we regard the image in the orthogonal group of the discriminant group. Since |$\overline {O(C)}$| is normal, we can turn this into the single coset

$$ \begin{align}& O(q_C) \; /\; \left(\overline{O(C)} \cdot \phi \overline{T} \phi^{-1} \right). \end{align}$$

(12)

Now |$\overline {O(C)}$| is computed by the exact sequence [38, VIII Thm. 5.1]

$$ \begin{align*} &O(C) \rightarrow O(q_C) \xrightarrow{\sigma} \Sigma(L)/(\Gamma_S\cap \Sigma(C))\cdot \Sigma^\#(C)\rightarrow 1,\end{align*}$$

where by abuse of notation we identify |$\sigma $| and the corresponding map in the sequence. We see that

$$ \begin{align*} & O(q_C)/ (\overline{O(C)} \cdot \phi \overline{T} \phi^{-1}) \cong \Sigma(L)/(\Gamma_S\cap \Sigma(C)\cdot \Sigma^\#(C))\cdot \sigma(\phi\overline{T}\phi^{-1}).\end{align*}$$

It remains to compute |$\phi \overline {T}\phi ^{-1}$|⁠. We pass to the discriminant group |$A_{\langle k \rangle } \oplus A_R$|⁠. Since |$O(R)\rightarrow O(q_R)$| is surjective, we may replace |$O(R)$| by |$O(q_R)$| and |$O(\langle k \rangle )$| by |$\{\pm {\operatorname {id}}_{q_{\langle k\rangle }}\}$|⁠. Looking at the three cases in Lemma 5.2, we see that |$\phi \overline {T} \phi ^{-1}= O(r) \times \{\pm {\operatorname {id}}_{-q}\}$|⁠.

We continue by computing the pieces making up |$\Sigma (C)$| and |$\Delta (C)$|⁠.

 

Lemma 5.7.

Let |$\ell $| be a prime, |$\ell \neq 2, p$|⁠.

$$ \begin{align}& \Sigma^\#(C_\ell)= \Gamma_{\ell,0}, \qquad \Sigma(C_\ell)= \begin{cases} \Gamma_{\ell,0} & \mbox{for }\nu_\ell(j) \equiv 0 \pmod 2\\ \Gamma_\ell &\mbox{for } \nu_\ell(j) \equiv 1 \pmod 2 \end{cases} \end{align}$$

(13)

$$ \begin{align}& \Sigma^\#(C_2)= \Gamma_{2,0}, \qquad \Sigma(C_2)= \begin{cases} \Gamma_{2,0} & \mbox{for }\nu_2(j) \equiv 1 \pmod 2\\ \Gamma_2 &\mbox{for } \nu_2(j) \equiv 0 \pmod 2 \end{cases} \end{align}$$

(14)

 

Proof.

This follows from [38, VII §12] with |${\operatorname {rk}} C \geq 3$| and |$l(A_C)_\ell \leq 1$|⁠.

 

Lemma 5.8.

Let |$q_C \cong -q \oplus r$| with |$q,r$| as in Lemma 5.2. Let |$\delta = \det C / \det (q_C)_p$|⁠.

$$ \begin{align}& \Sigma^\#(C_p)= \begin{cases} \langle (1,1)\rangle & \mbox{for } \quad l(A_C) = {\operatorname{rk}} C\\ \langle (-1,2\delta )\rangle& \mbox{for }\quad l(A_C) + 1 = {\operatorname{rk}} C\\ \Gamma_{p,0}& \mbox{for } \quad l(A_C) + 2 \leq {\operatorname{rk}} C. \end{cases} \end{align}$$

(15)

 

Proof.

This is a translation of [38, VII Thm. 12.1] to our notation.

Recall that |$|r|=p^m$| and |$|q|=j= j^{\prime}p^b$|⁠.

 

Lemma 5.9.

Let |$q_C \cong -q \oplus r$| as in Lemma 5.2. Let |$j=j^{\prime} p^b$|⁠, |$b=\nu _p(j)$|⁠. Let |$0 \neq u \in {\mathbb {E}}_p$|⁠.

$$ \begin{align*} &\Sigma(C_p) = \begin{cases} \Gamma_{p,0} &\mbox{for }\quad m=0 \wedge b \equiv_2 0,\\ \langle (1,u),(-1,p)\rangle &\mbox{for }\quad l(A_C)={\operatorname{rk}} C \wedge(b=0 \vee b \equiv_2 1),\\ \langle (-1,1),(-1,p)\rangle &\mbox{for }\quad m=1 \wedge b>1 \wedge {\operatorname{rk}} C=3 \wedge \left( \frac{-j^{\prime}}{p}\right)=\left(\frac{2}{p}\right)\ ({\ast}),\\ \Gamma_p &\mbox{else.} \end{cases}\end{align*}$$

 

Proof.

This is a translation of [38, VII Thms. 12.5, 12.7, 12.9] to our situation. The only nontrivial computation is (⁠|$\ast $|⁠).

Assume that |$m=1$|⁠, |$b>1$|⁠, and |${\operatorname {rk}} C = 3$|⁠. Then the rank of |$L$| is |$4$|⁠. By our assumptions, the signature of |$L$| is either |$(2,2)$| or |$(3,1)$|⁠. Let |$1^{\epsilon _0} p^{\epsilon _1} (p^b)^{\epsilon _b}$| be the |$p$|-adic genus symbol of |$C_p$|⁠. Suppose that |$\textit {div}(x)=1$| so that the discriminant of |$L$| is |$p$|⁠. By the classification of |$p$|-elementary genera in Theorem 2.1, |$L$| must be in the genus |$\textrm {II}_{(2,2)}p^{\epsilon _1}$| (respectively, |$\textrm {II}_{(3,1)}p^{\epsilon _1}$|⁠) for |$p \equiv 1 \pmod 4$| (respectively, |$p \equiv 3 \pmod 4$|⁠). Let |$\textit {div}(x)=p$|⁠. Then |$L \in \textrm {II}_{(2,2)}p^{\epsilon _1^{\prime} 3}$| (respectively, |$\textrm {II}_{(3,1)}p^{\epsilon _1^{\prime} 3}$|⁠) for |$p \equiv 1 \pmod 4$| (respectively, |$p \equiv 3 \pmod 4$|⁠). By Lemma 5.2, |$\epsilon _1 =\left( \frac {-1}{p} \right)\epsilon _1^{\prime}$|⁠.

With Theorem 2.1, we compute that |$\epsilon _1 = \left (\tfrac {2}{p}\right )$|⁠. We have |$\det C = (-1)^{l_-}j^{\prime} p^{b+m}= \left (\tfrac {-1}{p}\right ) j^{\prime} p^{b+m}$| and |$\epsilon _b = \left (\tfrac {-j^{\prime}}{p}\right )$|⁠. By comparing the determinant with the symbol, we obtain the relation

$$ \begin{align}& \left(\frac{-j^{\prime}}{p}\right)=\chi_p(\det C)=\epsilon_0 \epsilon_1 \epsilon_b=\epsilon_0 \epsilon_1 \left(\frac{-j^{\prime}}{p}\right). \end{align}$$

(16)

Hence, |$\epsilon := \epsilon _0 = \epsilon _1$|⁠. The conditions of [38, VII Thm. 12.5] hold if and only if |$\epsilon _b=\epsilon $| giving (⁠|$\ast $|⁠). It remains to compute the spinor norms with [38, VII Thms. 12.5, 12.9]. Let |$\alpha \in {\mathbb {E}}_p$| be defined by |$\left (\tfrac {\alpha }{p}\right )=\epsilon =\left (\tfrac {2}{p}\right )$|⁠. Then |$\Sigma (C_p) =\langle (-1,2\alpha ),(-1,2 \alpha p)\rangle $|⁠. We conclude with |$\left (\tfrac {2 \alpha }{p}\right )=1$|⁠.

 

Lemma 5.10.

Write |$q_C = -q \oplus r$| as in Lemma 5.2. Generators of

$$ \begin{align*} & \sigma\left(O(r)\times \{{\operatorname{id}}_{-q_j}\}\right) \leq \Sigma(C)/\Sigma^\#(C) \end{align*}$$

are given by

1. |$(-1,2\alpha p)_p$| for |$m=1$| and |$r \cong w_{p,1}^\epsilon $| where |$\epsilon = \left(\frac {\alpha }{p}\right)$|⁠,

2. |$(-1,p)_p, (1,u )_p$| for |$m \geq 2$| and |$1 \neq u \in {\mathbb {E}}_p$|⁠.

 

Proof.

We know that |$O(r)$| is generated by reflections. So one may just compute the spinor norms of all reflections. For |$m\geq 2,$| use that |$w_{p,1}^{1} \oplus w_{p,1}^{1} \cong w_{p,1}^{-1} \oplus w_{p,1}^{-1}$|⁠.

 

Lemma 5.11.

$$ \begin{align*} &\Delta(C)=\left(\Gamma_S \cap \Sigma(C)\right)\cdot \Sigma^\#(C) \cdot \Sigma(C_p) \end{align*}$$

 

Proof.

Combine Lemmas 5.8, 5.9, and 5.10 to obtain

$$ \begin{align}& \Sigma(C_p)= \Sigma^\#(C_p)\cdot \sigma_p\left(O(q_L)\times \{ {\operatorname{id}}_{-q}\}\right) \cdot \langle (1,t)_p \rangle \end{align}$$

(17)

where |$t=p$| if |$l(A_C)={\operatorname {rk}} C$|⁠, |$0<b\equiv 0 \pmod 2$| and else |$t=1$|⁠. If |$t\neq 1$|⁠, then |$\Sigma (C_p)=\Gamma _p$| and |$\Sigma ^\#(C_p)= \langle (1,1)\rangle $|⁠. We rewrite

$$ \begin{eqnarray*} \Delta(C) \cdot \langle (1,t)_p\rangle &=& \left(\Gamma_S \cap \Sigma(C)\right)\cdot \Sigma^\#(C) \cdot \sigma\left(O(r)\times \{ {\operatorname{id}}_{-q}\}\right) \cdot \sigma(\{\pm {\operatorname{id}}_{q_C}\})\cdot \langle (1,t)_p\rangle\\ &=& \left(\Gamma_S \cap \Sigma(C)\right)\cdot \Sigma^\#(C) \cdot \sigma\left(O(r)\times \{ {\operatorname{id}}_{-q}\}\right) \cdot\langle (1,t)_p\rangle\\ &=& \left(\Gamma_S \cap \Sigma(C)\right)\cdot \Sigma^\#(C) \cdot \Sigma(C_p) \end{eqnarray*}$$

where we use that |$-{\operatorname {id}}_{q_C} \in \overline {O(C)}$|⁠, that is, |$\sigma (-{\operatorname {id}}_{q_C}) \in (\Gamma _S \cap \Sigma (C))\cdot \Sigma ^\#(C)$| and Equation (17).

It remains to show that |$(1,t)_p \in \Delta (C)$|⁠. So suppose that |$t=p$|⁠. Then |$(1,p) \in \Gamma _S \cap \Sigma (C)$| and |$(1,p)_\ell \in \Gamma _{\ell ,0}= \Sigma ^\#(C_\ell )$| for |$\ell \neq p$|⁠. Thus, |$(1,p)_p \in (\Gamma _S\cap \Sigma (C))\cdot \Sigma ^\#(C)\subseteq \Delta (C)$|⁠.

 

Definition 5.12.

Define

$$ \begin{align*} &\mathcal{L}_1 = \{\ell \in {\mathbb{P}}\setminus \{2,p\} \mid \nu_\ell(j) \equiv_2 1\} \cup \{2 \mid 0 <\nu_2(j) \equiv_2 0 \}\end{align*}$$

$$ \begin{align*} &\mathcal{L}_0 = \{\ell \in {\mathbb{P}}\setminus \{2,p\} \mid 0< \nu_\ell(j) \equiv_2 0\} \cup \{2 \mid \nu_2(j) \equiv_2 1\}\end{align*}$$

so that the set of prime divisors of |$pj$| is the disjoint union |$\{p\} \sqcup \mathcal {L}_1 \sqcup \mathcal {L}_0$|⁠. For |$\ell \in \mathcal {L}_1$|⁠, we have |$\Sigma (C_\ell )=\Gamma _{\ell }$|⁠. For |$\ell \in \mathcal {L}_0$|⁠, we have |$\Sigma (C_\ell )=\Gamma _{\ell ,0}.$|

 

Proposition 5.13.

The index of |$[\Sigma (C): \Delta (C)]$| is one, unless (⁠|$\ast $|⁠) is true and |$\mathcal {L}_1$| contains a nonsquare modulo |$p$|⁠. In this case, the index is two.

 

Proof.

The set |$\mathcal {L}_1$| is defined so that for |$\ell \in \mathcal {L}_1$| the quotient |$\Sigma (C_\ell )/\Sigma ^\#(C_\ell ) = \Gamma _{\ell }/\Gamma _{\ell ,0}$| is of order |$2$| and generated by the class of |$(1,\ell )$|⁠. The map

$$ \begin{align*} &\phi\colon \Sigma(C)/\left(\Sigma^\#(C)\cdot \Sigma(C_p)\right) \longrightarrow {\mathbb{F}}_2^{\mathcal{L}_1}, \quad(d_\ell,s_\ell)_{\ell \in {\mathbb{P}}} \mapsto (\ell \mapsto \nu_\ell(s_\ell))\end{align*}$$

is an isomorphism of |${\mathbb {F}}_2$|-vector spaces. Hence, the index is given by

$$ \begin{align*} &[\Sigma(C): \Delta(C)]=[{\mathbb{F}}_2^{\mathcal{L}_1}: \phi(\Gamma_S \cap \Sigma(C))].\end{align*}$$

Let |$(d,s) \in \Gamma _S$|⁠. It lies in |$\Sigma (C)$| if and only if |$(d_p,s_p) \in \Sigma (C_p)$|⁠. Suppose that we are not in case (⁠|$\ast $|⁠). Then |$\{1\} \times {\mathbb {E}}_p \leq \Sigma (C_p)$|⁠. Hence, |$(1,\ell ) \in \Gamma _S \cap \Sigma (C)$| for all |$\ell \in {\mathcal {L}_1}$|⁠. Taking the images |$\phi (1,\ell )$|⁠, we obtain the standard basis of |${\mathbb {F}}_2^{\mathcal {L}_1}$|⁠.

If (⁠|$\ast $|⁠) holds, |$\Sigma (C_p)=\langle (-1,1),(1,p)\rangle $|⁠. Then |$(d,s)$| with |$s=s^{\prime}p^{\nu _p(s)}$| is in |$\Gamma _S \cap \Sigma (C)$| if and only if |$\left (\frac {s^{\prime}}{p}\right )=1$|⁠. This gives a linear condition on |${\mathbb {F}}_2^{{\mathcal {L}_1}}$|⁠, which is nontrivial if and only if there is at least one |$s \in {\mathcal {L}_1}$| with |$\left (\frac {s}{p}\right )=-1$|⁠.

Notice the isomorphism |$\Gamma _p \rightarrow {\mathbb {F}}_2^3$|⁠, |$(d,s) \mapsto (\epsilon _p(d),u_p(s),\nu _p(s))$|⁠, where |$d=(-1)^{\epsilon _p(d)}$| and |$u_p = \epsilon _p\circ \chi _p$|⁠.

 

Lemma 5.14.

We have |$|\mathfrak {g}(C)|=1$|⁠, unless (⁠|$\ast $|⁠) is true, |$p \equiv 1 \pmod 4$| and every element of |${\mathcal {L}_1}$| is a square modulo |$p$|⁠, in which case |$|\mathfrak {g}(C)|=2$|⁠.

 

Proof.

The order of |$\mathfrak {g}(C)$| is the order of the cokernel of

$$ \begin{align*} &\Gamma_S \rightarrow \Delta^{\prime} / \{(\pm 1, \pm 1)\}_S \end{align*}$$

where |$\Delta ^{\prime}:= \left (\prod _{\ell \mid 2jp} \Gamma _\ell / \Sigma (C_\ell )\right )$| (cf. [38, VIII Prop. 6.1]). For |$\ell \in \mathcal {L}_0$|⁠, we have |$\Gamma _\ell /\Sigma (C_\ell )=\Gamma _\ell /\Gamma _{\ell ,0}$| and for |$\ell \in \mathcal {L}_1$| the quotient is trivial. Let |$\pi _p\colon \Gamma _p / \Sigma (C_p) \xrightarrow {\sim } ({\mathbb {F}}_2)^\gamma $|⁠, |$\gamma \in \{0,1\}$| be the unique isomorphism. We obtain the isomorphism

$$ \begin{align*} & \psi \colon \Delta^{\prime} = \Gamma_p/\Sigma(C_p) \times \prod_{\ell \in \mathcal{L}_0} \Gamma_\ell / \Gamma_{\ell,0} \xrightarrow{\sim} {\mathbb{F}}_2^\gamma \times {\mathbb{F}}_2^{\mathcal{L}_0}, \quad (d,s) \mapsto (\pi_p(d,s),\ell\mapsto \nu_\ell(s)).\end{align*}$$

A basis |$B$| of |$\Gamma _S$| is given by

$$ \begin{align*} &\{(1,\ell) \mid \ell \in {\mathcal{L}_1} \cup \mathcal{L}_0\}\cup \{(1,p)\} \cup \{(\pm 1,\mp 1)\}.\end{align*}$$

We view |$\psi (B)$| as a matrix and compute its rank. The projection of |$\{\psi (1,\ell ) \mid \ell \in \mathcal {L}_0\}$| to |${\mathbb {F}}_2^{\mathcal {L}_0}$| is a basis of |${\mathbb {F}}_2^{\mathcal {L}_0}$|⁠. Thus, |$\psi (B)$| has rank at least |$|\mathcal {L}_0|$|⁠. It has rank |$|\mathcal {L}_0| + \gamma $| if and only if |$\gamma =0$| or |$\{\pi _p(1,s) \mid s \in {\mathcal {L}_1} \cup \{p\}\}\cup \{\pi _p(\pm 1,\mp 1)\}$| contains a nonzero element. If (⁠|$\ast $|⁠) does not hold and |$\gamma \neq 0$|⁠, then |$\pi _p \in \{\nu _p,\epsilon _p\cdot \nu _p\}$|⁠. Now |$\pi _p(1,p)\neq 0$| does the trick. If (⁠|$\ast $|⁠) holds, then |$\pi _p = u_p$|⁠. For |$p \equiv 3 \pmod 4$|⁠, we have |$u_p(1,-1)\neq 0$|⁠. But for |$p \equiv 1 \pmod 4$| we have |$u_p(\pm 1,\mp 1)= 0 =u_p(1,p)$|⁠. Thus, the map is not surjective if and only if |$p\equiv 1 \pmod 4$| and every |$\ell \in {\mathcal {L}_1}$| is a square mod |$p$|⁠.

 

Example 5.15.

For |$k=50$|⁠, |$L = H_5 \oplus U \in \textrm {II}_{(2,2)}5^{-1}$|⁠, and divisibility |$1,$| we have |$j=2$|⁠, |$b=2$|⁠, |$m=1$|⁠, |$\mathcal {L}_0=\{2\}$|⁠, |$\mathcal {L}_1=\emptyset $|⁠. Thus, the genus of |$C$| consists of the two classes

$$ \begin{align*} & \left(\begin{array}{rrr} 2 & 3 & 0 \\ 3 & -8 & 0 \\ 0 & 0 & -10 \end{array}\right), \left(\begin{array}{rrr} 2 & 1 & 0 \\ 1 & -2 & 10 \\ 0 & 10 & -90 \end{array}\right).\end{align*}$$

For a case where |$|\mathfrak {g}(C)|=1$| but |$[\Sigma (C): \Delta (C)]=2$|⁠, consider |$L = H_5 \oplus U$| and divisibility |$1$|⁠, so that |$j = k$|⁠. We need |$\left(\frac {-j^{\prime}}{5}\right)=\left(\frac {2}{5}\right)=-1$| and that |$\mathcal {L}_1$| contains a nonsquare modulo |$p$|⁠. The smallest example is |$k= 4 \cdot 3 \cdot 25 = 300$|⁠.

Theorem 5.5 now follows from Lemma 5.14 and the previous results.

 

Remark 5.16.

The number of |$O(L)$|-orbits of rank one primitive sublattices and primitive vectors is the same since |$-{\operatorname {id}}_L$| exchanges the two generators of the sublattice. For |$SO(L),$| this is certainly the case if |$\det (-{\operatorname {id}}_L)=1$|⁠, that is, the rank of |$L$| is even.

 

Lemma 5.17.

Let |$L$| be as in Theorem 5.5 and |$V \subset L$| a primitive sublattice of rank one. Then |$SO(L)V = O(L)V$|⁠.

 

Proof.

We have |$SO(L)V = O(L)V$| if and only if |$O(L) = SO(L)$| or |$\exists f \in O(L)$|⁠, |$\det f=-1,$| and |$fV=V$|⁠. If the rank of |$L$| is odd, then |$f = -{\operatorname {id}}_L$| suffices. If |${\operatorname {rk}}\ L$| is even, after replacing |$f$| by |$-f$| we can assume that |$f\vert _V = {\operatorname {id}}_V$|⁠. Such |$f$| exists if and only if there is |$g \in O(C)$| with |$g|_{-q}={\operatorname {id}}_{-q}$| (and then |$g=f|_C$|⁠). By Lemma 5.8 and 5.10 we find |$(-1,s) \in \sigma (\{{\operatorname {id}}_{-q}\} \times O(r)) \cdot \Sigma ^\#(C)$| for some |$s \in {\mathbb {Q}}$|⁠. Inspecting the proof of [38, VIII Thm. 2.3] shows that we can find |$g\in O(C)$| with |$\sigma (g) = (-1,s)$|⁠.

5.3 The definite case

Let |$L$| be an even positive definite lattice of even dimension |$n$|⁠. Its level is defined as the smallest natural number |$N$| with |$N L^\vee \subseteq L$|⁠. Its theta series is

$$ \begin{align*} &\Theta_L(z) = \sum_{x \in L }q^{x^2/2}= \sum_{k=0}^\infty a(k)q^k, \quad \mbox{with } q=\exp(2\pi i z).\end{align*}$$

It is a modular form of weight |$n/2$| with respect to the congruence subgroup |$\Gamma _0(N)$| of SL|$_2({\mathbb {Z}})$| and the character |$\chi (d)=\left(\frac {(-1)^{n/2}\det L}{d}\right)$| (see, e.g., [19, Thm. 3.2]).

The integer |$a(k) \!=\! \#\{x\! \in\! L \mid x^2 \!=\! 2k\}$| counts the number of vectors of length |$2k$| in |$L$|⁠. Define |$r(k) = \#\{x \in L \text { primitive} \mid x^2 = 2k\}$| as the number of primitive vectors of length |$2k$|⁠. We can compute |$r$| from |$a$| as follows. Since every vector is a multiple of a primitive vector, we can write

$$ \begin{align}& a(k) = \sum_{d^2 \mid k} r(k/d^2). \end{align}$$

(18)

Consider the space |${\mathbb {C}}^{\mathbb {N}}$| of complex sequences equipped with point wise addition and the Dirichlet convolution |$\ast $|⁠. It is a commutative ring with identity element |$\epsilon =(1,0,0,\dots )$|⁠. Denote by |$\mu $| the Möbius function, by |$\lambda $| the Liouville function, by |$1=1_{\mathbb {N}}$| the constant sequence |$(1,1,1,\dots )$| and by |$1_{sq}$| the indicator function of the set of square numbers. We note that |$1_{sq}=\lambda \ast 1$|⁠, |$\mu \ast 1 = \epsilon $| (Möbius inversion) and |$|\mu | \ast \lambda = \epsilon $|⁠. In this terminology, Equation (18) is |$a = 1_{sq} \ast r = (\lambda \ast 1) \ast r$|⁠. Thus, |$r = |\mu | \ast \mu \ast a$|⁠. This yields

$$ \begin{align}& r(n)= \sum_{d^2 \mid n}\mu(d)a(n/d^2) \end{align}$$

(19)

where we have used that |$(\mu \ast |\mu |)(d)=0$| if |$d$| is not a square and |$(\mu \ast |\mu |)(d)=\mu (\sqrt {d})$| if |$d$| is a square.

In the classification of non-symplectic monodromies of odd prime order conducted in Section 4 for manifolds of type |$K3^{\left [n\right ]}$| and |$\textrm {Kum}_n$|⁠, there are four positive definite invariant lattices of rank |$2$|⁠:

$$ \begin{align*} &A_2(-1)=\left(\begin{matrix}2 & 1 \\1 & 2\end{matrix}\right),\quad K_7=\left(\begin{matrix}2 & 1 \\1 & 4\end{matrix}\right),\quad F_{23a}=\left(\begin{matrix}2 & 1 \\1 &12\end{matrix}\right),\quad F_{23b}=\left(\begin{matrix}4 & 1 \\1 & 6\end{matrix}\right).\end{align*}$$

We refer to Figures 1a, 2, 3. In particular, the lattice |$K_7$| only occurs for manifolds of |$\textrm {Kum}_n$|-type, the lattices |$F_{23a}, F_{23b}$| only occur for manifolds of |$K3^{\left [n\right ]}$|-type, while |$A_2(-1)$| occurs for both.

Set |$\theta _2:= \sum _{m = -\infty }^{\infty } q^{(m+\tfrac {1}{2})^2}$| and |$\theta _3:= \sum _{m = -\infty }^{\infty } q^{m^2}$|⁠. Following [17, Ch. 4, §6.2], the 1st three lattices have theta series given by |$\theta _3(z)\theta _3(ez) + \theta _2(z)\theta _2(ez)$| for |$e=3,7,23$|⁠. For the last one |$F_{23b}$|⁠, we have |$\theta _3(z)\theta _3(ez) + \theta _2(z)\theta _2(ez) - 2\eta (z)\eta (23z)$| (see also [57]). To see this, note that the difference |$\Theta _{23a}-\Theta _{23b}$| is a cusp form of weight |$1$| and level |$23$|⁠. The space of such forms is of dimension one [35, Modular Form 23.1.b.a]. It is spanned by |$\eta (z)\eta (23z)$| where the Dedekind |$\eta $|-function is |$\eta =q^{1/24}\prod _{n=1}^\infty (1-q^n)$|⁠.

Denote by |$b_k$| the number of primitive vectors of length |$2k$| in |$L$| modulo the action of |$G \in \{SO(L),O(L)\}$|⁠. A primitive vector is stabilized by |$g \in G$| if and only if it is an eigenvector of eigenvalue |$1$| for |$g$|⁠. Since |$L$| is of rank |$2$| this eigenspace is of rank at most |$1$| if |$g\neq 1$|⁠. Thus, there are only finitely many primitive |$v \in L$| with nontrivial stabilizer in |$G$|⁠. This means that for |$k \notin S_G := \{v^2/2 \mid {\mathbb {Z}} v = \ker (1-g), g\in G, g\neq 1\}$| we have |$b_k = r(k)/|G|$|⁠. For the readers convenience, we list the 1st few terms in Table 2 (where we restrict to the groups |$G$| that are relevant to our classification).

 

Remark 5.18.

We note that |$r(k)$| and thus |$b(k)$| is unbounded.

For |$K \leq L$| a primitive sublattice, we define its divisibility |$\textit {div}(K)=|\det K| / [L: K \oplus K^\perp ]$|⁠.

 

Definition 5.19.

Let |$(X,G)$| be a pair consisting of an IHS manifold |$X$| of type |$K3^{\left [n\right ]}$| or |$\textrm {Kum}_n$| and |$G \leq {\operatorname {Aut}}(X)$| a group of non-symplectic automorphisms with |$\ker \rho _X \leq G$|⁠, |$\rho _X(G)$| of odd prime order |$p$|⁠. Fix a generator |$f \in \rho _X(G)$| acting by |$\zeta _p$| on the holomorphic |$2$|-form |$\omega _X$|⁠. Let |$(p,r,a)$| be the invariants of the extended action of |$f$| on the unimodular lattice |$M \supset H^2(X,{\mathbb {Z}})$| as in Section 4.2 and |$[I_f]$| the Steinitz class of the |${\mathbb {Z}}[\zeta _p]$|-module |$(H^2(X,{\mathbb {Z}})_f,f|_{H^2(X,{\mathbb {Z}})_f})$|

We say that |$G$| is ambiguous if |$(X,G)$| is not determined up to birational conjugation and deformation by |$(p,r,a)$|⁠, |$[I_f]$| and the divisibility of |$V \leq M$|⁠.

 

Remark 5.20.

If |$p<23$|⁠, then |$[I_f]$| is trivial (Remark 2.16). The invariants |$(p,r,a)$| and |$\textit {div}(V)$| are equivalent to the datum of the genera of the invariant and coinvariant lattices |$H^2(X,{\mathbb {Z}})^f$| and |$H^2(X,{\mathbb {Z}})_f$|⁠.

 

Corollary 5.21.

If a group of automorphisms of an IHS manifold of type |$K3^{\left [n\right ]}$| or |$\textrm {Kum}_n$| is ambiguous, then its invariants are as in Tables 3 and 4.

 

Remark 5.22.

Differently from the case of K3 surfaces, for IHS manifolds of higher dimension it is possible to have non-symplectic automorphisms with birationally conjugate deformations, but whose geometrical fixed loci are topologically different. An example for manifolds of type K3|$^{[2]}$| is discussed in [9, Rmk. 7.7] and [10, §7.2]. However, for a manifold |$X$| of type K3|$^{[2]}$| with an automorphism |$g$| of odd prime order |$p \leq 19$| the numerical invariants of the coinvariant lattice determine the dimension of the mod |$p$| cohomology |$h^*(X^g, \mathbb {F}_p)$| and the Euler characteristic |$\chi (X^g)$|⁠, and vice-versa (see [12, Thm. 2] and [9, §3.2, Appendix A]).

6 Embeddings of |$A_2(-1)$| and OG|$_{10}$|

In this section, we compute the number of orbits of primitive sublattices isomorphic to |$A_2(-1)$| into the invariant lattices of prime order isometries of |$\textrm {II}_{(5,21)}$|⁠.

 

Proposition 6.1.

Let |$L \in \textrm {II}_{(3,l_-)}p^{\epsilon n}$| with |$l_-> 0$|⁠, |$l_- \equiv 1 \pmod 2$|⁠. Then |$A_2(-1)$| embeds primitively into |$L$| with divisibility |$1$| if and only if one of the following holds

• (I-1)

  |$p\neq 3$|⁠, |${\operatorname {rk}}\ L - n> 2$|⁠;

• (II-1)

  |$p\neq 3$|⁠, |${\operatorname {rk}}\ L - n = 2$| and |$\epsilon = \left(\frac {-3}{p}\right)$|⁠;

• (III-1)

  |$p = 3$|⁠, |${\operatorname {rk}}\ L - n> 3$|⁠;

• (IV-1)

  |$p = 3$|⁠, |${\operatorname {rk}}\ L - n = 3$| and |$\epsilon = -1$|⁠.

The lattice |$A_2(-1)$| embeds primitively into |$L$| with divisibility |$3$| if and only if |$p=3$| and one of the following holds

• (I-3)

  |$n=1$| and |$\epsilon =-1$|⁠;

• (II-3)

  |$n>1$|⁠, |${\operatorname {rk}}\ L - n> 1$|⁠;

• (III-3)

  |$n>1$|⁠, |${\operatorname {rk}}\ L - n = 1$| and |$\epsilon = 1$|⁠.

The embedding of |$A_2(-1)$| is determined up to the action of |$O(L)$| by its divisibility.

Fig. 4.

Isometries of |$\textrm {II}_{(5,21)}$| and OG|$_{10}$|

 

Proof.

Recall the notation used in Section 5.2. We use Lemma 5.2 with |$q_{\langle k \rangle }$| replaced by |$q_V$|⁠. Note that |$A_2(-1)$| embeds into |$L$| if and only if |$l(q_C)< {\operatorname {rk}}\ L -2$| or |$l(q_C)={\operatorname {rk}}\ L-2$| and

$$ \begin{align*} &\chi_p((-1)^{l_-} 3 p^n) =\chi_p(-3)= \chi_p(q_C).\end{align*}$$

If |$p = 3$|⁠, this yields |$-1=\chi _3(q_C)$|⁠. Suppose that the divisibility is |$1$|⁠. Then |$q_C=q_{A_2} \oplus q_L=w_{3,1}^1 \oplus q_L$|⁠. If |$p \neq 3$|⁠, then |$l(q_C)=n$| and |$\chi _p(q_C)=\chi _p(q_L)=\epsilon $| giving (I-1) and (II-1). If |$p = 3$|⁠, then |$l(q_C)=n+1$| and |$\chi _3(q_C)=\epsilon $| giving (III-1) and (IV-1). Let |$\textit {div}(V)=3$|⁠, then |$p=3$| and we can write |$q_C= r$| where |$q_L = w_{3,1}^{-1} \oplus r$| if and only if |$n>1$| or |$n=1$| and |$q_L=w_{3,1}^{-1}$|⁠, i.e. |$\epsilon =-1$|⁠. Then |$l(q_C)=n -1$| and |$\chi _3(q_C)=-\epsilon $| giving (I-3), (II-3), (III-3).

The analogue of (⁠|$\ast $|⁠) in this situation never holds since |$b \leq 1$|⁠. So we have uniqueness of the embedding for |$l_->1$|⁠. If |$l_-=1$|⁠, then |${\operatorname {rk}} C=2$| so the previous reasoning does not apply. However, there are just three cases to consider. We leave them to the reader.

 

Corollary 6.2.

Let |$X$| be an IHS manifold of type OG|$_{10}$| and |$G \leq {\operatorname {Aut}}(X)$| a group of odd prime order |$p$| of non-symplectic automorphisms. Let |$f \in \rho _X(G)$| be a generator acting by |$\zeta _p$| on the holomorphic |$2$|-form |$\omega _X$|⁠. Then |$(X, G)$| is determined up to deformation and birational conjugation by the isomorphism classes of |$H^2(X,{\mathbb {Z}})^f$|⁠, |$H^2(X,{\mathbb {Z}})_f$| and additionally, if |$p=23$|⁠, by the Steinitz class |$[I_f]$| of the |${\mathbb {Z}}[\zeta _p]$|-module |$(H^2(X,{\mathbb {Z}})_f,f|_{H^2(X,{\mathbb {Z}})_f})$|⁠.

Figure 4 provide the triples |$(p, r, a)$| for isometries |$f$| of |$M \in \textrm {II}_{(5,21)}$| of odd prime order |$p$|⁠, with |$M^f$| of rank |$r$| and discriminant |$p^a$|⁠, and |$M_f$| of signature |$(2, \ast )$|⁠. We highlight the cases where |$M^f$| contains a primitive copy of |$V = A_2(-1)$|⁠, and its divisibility.

Funding

A. C. is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 286237555 – TRR 195. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 286237555 – TRR 195.

Acknowledgments

The authors thank Samuel Boissière, Chiara Camere, Alessandra Sarti, and Davide Veniani for sharing their insights, Markus Kirschmer for explaining his work and how to prove Proposition 2.12, Rainer Schulze-Pillot for pointing out the work of Quebbemann, suggesting the use of Möbius inversion and the theta series of |$F_{23b}$|⁠. We thank the anonymous referees for their suggestions on how to improve the presentation. Part of this work arose at the 2019 Japanese-European Symposium on Symplectic Varieties and Moduli Spaces at ETH Zürich.

Communicated by Prof. Enrico Arbarello