Quantitative reduction theory and unlikely intersections

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila--Zannier strategy to the Zilber--Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions.


Introduction
Reduction theory is concerned with finding small representatives for each orbit in actions of arithmetic groups, for example through constructing fundamental sets. It began with the study of the action of SL n (Z) on quadratic forms, which was described by Siegel in terms of a fundamental set for SL n (Z) in SL n (R). Borel and Harish-Chandra generalised this to arithmetic lattices in arbitrary semisimple Lie groups. This theory has had wide-ranging applications in areas such as the theory of automorphic forms and locally symmetric spaces [AMRT10], the arithmetic of algebraic groups [PR94] and finiteness theorems for abelian varieties [Mil86].
The first goal of this paper is to prove quantitative bounds for the group elements used in Borel and Harish-Chandra's construction of fundamental sets. These bounds are polynomial in terms of suitable input parameters, although they are not fully effective. They generalise the polynomial bounds of Li and Margulis for the reduction theory of quadratic forms [LM16] and complement the second-named where B u ⊂ Γ is a finite set such that every b ∈ B u satisfies We apply Theorem 1.2 to the Zilber-Pink conjecture on unlikely intersections. We prove the Zilber-Pink conjecture for A 2 , the (coarse) moduli space of principally polarised abelian surfaces over C, subject to a large Galois orbits conjecture (Conjecture 6.2), which is stated in section 6.
We recall that A 2 is a Shimura variety of dimension 3 associated with the group GSp 4 . The Zilber-Pink conjecture predicts that an irreducible algebraic curve C ⊂ A 2 that is Hodge generic (that is, not contained in any proper special subvariety) contains only finitely many points of intersection with the special subvarieties of A 2 having dimension 1 or 0 (see [Pin05,Conjecture 1.3] for the most relevant formulation of the Zilber-Pink conjecture).
Pila and Tsimerman's proof of the André-Oort conjecture for A 2 [PT13] shows that C contains only finitely many special points. (A special point on A 2 is a point associated with an abelian surface with complex multiplication.) Therefore, in order to prove the Zilber-Pink conjecture for A 2 , it suffices to show that C contains only finitely many non-special points belonging also to a special curve.
The special curves in A 2 are of three types: (1) curves parametrising abelian surfaces with quaternionic multiplication (we refer to these as quaternionic curves); (2) curves parametrising abelian surfaces isogenous to the square of an elliptic curve ("E 2 curves"); (3) curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication ("E × CM curves").
In this paper, we study intersections with the quaternionic and E 2 curves. Let Σ Quat (resp. Σ E 2 ) denote the set of points of A 2 which are Hodge generic in some quaternionic (resp. E 2 ) curve. Our first main result on unlikely intersections is the following.
Theorem 1.3. Let Σ denote Σ Quat or Σ E 2 and let C ⊂ A 2 denote an irreducible Hodge generic algebraic curve. If C satisfies Conjecture 6.2 for Σ, then C ∩ Σ is finite.
Combined with our previous work [DO], in which we study intersections with the E × CM curves, and Pila and Tsimerman's proof of the André-Oort conjecture for A 2 , this completes the proof of the Zilber-Pink conjecture for A 2 , subject to Conjecture 6.2 (a large Galois orbits conjecture). As in [DO], the general strategy follows the proof of [DR18,Theorem 14.2], which was an application of the so-called Pila-Zannier method to the Zilber-Pink conjecture for general Shimura varieties. However, we will have to make several modifications, and the end of the proof is closer to [Orr21,Proposition 3.5].
Finally, we show that the large Galois orbits conjecture holds for Σ = Σ Quat when the curve under consideration satisfies a multiplicative reduction hypothesis at the boundary of the moduli space. We proved the analogous result for intersections with E × CM curves in [DO].
Theorem 1.4. Let C ⊂ A 2 denote an irreducible Hodge generic algebraic curve defined over Q such that the Zariski closure of C in the Baily-Borel compactification of A 2 intersects the 0-dimensional stratum of the boundary.
Then C satisfies Conjecture 6.2 for Σ Quat , and so C ∩ Σ Quat is finite.
Theorem 1.4 follows from a result of André [And89, Ch. X, Thm. 1.3] and the Masser-Wüstholz isogeny theorem. We have not been able to prove an analogue of Theorem 1.4 for Σ = Σ E 2 because the result of André does not apply to abelian surfaces isogenous to the square of an elliptic curve.
We note that the results on quantitative reduction theory in this paper will be an important tool for proving the Zilber-Pink conjecture for other Shimura varieties, which will be the subject of future work by the authors. We expect these results to have further applications, for example, a uniform version of the second-named author's bounds for polarisations and isogenies of abelian varieties [Orr17] and bounds for the heights of generators of arithmetic groups by combining them with some of the techniques of homogeneous dynamics from [LM16].
1.1. Outline of the paper. In section 2, we give some background on reduction theory to put our Theorems 1.1 and 1.2 into context.
In section 3, we define notation to be used throughout the paper. We state the various (equivalent) definitions of a Cartan involution that exist in the literature, and we define the notion of a Siegel set.
In section 4, we prove our main theorems on quantitative reduction theory, namely, Theorems 1.1 and 1.2. The proof follows the stategy of [BHC62] but, in order to obtain a quantitative result, it is necessary to replace "soft" topological ingredients in the former, notably [BHC62,Prop. 5.2], with results from elsewhere.
In order to apply Theorem 1.2 to a specific situation, we must construct a representation ρ of the ambient reductive group and show that the vectors v u can be chosen suitably bounded. This is the topic of section 5 -we construct a representation of GSp 4 with the properties required to apply Theorem 1.2 to the subgroups associated with quaternionic and E 2 curves.
In section 6, we prove our theorems on unlikely intersections, namely, Theorems 1.3 and 1.4. The strategy for the former is to parametrise the unlikely intersections by integral vectors of suitably bounded length (or, equivalently, height). This parametrisation is obtained using the results of sections 4 and 5. Then, as in all versions of the Pila-Zannier method, we consider the parameters for unlikely intersections as a set definable in an o-minimal structure and apply a Pila-Wilkie counting theorem to control the number of such points in terms of their height. Theorem 1.3 follows by comparing this upper bound with the lower bound of the large Galois orbits conjecture.
We emphasise that sections 2-5 require no knowledge of Shimura varieties and may be read independently of section 6, while the results of section 2 are not used elsewhere in the paper.

Background on reduction theory
In this section we outline some of the history of reduction theory, focussing on quantitative results and on the work of Borel and Harish-Chandra, and explain how our Theorems 1.1 and 1.2 fit into this theory. The results of this section are not used later in the paper.
2.1. Reduction theory for quadratic forms. The group SL n (Z) acts on the set of integral quadratic forms in n variables via its natural action on the variables. The classical reduction theory of quadratic forms defines a set of reduced quadratic forms with the following properties.
(i) Each SL n (Z)-orbit of non-degenerate integral quadratic forms contains at least one reduced form. (ii) Each SL n (Z)-orbit of non-degenerate integral quadratic forms contains only finitely many reduced forms.
A variety of definitions of reduced quadratic forms are used, possessing varying properties in addition to Properties 2.1. The most important definitions are due to Lagrange and Gauss for binary quadratic forms, and to Hermite, Minkowski and Siegel for quadratic forms in any number of variables. In these definitions (except some of Hermite's definitions), the reduced quadratic forms can be defined by finitely many polynomial inequalities in the coefficients of the forms.
Reduction theory behaves better for positive (or negative) definite quadratic forms than for indefinite forms. Definite quadratic forms satisfy a much stronger version of Property 2.1(ii) called the Siegel property. This guarantees that there is a uniform bound on the number of reduced quadratic forms in each SL n (Z)-orbit (for fixed n). In the nicest case of all, definite binary quadratic forms using Gauss's definition of reduced forms, each SL 2 (Z)-orbit contains exactly one reduced form.
Quantitative reduction theory for quadratic forms. The discriminant of a quadratic form is invariant under the action of SL n (Z). A form is non-degenerate if and only if its discriminant is non-zero. Hence the following lemma implies Property 2.1(i).
Lemma 2.2. For each integer ∆ = 0, there are only finitely many reduced integral quadratic forms in n variables of discriminant ∆.
The following quantitative version of Lemma 2.2 is classical for anisotropic quadratic forms [Cas78, p. 287, Cor. 1], using Hermite and Minkowski's definitions of reduction (note that anisotropic forms are always definite when n ≥ 5). Indeed, for binary anisotropic forms, it goes back to Lagrange. For Siegel reduced forms (definite or indefinite), it can be proved by adapting the proof of Lemma 2.2 found on [Cas78, pp. 322-324] but we are unsure whether this was classically known. Proposition 2.3. For each positive integer n, there exists a positive real number c 5 (n) such that, for every integer ∆ = 0, all reduced integral n-ary quadratic forms of discriminant ∆ have coefficients with absolute values at most c 5 (n)|∆|.
Lemma 2.2 also implies the following lemma, which does not mention reduced forms, and which was historically one of the most important consequences of reduction theory.
Lemma 2.4. For each integer ∆ = 0, there are only finitely many SL n (Z)-orbits of integral quadratic forms in n variables of discriminant ∆.
Li and Margulis have proved a quantitative version of Lemma 2.4. The bound is stronger than in Proposition 2.3, but it applies only to at least one form in each SL n (Z)-orbit, rather than to all reduced forms.
Proposition 2.5. [LM16,Theorem 3] For each integer n ≥ 3, there exists a constant c 6 (n) such that every SL n (Z)-orbit of indefinite quadratic forms in n variables of discriminant ∆ = 0 contains a form whose coefficients have absolute value at most c 6 (n)|∆| 1/n .

Siegel sets.
Siegel shifted the emphasis in reduction theory from quadratic forms to arithmetic groups. There is a direct link between the reduction theory of definite quadratic forms and fundamental sets for SL n (Z) in SL n (R).
Let v 0 denote the standard positive definite quadratic form in n variables: v 0 (x 1 , . . . , x n ) = x 2 1 + · · · + x 2 n . If F is a fundamental set of positive definite real quadratic forms in n variables (that is, a set which satisfies the generalisations of Properties 2.1 for positive definite real forms), then S = {g ∈ SL n (R) : gv 0 ∈ F} is a fundamental set in SL n (R) for the action of SL n (Z) by multiplication on the left (that is, every right SL n (Z)-coset intersects S in at least one, and at most finitely many, elements). Conversely, if S ⊂ SL n (R) is a fundamental set for SL n (Z) which is invariant under right multiplication by SO n (R) = Stab SLn(R) (v 0 ), then R >0 Sv 0 is a fundamental set of positive definite real quadratic forms.
Siegel defined a family of sets S = S t,u ⊂ SL n (R), depending on two parameters t, u ∈ R >0 . The set S t,u is a fundamental set for SL n (Z) whenever t ≤ √ 3/2 and u ≥ 1/2 (according to the conventions used in this paper). We call S 0 = S √ 3/2,1/2 the standard Siegel set in SL n (R). Using the construction described in the previous paragraph, we obtain a fundamental set of positive definite real quadratic forms, namely R >0 S 0 v 0 . We say that a positive definite quadratic form is Siegel We also say that an indefinite quadratic form of signature (p, The Siegel reduced indefinite integral quadratic forms satisfy Properties 2.1. However the set of Siegel reduced indefinite real quadratic forms is not a fundamental set because it does not satisfy the generalisation of Property 2.1(ii) to real forms.
Borel and Harish-Chandra generalised the notion of Siegel set from SL n (R) to all reductive Lie groups [BHC62,sec. 4.1] (which they called Siegel domains). However these Siegel domains are not always fundamental sets for arithmetic subgroups -in general, one can only say that there is a fundamental set contained in a finite union of translates of Siegel domains [BHC62, Thm. 6.5, Lemma 7.5].
Borel subsequently gave a new definition of Siegel sets for reductive Q-algebraic groups [Bor69,12.3], taking into account the Q-algebraic group structure and not just the Lie group structure. For each reductive Q-algebraic group G and each arithmetic subgroup Γ ⊂ G(Q), there is a finite union of G(Q)-translates of a Siegel set which forms a fundamental set for Γ in G(R) [Bor69,Thm. 13.1] (this is a consequence of Theorem 2.7 below). In this paper we shall use a minor modification of Borel's definition of Siegel sets, described in section 3.3.
2.3. Reduction theory for representations of reductive groups. The following result is a key step in Borel and Harish-Chandra's construction of fundamental sets for arithmetic groups.
Theorem 2.6. [BHC62,Lemma 5.4] Let G be a reductive Q-algebraic group whose Q-rank is equal to its R-rank. Let S ⊂ G(R) be a Siegel set. Let ρ : The restriction on the Q-rank of G in Theorem 2.6 can be removed with only minor alterations to the proof, provided we use the definition of Siegel sets from [Bor69,12.3] (or the definition in section 3.3 of this paper) instead of the definition of Siegel domains from [BHC62, 4.1].
As noted in [BHC62, Example 5.5], Theorem 2.6 implies Lemma 2.2, by applying it to the representation of SL n (R) on the vector space of real quadratic forms in n variables, with v = λv (p,q) 0 where λ ∈ R >0 and v (p,q) 0 is defined by equation (1). Then the orbit SL n (R)v is the set of all quadratic forms of signature (p, q) and discriminant (−1) q λ n , and Sv ∩ Λ is the set of Siegel reduced integral quadratic forms of given signature and discriminant.
In general, we may think of ρ(S)v as a set of "reduced vectors" in the representation V R . However we should note that this set depends on v.
Quantitative reduction theory for representations. Theorem 1.1 is a quantitative version of Theorem 2.6, bounding the length of "reduced integral vectors", that is elements of the finite set ρ(S)v ∩ Λ, in terms of v (for fixed group G and representation ρ). We are not able to prove such a bound for all v ∈ V R : we must restrict to a set of v for which G(R) acts "in a similar way" on all of the permitted vectors v. This is achieved through the condition that v must lie in the Aut ρ(G) (V R )-orbit of a fixed vector.
For an example application of Theorem 1.1, let ρ be the representation of G = SL n on the quadratic forms in n variables, and let v 0 = v (p,q) 0 . Noting that every scalar λ ∈ R × is in Aut ρ(G) (V R ), we deduce that there are constants c 7 and c 8 (depending on n) such that |w| < c 7 |λ| c 8 for all λ ∈ R × and w ∈ Sλv 0 ∩ L.
Thus Theorem 1.1 implies a weakened version of Proposition 2.3 -Proposition 2.3 is stronger because it gives a bound which is linear in the discriminant, while the constants in Theorem 1.1, even the exponent, are ineffective (see Remark 4.3).
In this example, the closed orbits in V R are those which consist of non-degenerate quadratic forms. These orbits are parameterised by their signature and discriminant, so every closed orbit intersects Aut ρ(G) (V R )v (p,q) 0 for some signature (p, q). Thus, in this case, Theorem 1.1 is sufficient to give a polynomial bound for integral elements of a reduced set in every closed orbit. In general, however, there is no finite subset of V R whose Aut ρ(G) (V R )-orbit intersects every closed G(R)-orbit, and then Theorem 1.1 does not allow us to compare all closed orbits.
In this example, the representation ρ is absolutely irreducible so its only endomorphisms are scalars. In general, there may be more endomorphisms of ρ, and it will be important for our applications that we allow v to be any element of Aut ρ(G) (V R )v 0 , not just a scalar multiple of v 0 .
2.4. Fundamental sets for arithmetic groups. The central result of Borel and Harish-Chandra's reduction theory was the construction of fundamental sets for Γ H \H(R), where H is a reductive Q-algebraic group and Γ H ⊂ H(Q) is an arithmetic subgroup. These fundamental sets are constructed by embedding H into some GL n , where we already know how to obtain fundamental sets using standard Siegel sets. (Note that [BHC62, Thm. 6.5] used the notation G where we write H in this theorem.) Theorem 2.7. [BHC62,Thm. 6.5] Let H be a reductive Q-algebraic subgroup of GL n,Q and let Γ H = GL n (Z) ∩ H(R). Let S 0 be the standard Siegel set in GL n (R). Let u ∈ GL n (R) be such that u −1 H(R)u is self-adjoint.
Then there exists a finite set B ⊂ GL n (Z) such that is a fundamental set for Γ H in H(R).
The ambient group GL n in Theorem 2.7 can be replaced by an arbitrary reductive Q-algebraic group G containg H with only minor alterations to the proof, where S 0 is replaced by a sufficiently large Siegel set in G(R).
Quantitative fundamental sets for arithmetic groups. Theorem 1.2 is a quantitative version of Theorem 2.7, where H varies over the Q-algebraic members of a G(R)conjugacy class of subgroups of some fixed reductive group G. This theorem is "quantitative" in the sense that it controls a measure of the size of the elements of the finite set B. Ideally we would like to bound the height of elements of B but we have not yet achieved this (it may be possible by combining the methods of this paper with tools of homogeneous dynamics as in [LM16]). Instead we measure the size of elements of B in terms of how they act on a vector v u (whose stabiliser is H) in a suitable representation of G. This turns out to be sufficient for our applications to unlikely intersections.
The theorem will only apply to subgroups H ⊂ G which are defined over Q because these are the subgroups for which Γ ∩ H(R) is a lattice in H(R). However it is very important that H varies over a G(R)-conjugacy class, not just a G(Q)conjugacy class, because this allows the Q-algebraic subgroups in the conjugacy class to belong to more than one isomorphism class over Q. A striking consequence of allowing this is that the conjugacy class may contain both Q-anisotropic and Q-isotropic groups, so the fundamental set in H(R) is sometimes compact and sometimes not compact, yet the same bounds apply to fundamental sets for all H in the conjugacy class. For example SL 2,Q and unit groups of quaternion algebras can be found in the same SL 4 (R)-conjugacy class of subgroups of SL 4 .
Note also that the semisimple subgroups of G belong to only finitely many G(R)conjugacy classes [BDR, Cor. 0.2]. This is not true for reductive subgroups, as may be seen by considering the torus G 2 m , which contains infinitely many non-conjugate subgroups isomorphic to G m -see [BDR, Remark 1.2].
For an example application of Theorem 1.2, consider the case where G = GL n and H 0 ⊂ GL n is the orthogonal group of the quadratic form v p,q 0 . The representation ρ : G → GL(Λ Q ), where Λ is the Z-module of integral quadratic forms of signature (p, q), satisfies the conditions of Theorem 1.2. In particular, (iii) holds because if H u is defined over Q then it is the orthogonal group of the integral quadratic form v = ρ(u)λv (p,q) 0 for some λ ∈ R × . As noted in [BHC62, 6.7], the space of Hermite majorants of v is Theorem 1.2 allows us to control the set B u used to construct this fundamental set in that for each b ∈ B u , the coefficients of the quadratic form ρ(b −1 )v are polynomially bounded in terms of disc(v). A related result can be found in [LM16, §9.5], which bounds the entries of the matrices b ∈ B u (stronger than bounding ρ(b −1 )v), although its bound involves the coefficients of v as well as the discriminant. As with Theorem 1.1, the constants in Theorem 1.2 are ineffective.

Preliminaries
3.1. Notation. If Λ is a Z-module, we write Λ Q for Λ ⊗ Z Q and Λ R for Λ ⊗ Z R. If Λ is free and finitely generated and φ : Λ Q × Λ Q → Q is a Q-bilinear form, we denote by disc(Λ, φ) the determinant of the matrix (φ(e i , e j )) i,j where {e 1 , . . . , e n } is a Z-basis for Λ (the determinant is independent of the choice of basis).
If R is an order in a semisimple Q-algebra D, then we write disc(R) for the discriminant of the Z-module R with respect to the bilinear form φ(x, y) = Tr D/Q (xy) where Tr D/Q is the (non-reduced) trace of the regular representation of D. See section 5.4 for more details.
If V = Λ Q (or V = Λ R ) and G is an algebraic group over Q (or R, respectively), then, for any representation ρ : G → GL(V ) and v ∈ V , we write Stab G,ρ (v) for the stabiliser of v in G with respect to ρ, that is, If W is a subspace of V , we write Stab G,ρ (W ) for the subgroup preserving W . Similarly, we write End G,ρ (V ) and Aut G,ρ (V ) for the endomorphisms and automorphisms, respectively, of V commuting with ρ(G), and we also write End G,ρ (Λ) for the endomorphisms of Λ commuting with ρ(G). If ρ is an inclusion G → GL(V ), then we omit it from the subscripts. If D is a ring acting on V , we denote by End D (V ) the endomorphisms of V commuting with the action of D.
We denote by G der the derived subgroup of G, by Z(G) the centre of G, and by G(R) + the connected component of G(R) (in the archimedean topology) containing the identity. If S is a split Q-subtorus of G, we denote by Z G (S) the centraliser of S in G and by X * (S) the character group of S.
If V = Λ R , we write |·| for a norm on V . Unless otherwise specified, it does not matter which norm we choose, except that the values of constants will depend on the norm. Whenever the statement of a theorem involves a norm |·|, we implicitly assume that such a norm has been chosen, and the constants in the theorem implicitly depend on this choice. We write · for the associated operator norm on End R (V ). In other words, for f ∈ End R (V ), 3.2. Cartan involutions. The theory of Cartan involutions is well-known for connected semisimple groups. However, for reductive real algebraic groups, several definitions of Cartan involutions are used in the literature. The following seems to us to be the most elegant definition.
Definition. Let G be a reductive R-algebraic group. A Cartan involution of G is an involution θ : G → G in the category of R-algebraic groups such that the set of fixed points of θ in G(R) is a maximal compact subgroup of G(R).
The fundamental example is the standard Cartan involution x → (x t ) −1 on GL n , whose real fixed point set is O n (R).
The commonly used definitions of Cartan involutions for reductive real algebraic groups are all equivalent to this one, but the equivalences are not obvious and it is difficult to find proofs for all of the equivalences. For convenience, we provide a list of equivalent definitions, and we will post a self-contained proof of this lemma on arXiv. In the following lemma: (ii) is the definition of Cartan involution used in [Bor69,11.17]; (iii) is the definition used in [BHC62] and [RS90]; while (iv) is the definition from [Del79,p. 255], commonly used in the study of Shimura varieties.

g} is compact and intersects every connected component of G(C), where· denotes complex conjugation. Furthermore, for each maximal compact subgroup K ⊂ G(R), there is a unique Cartan involution of G whose set of real fixed points is K.
Given a reductive R-algebraic group G and a Cartan involution θ of G, we say that an algebraic subgroup H ⊂ G is self-adjoint (with respect to θ) if θ(H) = H.
In several of our theorem statements (including Theorems 1.1 and 1.2), we are given a reductive Q-algebraic group G and a Siegel set S ⊂ G(R). It will be seen in section 3.3 that the definition of a Siegel set involves the choice of a maximal compact subgroup K ⊂ G(R). In such a situation, we say that a subgroup of G is self-adjoint if it is self-adjoint with respect to the Cartan involution whose fixed point set is the K used in the construction of the Siegel set. Let G be a reductive Q-algebraic group. In order to define a Siegel set in G(R), we begin by making choices of the following subgroups of G: (1) P a minimal parabolic Q-subgroup of G; (2) K a maximal compact subgroup of G(R).
As a consequence of [AMRT10, Ch. II, Lemma 3.12], there is a unique R-torus S ⊂ P satisfying the following conditions: (i) S is P(R)-conjugate to a maximal Q-split torus in P. (ii) S is self-adjoint with respect to the Cartan involution associated with K. These conditions could equivalently be stated as: (i) S is a lift of the unique maximal Q-split torus in P/U, where U denotes the unipotent radical of P. (ii) Lie S(R) is orthogonal to Lie K with respect to the Killing form of G.
Define the following further pieces of notation: (1) M is the preimage in Z G (S) of the maximal Q-anisotropic subgroup of P/U. (Note that by [BT65,Corollaire 4.16], Z G (S) is a Levi subgroup of P and hence maps isomorphically onto P/U.) (2) ∆ is the set of simple roots of G with respect to S, using the ordering induced by P. (The roots of G with respect to S form a root system because S is conjugate to a maximal Q-split torus in G.) (1) Ω is a compact subset of U(R)M(R) + ; and (2) t is a positive real number. We say that a set Ω ⊂ G(R) is a fundamental set for Γ if the following conditions are satisfied: (F1) ΓΩ = G(R); and (F2) for every g ∈ G(Q), the set {γ ∈ Γ : γΩ ∩ gΩ = ∅} is finite (the Siegel property). The following two theorems show that, if we make suitable choices of Siegel set S ⊂ G(R) and finite set C ⊂ G(Q), then CS is a fundamental set for Γ in G(R).

Quantitative reduction theory
In this section, we will prove Theorems 1.1 and 1.2. The proof follows the same strategy as that of [BHC62, Lemma 5.3 and Thm. 6.5]. We replace the purely topological proof of [BHC62, Prop. 5.2] by an orbit growth bound of Eberlein [Ebe14] using Riemannian geometry in GL n (R) + . We also prove a lemma bounding the norm of τ ∈ Aut ρ(G) (V R ) in terms of the length of v = τ (v 0 ) -this is a calculation in a semisimple R-algebra. For the rest, the proof closely follows the method of Borel and Harish-Chandra, keeping track of quantitative information and the action of Aut ρ(G) (V R ) throughout and with some small adaptations to generalise to reductive groups whose R-rank is greater than their Q-rank.
4.1. Bound for orbits of real reductive groups. We begin by proving the following bound for orbits in representations of real reductive groups, not yet considering any arithmetic subgroup.
Proposition 4.1. Let G be a reductive R-algebraic group and let ρ : G → GL(V R ) be an R-algebraic representation. Let v 0 ∈ V R be a non-zero vector whose orbit ρ(G(R))v 0 is closed. Then there exist constants c 9 and c 10 (depending on G, ρ Here we show that the operator norm of elements of ρ(Ω) is polynomially bounded with respect to the length of vectors in Q.
We define a Riemannian metric on GL n (R) + as follows. The positive definite bilinear form (A, B) → tr(AB t ) on M n (R), which is the Lie algebra of GL n (R), induces a right-invariant Riemannian metric on the Lie group GL n (R) + . Let d R denote the distance function on GL n (R) + induced by this Riemannian metric.
Eberlein's theorem relates |ρ(g)v 0 | to the Riemannian distance between g and the stabiliser of v 0 . We will combine this with the following lemma bounding ρ(g) in terms of the Riemannian distance.
Lemma 4.2. Let I denote the identity matrix in GL n (R). There exists a constant c 11 (n) such that every g ∈ GL n (R) + satisfies Proof. Let |g| F denote the Frobenius norm of g, that is, Using the Cartan decomposition, we can write g = k exp(X) for some k ∈ SO n (R) and some symmetric matrix X ∈ M n (R). Let λ max denote the largest eigenvalue of X (note that X is diagonalisable and all its eigenvalues are real because it is symmetric).
Since |·| F and · are norms on the finite-dimensional vector space M n (R), they are equivalent, so this proves the lemma.
Proof of Proposition 4.1. Let G = G(R) + . Fix a finite list of representatives a 1 , . . . , a r for the connected components of G(R). Then, given w ∈ ρ(G(R))v 0 , we can write w = a i w for some i ≤ r and some w ∈ ρ(G)v 0 . Hence it suffices to prove the proposition for w ∈ ρ(G)v 0 .
By [Mos55] we can choose an inner product on V R with respect to which ρ(G) is self-adjoint. Since all norms on a finite-dimensional vector space are equivalent, it suffices to prove the proposition under the assumption that the norm on V R is induced by such an inner product. In particular, the stabiliser of the norm in G is a maximal compact subgroup K and if p denotes the −1 eigenspace of the associated Cartan involution θ on Lie(G), then for every X ∈ p, dρ(X) is self-adjoint. Thus the conditions of [RS90, sec. 3] are satisfied.
Since ρ(G)v 0 is closed it contains a minimal vector, that is, a vector whose length is minimal among all elements of the orbit. (Indeed, a theorem of Richardson and Slodowy [RS90,Thm. 4.4] states that the two properties are, in fact, equivalent).
Replacing v 0 by another vector in its orbit changes the element g such that w = ρ(g)v 0 by a fixed element of G, so we may assume that v 0 itself is a minimal vector. Let . Note that H 0 is self-adjoint with respect to our chosen inner product on V R (see [RS90,Thm. 4.3], for example). If The elements g ∈ Ω satisfy a uniform bound for max( ρ(g) , ρ(g −1 ) ), proving the proposition in this case since |w| ≥ |v 0 | > 0.
From now on, assume that the orbit Hence there exists a constant c 14 > 0 (depending only on G, ρ and v 0 ) such that which is a positive constant.
Combining the above two inequalities, we deduce that there is a positive constant c 15 such that the following inequality holds for all g ∈ G: or in other words, Thus (3) (applied to g ) becomes Applying Lemma 4.2 to both ρ(g) and ρ(g −1 ) completes the proof of the proposition. (1) It depends on the value of λ − (v 0 ). We do not know a general method for calculating this value, although it seems to be feasible to calculate it in particular cases.
(2) The value c 14 depends on the speed of convergence of the limit in [Ebe14, Thm. 3.1], which is ineffective.

Quantitative reduction theory for representations.
We now prove Theorem 1.1. The proof follows that of [BHC62, Lemma 5.3], keeping track of quantitative information and some minor generalisations. For the sake of clarity, we have broken it down into a series of lemmas, each proved by a short calculation. We use the notation for Siegel sets from section 3.3. Then S is P(R)-conjugate to a maximal Q-split torus T in P. Since P(R) = Z G (S)(R).U(R), we can choose n in U(R) such that S = nTn −1 . We also adopt some notation from the proof of [BHC62, Lemma 5.3] (bearing in mind that we have reversed the order of multiplication in our Iwasawa decomposition relative to [BHC62]). By the Iwasawa and Langlands decompositions, the multiplication map Since all norms on the finite-dimensional vector space V R are equivalent, we may assume without loss of generality that the norm is chosen so that the spaces V χ are orthogonal to each other.
In the lemmas which follow, τ denotes an element of Aut ρ(G) (V R ). Constants labelled c n depend only on G, S, ρ, Λ and v 0 , and not on τ , x, v or w.
Since V ψ = 0 for only finitely many characters ψ ∈ X * (T), it is possible to choose a single constant c 17 > 0 which works for every ψ.
Since τ commutes with ρ(G(R)), it preserves the eigenspaces V χ and hence commutes with π χ . Therefore, either Lemma 4.5. There exists a constant c 18 such that, for all x ∈ S, we have Proof. Let χ ∈ X * (S). From the definitions of y x and z x , we can calculate Since V R is the orthogonal direct sum of the V χ , the lemma follows by squaring and summing over χ.
Let θ denote the Cartan involution of G whose set of real fixed points is K.
This is not a unique decomposition, but the definition of Siegel set guarantees that we can choose m x in a fixed compact subset of M(R) + . (Recall that, by definition, M commutes with S.) By definition, θ acts trivially on K and stabilises S(R). Since S is an R-split torus, the latter implies that θ(a) = a −1 for all a ∈ S(R). Hence Since m x and n x lie in compact sets independent of x, this proves the lemma.
We are now ready to prove a version of Theorem 1.1 in which the bound is expressed in terms of the operator norm of Then there exist constants c 23 , c 24 such that, for every τ ∈ Aut ρ(G) (V R ) and Then we get g as in Lemma 4.7.
By [BHC62,Prop. 13.5], there is a Cartan involution θ of GL(V R ) such that θ • ρ = ρ • θ. With respect to a suitable basis of V R , θ is given by g → (g −1 ) t . The norms X and X t on the finite dimensional vector space End(V R ) are equivalent, so there exists a constant c 25 such that ρ(θ(g)) ≤ c 25 ρ(g −1 ) .
We have g = a −1 x k x h where h ∈ H 0 = Stab G(R),ρ (v 0 ). Hence by Lemma 4.8, we get which is polynomially bounded with respect to τ , as required.
To conclude, we show that it is possible to choose τ such that τ (v 0 ) = v and τ is bounded in terms of |v|. Theorem 1.1 follows by combining Proposition 4.9 with Lemma 4.10, applied to E = End ρ(G) (V R ).
Lemma 4.10. Let V R be a real vector space and let E be a semisimple R-subalgebra Then there exists a constant c 30 such that, for every v ∈ E × v 0 , there exists e ∈ E × satisfying v = ev 0 and e ≤ c 30 |v|.
Note that E × is the group of R-points of a reductive R-algebraic group. However, this lemma does not follow from Proposition 4.1 because the orbit E × v 0 is not closed.
Because all norms on a finite-dimensional real vector space are equivalent, we may assume without loss of generality that the norm of each element of V R is the maximum of the norms of its projections to the V i . Then the operator norm satisfies e = max{ e i : 1 ≤ i ≤ m}. Hence it suffices to prove the lemma for each pair (E i , V i ). In other words, we may assume that E is a simple R-algebra.
Then E = M n (D) for some positive integer n, where D is a division algebra isomorphic to R, C or H. There is a unique simple E-module, namely D n , such that we can identify V R (as a left E-module) with (D n ) r for some positive integer r.
Again, since all norms on a finite-dimensional real vector space are equivalent, we may assume that the norm on V R ∼ = D nr is induced by a norm on D by letting the norm of an element of D nr be the maximum of the norms of its coordinates.
Via this identification, write v 0 = (x 1 , . . . , x r ), v = (y 1 , . . . , y r ) where x 1 , . . . , x r , y 1 , . . . , y r ∈ D n . Reordering x 1 , . . . , x r (and the corresponding y 1 , . . . , y r ), we may assume that {x 1 , . . . , x s } forms a maximal right D-linearly independent subset of {x 1 , . . . , x r } for a suitable positive integer s ≤ r. Note that s ≤ n. Then there exist a ij ∈ D (1 ≤ i ≤ s < j ≤ r) such that By hypothesis, there exists e ∈ E × such that v = e v 0 or in other words y i = e x i for all i. Consequently Since the set {x 1 , . . . , x s } is right D-linearly independent, it can be extended to form a right D-basis of D n . Let h ∈ M n (D) denote the matrix formed using such a basis as its columns. Then h is invertible and hb i = x i for 1 ≤ i ≤ s, where {b 1 , . . . , b n } denotes the standard D-basis of D n . Note that the choices made in constructing h can be made depending only on v 0 , not on v.
Note that, since e ∈ E × = GL n (D), the set {y 1 , . . . , y s } is also right D-linearly independent. Hence, it can be extended to a right D-basis of D n and we can assume, by scaling if necessary, that the norms of the additional vectors are at most |y 1 | + · · · + |y s |.
Let f ∈ M n (D) be the invertible matrix that has this basis as its columns, the first s equal to the y 1 , . . . , y s . Then Using (5) and (6), we deduce that also In other words, e = f h −1 ∈ E × satisfies ev 0 = v.
Since h is independent of v, the proof is complete.
Thanks to [BHC62,Cor. 6.3], we may enlarge the lattice Λ ⊂ Λ Q so that it is ρ(Γ)-stable. For each c ∈ C, c −1 Λ is a lattice in Λ Q . Hence we can choose a lattice Λ ⊂ Λ Q such that c −1 Λ ⊂ Λ for all c ∈ C.
By Theorem 1.1, every w ∈ ρ(S)v u ∩ Λ has length polynomially bounded with respect to |v u |. In particular, for each c ∈ C, the set c,1 , . . . , b c,mc }, which is a finite subset of Γ.
By Theorem 1.1, we have for all c ∈ C and i ≤ m c . Since c comes from a fixed finite set, we deduce that . It remains to show that F Hu is a fundamental set for Γ u in H u (R).
Let h ∈ H u (R) ⊂ G(R). By hypothesis, CS is a fundamental set for Γ in G(R) so we can write hu = γcs where γ ∈ Γ, c ∈ C and s ∈ S. Since h ∈ H u (R) = Stab G(R),ρ (ρ(u)v 0 ), we obtain Thus the Γ u -translates of F Hu cover H u (R). The fact that there are only finitely many γ ∈ Γ u for which γF Hu ∩ F Hu = ∅ follows from the Siegel property for S (and indeed this implies that F Hu also satisfies the Siegel property). Thus F Hu is a fundamental set for Γ u in H u (R).

Quantitative reduction theory for quaternion algebras
In order to apply Theorem 1.2, it is necessary to choose a representation ρ and a vector v 0 having the properties described in the theorem. In this section, we will explain how to construct a suitable representation for our application to unlikely intersections with E 2 and quaternionic curves. This illustrates a method for constructing representations which will be useful for applying Theorem 1.2 to other problems of unlikely intersections in the future, while avoiding many technical complications which occur in more general situations.
Borel and Harish-Chandra's reduction theory considered only a fixed reductive subgroup H 0 ⊂ G (and not its conjugates uH 0 u −1 ). As such, [BHC62, Thm. 3.8] constructs a representation satisfying the properties (i) and (ii) of Theorem 1.2 but does not construct the vectors v u . Another construction of representations satisfying (i) is given by [Del82,Prop. 3.1] (based on [Che58, Exp. 10, Prop. 5]), and Deligne's construction can easily be modified to yield the vectors v u .
However, it is not enough to know just that the vectors v u exist. Theorem 1.2 gives bounds in terms of |v u | so, in order to apply these, we need to control the length |v u | in terms of some more intrinsic quantity attached to the subgroup uH 0 u −1 . For example, in our application the subgroups uH 0 u −1 will be associated with quaternion algebras and we will bound |v u | in terms of the discriminants of (orders in) these algebras.
5.1. The set-up: quaternionic subgroups of GSp 4 . Let G = GSp 4 , the algebraic group whose Q-points are the invertible linear transformations of Q 4 which multiply the standard symplectic form by a scalar. For the standard symplectic form, we use ψ : Q 4 × Q 4 → Q represented by the matrix The subgroup H 0 is equal to GL 2 , embedded block diagonally in GSp 4 : If a G(R)-conjugate uH 0,R u −1 is defined over Q, then its Q-points form the multiplicative group of a (perhaps split) indefinite quaternion algebra over Q. We shall prove the following proposition.
where R u denotes the order End Hu (L) of the quaternion algebra End Hu (L Q ).
The condition v u ∈ Aut ρ(G) (Λ R )v 0 in Proposition 5.1(iii)(a) ensures that the element ρ(u)v u ∈ Λ satisfies Stab G,ρ (ρ(u)v u ) = H u . Proposition 5.1(iii)(a) is the bound we need to apply Theorem 1.2. Proposition 5.1(iii)(b) is not required for our application to unlikely intersections, but may be useful in its own right -we can replace u by γuh if we replace H u by γH u γ −1 , a subgroup of G which gives rise to the same special subvariety of A 2 as H u .
The proof of Proposition 5.1 will proceed in three steps: first we construct ρ and v 0 satisfying property (i), then we show that the representation we have constructed possesses property (ii) and then (iii). The proofs of properties (ii) and (iii) are independent of each other, while (iii)(b) is a by-product of the proof of (iii)(a).

Construction of representation of GSp 4 .
We construct the representation ρ of GSp 4 and the vector v 0 , and define notation which we shall use throughout the rest of the section.
Let W = M 4 (Q), considered as a Q-vector space. Define two representations σ L , σ R : G = GSp 4 → GL(W ) by multiplication on the left and on the right: (The inverse in the formula for σ R is so that σ R is a left representation of G.) Let E 0 = M 2 (Q) and define ι 0 : E 0 → M 4 (Q) by Similarly, Stab G,σ R (Z) = H 0 but we shall not need this latter fact.
Let V = 4 W and let ρ L , ρ R : G → GL(V ) be the representations Then 4 Z is a 1-dimensional Q-linear subspace of V , with The action of GL 2 (Q) on Z via σ L •ι 0 is the restriction of the left regular representation of M 2 (Q). Hence the action of GL 2 (Q) on 4 Z via 4 σ L • ι 0 is multiplication by (det GL 2 ) 2 = det GL 4 • ι 0 . Therefore the action of H 0 on 4 Z via ρ L is trivial, so each non-zero vector in 4 Z has stabiliser equal to H 0 . Let Λ = 4 M 4 (Z) ⊂ V . For later use, we choose a specific element v 0 ∈ ( 4 Z) ∩ Λ. Let e 1 , e 2 , e 3 , e 4 denote the following Z-basis for M 2 (Z): Then ι 0 (e 1 ), ι 0 (e 2 ), ι 0 (e 3 ), ι 0 (e 4 ) form a Z-basis for Z ∩ M 4 (Z), so is a generator of the rank-1 Z-module ( 4 Z) ∩ Λ. Then ρ L and v 0 satisfy Proposition 5.1(i).
Given u ∈ G(R), we can easily find a vector v u ∈ Aut ρ L (G) (Λ R )v 0 such that ρ L (u)v u ∈ Λ (that is, the first part of Proposition 5.1(iii)(a)). The vector If the subgroup uH 0,R u −1 ⊂ G R is defined over Q, then so is the linear subspace uZ R u −1 ⊂ W R . (This follows from the fact that Z is the Q-linear span of H 0 (Q).) Consequently ( 4 uZ R u −1 ) ∩ Λ is non-empty, so there exists d u ∈ R × such that If we choose W to be stable under left as well as right translations (denoting the representations by ρ L and ρ R respectively), then the same argument as in the special case above shows that the line R × ρ L (u)ρ R (u)v 0 is defined over Q whenever uH 0,R u −1 is defined over Q, and so this line contains non-zero rational vectors.
Comparing this general construction with our special case of G = GSp 4 , H 0 = ι 0 (GL 2 ), we note that in the special case I(H 0 ) is generated by linear functions on M 4 . Thus following Chevalley's method, we could choose W to be the linear dual of M 4 (Q). In fact, we chose W to be M 4 (Q) itself, and Z to be the linear subspace of M 4 (Q) which is annihilated by I(H 0 ) ∩ M 4 (Q) ∨ . The choice of M 4 (Q) instead of its dual is a matter of convenience.
The representations constructed by Chevalley's method do not necessarily contain a closed orbit ρ L (G(R))v 0 , although this can often be achieved by carefully choosing W ⊂ Q[G] and perhaps making some minor modifications using linear algebra constructions. On the other hand, finding a suitable v u with bounded length requires much more detailed arithmetic information about the groups H u .

Closed orbit.
We now show that Proposition 5.1(ii) holds, that is, the orbit ρ L (G(R))v 0 is closed in V R . By [BHC62, Prop. 2.3], it suffices to prove that ρ L (G(C))v 0 is closed in V C .
We use the following definitions. If V C is a vector space over C, we say that a subset of V C is homogeneous if it is non-empty and stable under multiplication by scalars. In other words, a subset of V C is homogeneous if and only if it is the cone over some subset of P(V C ). For a non-negative integer d, a set-theoretic function between vector spaces f : Homogeneous sets and homogeneous maps are useful because of the following lemma, which is equivalent to the fact that a morphism of projective algebraic varieties maps Zariski closed sets to Zariski closed sets. Let U = C 4 . We define a sequence (u 1 , u 2 , u 3 , u 4 ) ∈ U 4 to be quasi-symplectic if it satisfies the conditions If (u 1 , u 2 , u 3 , u 4 ) is a quasi-symplectic sequence then either: (1) ψ(u 1 , u 2 ) = ψ(u 3 , u 4 ) = 0, in which case (u 1 , u 2 , u 3 , u 4 ) is a non-zero scalar multiple of a symplectic basis for (U, ψ); or (2) ψ(u 1 , u 2 ) = ψ(u 3 , u 4 ) = 0, in which case u 1 , u 2 , u 3 , u 4 are contained in an isotropic subspace of U for ψ; in particular they are linearly dependent. Let Q = {g ∈ M 4 (C) : the columns of g form a quasi-symplectic sequence} = {g ∈ M 4 (C) : ∃ν(g) ∈ C such that ψ(gx, gy) = ν(g)ψ(x, y)} The set Q is closed under multiplication, but not all of its elements are invertible so it is not a group. We have Q ∩ GL 4 (C) = GSp 4 (C). Let σ L denote the action of M 4 (C) on W C = End C (U ) by left multiplication (this extends our earlier definition of σ L as a representation of G = GSp 4 ). Let ρ L denote the induced action 4 σ L of M 4 (C) on V C = 4 W C (this is a representation of M 4 (C) as a multiplicative monoid but not as a C-algebra). Note that ρ L = ρ L ⊗ det −1 , but ρ L (g) is only defined for g ∈ GL 4 (C) while ρ L (g) is defined for all g ∈ M 4 (C). In particular ρ L is defined on Q.
In order to prove that ρ L (G(C))v 0 is closed, we find a homogeneous Zariski closed set X ⊂ 2 U 2 such that ρ L (Q)v 0 is the image of X under a homogeneous morphism of varieties ζ. Hence ρ L (Q)v 0 is Zariski closed. We conclude by showing that ρ L (G(C))v 0 is the intersection of ρ L (Q)v 0 with a hyperplane in V C .
Proof. Let 2 dec U 2 denote the set of decomposable vectors in 2 U 2 : 2 dec U 2 = {x ∧ y : x, y ∈ U 2 }. This is the cone over the Grassmannian Gr(2, U 2 ) (embedded in P( 2 U 2 ) via the Plücker embedding), so it is a homogeneous Zariski closed subset of 2 U 2 .
Define a quadratic form q : where x, y denotes the linear subspace of U 2 spanned by x and y. Then X is the cone over the orthogonal Grassmannian OGr(2, U 2 , q) ⊂ P( 2 U 2 ), so it is a homogeneous Zariski closed subset of 2 dec U 2 . For an element (u 1 , u 3 ) ∧ (u 2 , u 4 ) ∈ 2 dec U 2 , we have: It follows that we can define linear maps Ψ ij : X → C for i = 1, 3 and j = 2, 4 by Ψ ij ((u 1 , u 3 ) ∧ (u 2 , u 4 )) = ψ(u i , u j ). (When i = 1 and j = 2 or i = 3 and j = 4, the map Ψ ij is defined on the whole of 2 U 2 . When i = 1 and j = 4 or i = 3 and j = 2, we require (9).) Using (9), we conclude that the set X from the statement of the lemma is equal to Thus X is homogeneous and Zariski closed.
Lemma 5.5. There exists a linear map s : V C → C such that Proof. We continue to use the functions β 1 and β 2 from the proof of Lemma 5.4.
Let δ : W C → U be the linear map which sends the matrix with columns C 1 C 2 C 3 C 4 to the sum C 1 + C 4 . This map is equivariant with respect to multiplication by M 4 (C) on the left and the compositions δ • β 1 , δ • β 2 : U 2 → U are the projections onto the two copies of U .
Taking the fourth exterior power, δ induces a linear map By (10) and the descriptions of δ • β 1 , δ • β 2 , we have Since z 1 , z 2 , z 3 , z 4 form a basis for U , s(v 0 ) = 0. Hence we can choose the isomorphism 4 U ∼ = C so that s(v 0 ) = 1. The linear map δ is M 4 (C)-equivariant with respect to left multiplication. Consequently s is M 4 (C)-equivariant with respect to ρ L on V C and multiplication by the determinant on 4 U . Twisting by det −1 , we deduce that s is GL 4 (C)-equivariant with respect to ρ L on V C and the trivial action on 4 U . Thus s(ρ L (g)v 0 ) = s(v 0 ) = 1 for all g ∈ G(C).

Discriminants and orders.
Before the proof of Proposition 5.1(iii), we prove some results on discriminants, involutions and orders in semisimple algebras.
Let D be a semisimple Q-algebra. We define a symmetric Q-bilinear form φ : D × D → Q by φ(x, y) = Tr D/Q (xy), where Tr D/Q is the (non-reduced) trace of the regular representation of D. This form is non-degenerate by [KMRT98, Ch. I, Prop. (1.8)] (note that [KMRT98] refers to the reduced trace, which is a non-zero multiple of Tr D/Q on each simple factor). For any order R in D, we define the discriminant of R, denoted disc(R), to be disc(R, φ).
For any involution † of D, we define another Q-bilinear form φ † : Lemma 5.6. Let R be an order in a semisimple Q-algebra D and let † be any involution of D. Then disc(R, φ † ) = ± disc(R).
Proof. This is based on the proof of [GR14, Prop. 2.9]. Let {d 1 , . . . , d n } be a Z-basis for R and let A ∈ GL n (Q) be the matrix such Since † is Q-linear and an involution, we have A 2 = I, hence, The following lemma is restricted to quaternion algebras because its proof makes use of the fact that the reduced norm is a quadratic form on a quaternion algebra.
Lemma 5.7. There exists an absolute constant c 39 with the following property.
Let R be an order in a quaternion algebra D over Q. Let L be a left R-module such that L Q is isomorphic to the left regular representation of D.
Then there exists a left R-ideal I ⊂ R such that I is isomorphic to L as a left R-module and Proof. This is a generalisation to quaternion algebras of Minkowski's bound for ideal classes in a number field, and the proof is similar.
Choose an isomorphism of D-modules η 1 : D → L Q and let I 1 = η −1 1 (L). Since D is a quaternion algebra, it possesses a canonical involution * defined by where we write Trd D/Q for the reduced trace. The canonical involution has the property that By [Bla48, Lemma 1], there is an element s ∈ I 1 satisfying 0 < |φ * (s, s)| ≤ c 40 |disc(I 1 , φ * )| 1/4 (the exponent is 1/ rk Z (I 1 )). Hence by (13), there is a constant c 41 such that Since Nm D/Q (s) = 0, s is invertible in D. Let I 2 = I 1 s −1 ⊂ D. Then I 2 is a left R-module isomorphic to L. Since 1 = ss −1 ∈ I 2 , we have R ⊂ I 2 .
Using (14), we can calculate . Consequently, using Lemma 5.6, In the following lemma, we note that S = End R (L) is an order in End D (L Q ) ∼ = End D (D) ∼ = D op . Thus S is an order in a quaternion algebra, so disc(S) is defined. This is a Q-algebra isomorphism D op → End D (D) [Sch85, Ch. 8, Lemma 1.10].
If r ∈ R and x ∈ I, then we have µ(r)x = xr ∈ R since I ⊂ R and R is closed under multiplication. Hence Let φ S denote the trace form on End D (D) = S Q . Since µ is an algebra isomorphism, it pulls back φ S to the trace form on D op , which is equal to the trace form on D. Hence disc(µ(R), φ S ) = disc(R).
Since 5.5. Choice of v u , γ and h. Throughout this section, c n will denote absolute constants (in particular, independent of u).
We will now prove Proposition 5.1(iii). Thus, we are given u ∈ G(R) = GSp 4 (R) such that the algebraic group H u = uH 0,R u −1 ⊂ G R is defined over Q. Multiplying u by a scalar does not change H u , so we may assume that u multiplies the symplectic form ψ by ±1; consequently det(u) = 1.
Since H u is defined over Q, the R-vector space uι 0 (E 0,R )u −1 is also defined over Q.
Let D 0 and D denote the centralisers in M 4 (Q) of ι 0 (E 0 ) and E, respectively. Then D R = uD 0,R u −1 , so there is an isomorphism of R-algebras α : D R → D 0,R defined by α(d) = u −1 du.

Note that
Hence the "transpose" involution of M 4 (Q) restricts to an involution of D 0 , which we denote by t. This involution is positive, which is to say that the corresponding trace form (see (12)) is positive definite.
, which is the order in D consisting of those elements preserving L.
Lemma 5.9. The quadratic form φ † takes integer values on R.
Proof. Thanks to our choice of symplectic form ψ on Q 4 , we have ψ(d 0 x, y) = ψ(x, d t 0 y) for all x, y ∈ R 4 and d 0 ∈ D 0,R . Using this, the fact that u multiplies ψ by ±1, and the definition of †, we can calculate, for d ∈ D R , Since ψ is a perfect pairing on L, this implies that † maps R into R. It follows that φ † (x, y) = Tr D/Q (xy † ) ∈ Z for all x, y ∈ R.
Observe that L Q is isomorphic to the left regular representation of D 0 . Hence, α induces an isomorphism between L R and the left regular representation of D R and it follows easily that this isomorphism can be scaled to produce an isomorphism between L Q and the left regular representation of D. Therefore, by Lemma 5.7, there is a left R-ideal I ⊂ R which is isomorphic to L as a left R-module, such that [R : I] ≤ c 39 |disc(R)| 3/2 . Choose a left R-module isomorphism η : I → L.
Lemma 5.10. There exists h ∈ H 0 (R) such that ηα −1 η −1 0 = uh in Aut(R 4 ). Proof. We can view D R as a left D 0,R -module with the action given by Now α −1 η −1 0 : R 4 → D R is an isomorphism of left D 0,R -modules with respect to the natural action on R 4 and the action (15) on D R .
Since η : D R → R 4 is an isomorphism of D R -modules, it is also an isomorphism of D 0,R -modules with respect to the action (15) on D R and the natural action conjugated by u on R 4 . (We use here the fact that α −1 is conjugation by u.) Finally u −1 : R 4 → R 4 is an isomorphism of D 0,R -modules with respect to the natural action conjugated by u on the domain and the natural action on the target.
Composing these, we deduce that u −1 ηα −1 η −1 0 is an automorphism of R 4 with its natural action of D 0,R . In other words, u −1 ηα −1 η −1 0 lies in the centraliser of D 0,R in M 4 (R). By the double centraliser theorem, this centraliser is equal to ι 0 (E 0,R ) and so its group of invertible elements is equal to ι 0 (GL 2 (R)) = H 0 (R).
Lemma 5.11. The absolute value of det(h) is uniformly bounded.
By Lemma 5.9, φ † takes integer values on R and hence on I. So disc(φ † , I) is a positive integer. So det(h) 2 ≤ disc(φ L , L), which is a constant. Finally, by Lemmas 5.6 and 5.7, we have Meanwhile, |det(γ )| = det(u) = 1 so |det(A)| = |det(h)|. By Lemma 5.11 we deduce that |det(A −1 )| is bounded below by a positive constant. Hence by Cramer's rule, the entries of the matrix A are also polynomially bounded in terms of |disc(R)|.
The following lemma establishes Proposition 5.1(iii)(b). Note that it is not required for the proof of Proposition 5.1(iii)(a): the subsequent arguments proving Proposition 5.1(iii)(a) do not use the fact that γ ∈ Sp 4 (Z), so they would still work with γ instead of γ.
Let γ ∈ GL 4 (Z) be the matrix such that i = γs i for each i. Since {s 1 , s 2 , s 3 , s 4 } is a symplectic basis, we have γ ∈ Γ. We have just shown that the coordinates of γ s i = γ γ −1 i are polynomially bounded. In other words, the entries of the matrix γ γ −1 are polynomially bounded in terms of |disc(R)|.
Multiplying (γ uh) −1 by γ γ −1 and applying Lemma 5.13, we deduce that the entries of (γuh) −1 are polynomially bounded in terms of |disc(R)|. Thanks to Lemma 5.11, |det((γuh) −1 )| is bounded below by a positive constant, so it follows that the entries of γuh are also polynomially bounded in terms of |disc(R)|.
We shall use this v u to prove Proposition 5.1(iii)(a). Note first that d u ρ R (γu) ∈ Aut ρ L (Λ R ).
Proof. Since ρ R (γ) stabilises Λ, in order to show that ρ L (u)v u ∈ Λ, it suffices to show that Since v 0 is a generator for the rank-1 Z-module ( 4 Z) ∩ Λ = 4 S 0 , it follows that ρ R (u)ρ L (u)v 0 is a generator for 4 uS 0 u −1 ⊂ 4 E R .
Consider a matrix B ∈ GL(E R ) such that B(uS 0 u −1 ) = S. Conjugation by u maps the trace form φ E 0 on ι 0 (E 0,R ) to the trace form φ E on E R , so we have Consequently det(B) 2 = disc(S)/ disc(S 0 ). In other words, det(B) = ±d u .
Proof. The action of H 0 on the line 4 Z via ρ R is trivial, for the same reasons as the action via ρ L is trivial. Therefore By Lemma 5.8, d u is polynomially bounded in terms of |disc(R)|. By Lemma 5.14, the entries of (γuh) −1 are polynomially bounded in terms of |disc(R)|. We conclude that the coordinates, and hence the length, of v u are polynomially bounded in terms of |disc(R)|.
Lemmas 5.15 and 5.16 complete the proof of Proposition 5.1(iii)(a).

Unlikely intersections in A 2
In this section, we prove Theorems 1.3 and 1.4. We first need some preliminary material.
6.1. Realising A 2 as a Shimura variety. Recall that A 2 denotes the (coarse) moduli space of principally polarised abelian surfaces. To realise A 2 as a Shimura variety, we let (G, X) denote the Shimura datum for which G = GSp 4 and X is isomorphic to H 2 ∪ H − 2 , where H 2 and H − 2 are respectively the Siegel upper and lower half-spaces of genus 2. (Recall that X is a G(R)-conjugacy class of morphisms S → G R , where S = Res C/R G m,C . We will henceforth identify X with H 2 ∪ H − 2 .) We let K = G(Ẑ), whereẐ = p Z p , the product ranging over all finite primes p. Then K is a compact open subgroup of G(A f ), where A f denotes the finite rational adeles, and A 2 is equal to the Shimura variety whose complex points are given by Sh K (G, X) = G(Q)\X × G(A f )/K. As is easily seen, this is isomorphic to the quotient Sp 4 (Z)\H 2 . 6.2. Quaternionic curves and E 2 curves. Quaternionic curves and E 2 curves are the images in A 2 of Shimura varieties of PEL type, via maps induced by morphisms of Shimura data. We recall the construction of Shimura varieties of PEL type attached to quaternion algebras over Q, following [Mil05, sec. 8].
Let B denote a quaternion algebra over Q such that B ⊗ Q R is isomorphic to M 2 (R) and let † be a positive involution of B. (Positive involutions exist for any such B, as explained in [Mum74,.) As explained in [Mum74,p. 196 By [Mil05,Prop. 8.14], there is a unique Shimura datum (H B , X B ) such that each h ∈ X B defines a complex structure on B ⊗ Q R for which the symmetric form ψ(x, h(i)y) is positive or negative definite. As a Hermitian symmetric domain, X B is isomorphic to the union of the upper and lower half-planes in C.
Choosing a symplectic Q-basis for (B, ψ α ), the tautological action of H B on B gives rise to an injective group homomorphism H B → G. Thanks to the properties of X B given to us by [Mil05,Prop. 8.14], this induces an embedding of Shimura of algebraic varieties. The irreducible components of the images of such morphisms are, by definition, special curves in A 2 . If B is isomorphic (over Q) to M 2 (Q), we obtain E 2 curves, and otherwise we obtain quaternionic curves. Any such curve parametrises abelian surfaces with multiplication by an order in B.
The Shimura data (G, X) and (H B , X B ) all have reflex field Q. Therefore, Sh K B (H B , X B ), Sh K (G, X) and Sh K B (H B , X B ) → Sh K (G, X) are all defined over Q, but Sh K B (H B , X B ) often has geometrically irreducible components which are not defined over Q. Hence, the action of Aut(C/Q) on A 2 (C) preserves the image of Sh K B (H B , X B ) → Sh K (G, X) but permutes its irreducible components and so acts on the set of quaternionic curves and on the set of E 2 curves in A 2 . From the theory of complex multiplication of abelian varieties, we know that Aut(C/Q) acts on the set of special points in A 2 . Consequently, Aut(C/Q) acts on where S denotes the set of quaternionic curves in A 2 and Z sp denotes the set of the special points contained in Z. Similarly, Aut(C/Q) acts on Σ E 2 .
Another way to obtain these families of special subvarieties is as follows. Let B 0 = M 2 (Q). Let B 0 act on Q 4 via the left regular representation, with respect to the basis given by (8) (which is a symplectic basis with respect to the form ψ α where α = 0 −1 1 0 ). Let H 0 ⊂ G be the centraliser of this action of B 0 in G. Then H 0 is equal to the image of GL 2 embedded block diagonally, as in (7). Let Then (H 0 , X ± 0 ) is the unique Shimura subdatum of (G, X) with underlying group H 0 , and X 0 is the only connected component of X ± 0 contained in H 2 . We obtain a morphism of Shimura varieties A 1 → A 2 (where A 1 denotes the moduli space of elliptic curves), which, in terms of moduli, sends an elliptic curve E with its principal polarisation λ to the principally polarised abelian surface (E × E, λ × λ).
For any point x 0 ∈ X 0 , we have X 0 = H der 0 (R)x 0 (recall that H der 0 (R) = SL 2 (R) is connected) and its image in A 2 is an E 2 curve. For any g ∈ G(R) such that H = gH 0,R g −1 is defined over Q the image of gX 0 in A 2 is a special curve, and H is isomorphic (as a Q-group) to H B for some quaternion algebra B as above. If H is isomorphic to GL 2,Q , then we obtain an E 2 curve, and, otherwise, we obtain a quaternionic curve.
Lemma 6.1. Every quaternionic or E 2 curve in A 2 is the image of gX 0 for some g ∈ G(R) such that gH 0,R g −1 is defined over Q.
Proof. Let Z be a quaternionic or E 2 curve in A 2 . Let B be the generic endomorphism algebra of the abelian surfaces parametrised by Z, and let † be the Rosati involution of B. Choose an analytic irreducible component Y of the preimage of Z in X.
The inclusion G → GL 4 induces a variation V of Q-Hodge structures on X with trivial underlying local system X × Q 4 . The restriction V |Y has endomorphism algebra B and its generic Mumford-Tate group H ⊂ G is the centraliser of B in G. Thus H is the image of one of the homomorphisms H B → G defined above.
The choice of basis (8) induces an isomorphism of Q-vector spaces Q 4 → B 0 . Choose an isomorphism of R-algebras with involutions (B 0 ⊗ Q R, t) → (B ⊗ Q R, †). The action of B on V |Y gives rise to a B-module structure on Q 4 , and this is isomorphic to the left regular representation of B on itself. Thus we get an isomorphism of B-modules B → Q 4 . Composing these isomorphisms (after extending scalars to R), we get an isomorphism of R-vector spaces or in other words an element g ∈ GL 4 (R). Via the isomorphism R 4 → B 0 ⊗ Q R, the standard symplectic form on R 4 satisfies ψ(bx, y) = ψ(x, b t y) for all b ∈ B 0 , and the isomorphism B ⊗ Q R → R 4 behaves similarly with respect to (B, †). Since the spaces of symplectic forms satisfying these conditions are one-dimensional, the composed isomorphism maps the standard symplectic form to a multiple of itself; in other words, g ∈ G(R).
Comparing the actions of B ⊗ Q R and B 0 ⊗ Q R, we see that H R = gH 0,R g −1 . It follows that g −1 Y is a connected component of a Shimura subdatum of (G, X) with underlying group H 0 . The only such Shimura subdatum is (H 0 , X ± 0 ), so g −1 Y is a connected component of X ± 0 . If g −1 Y = X 0 , replace g by g diag (1, −1, 1, −1). Since diag(1, −1, 1, −1) ∈ H 0 (R) and it swaps the two connected components of X ± 0 , after this replacement we will still have H R = gH 0,R g −1 but now g −1 Y = X 0 .
Thus we get Y = gX 0 and Z is the image of Y in A 2 .
6.3. Complexity. As in [DR18], we will need to define a notion of complexity.
That is, to each E 2 or quaternionic curve Z in A 2 , we attach a natural number ∆(Z), which we refer to as the complexity of Z. The complexity is defined in terms of the generic endomorphism algebra of abelian surfaces parametrised by Z.
Let g ∈ G(R) be such that H = gH 0,R g −1 is defined over Q. Then the image Z of gX 0 in A 2 is an E 2 or quaternionic curve and, by Lemma 6.1 every E 2 or quaternionic curve is obtained this way. We define the complexity ∆(Z) of Z to be |disc(R)|, where R denotes the ring End H (Z 4 ) of Z-linear endomorphisms of Z 4 ⊂ Q 4 commuting with H(Q) ⊂ G(Q) ⊂ M 4 (Q). Note that this ring R is the generic endomorphism ring of the abelian surfaces parameterised by Z. Indeed, for every non-special point of Z, the associated abelian surface (over C) has endomorphism ring isomorphic to R.
We are now in a position to state the Galois orbits conjecture which appears in Theorem 1.3. Conjecture 6.2. Let Σ denote Σ Quat or Σ E 2 and let C ⊂ A 2 denote an irreducible Hodge generic algebraic curve. Let L be a finitely generated subfield of C over which C is defined.
There exist positive constants c 47 and c 48 such that, for any point s ∈ C ∩ Σ, if we let Z denote the (unique) special curve containing s, then # Aut(C/L) · s ≥ c 47 ∆(Z) c 48 .
6.4. The fixed data. We write Γ for the subgroup Sp 4 (Z) ⊂ G(Q). We let π : H 2 → A 2 denote the (transcendental) uniformisation map. We choose a Siegel set S ⊂ G(R) + (associated with the standard Siegel triple) such that, for some finite set C ⊂ G(Q), F G = CS is a fundamental set for Γ in G(R) + . We write F for F G x 0 , where x 0 ∈ H 2 is the point whose stabiliser in G(R) + is the maximal compact subgroup appearing in the definition of F G .
By Proposition 5.1, we can fix a finitely generated, free Z-module Λ, a representation ρ : G → GL(Λ Q ) such that Λ is stabilised by ρ(Γ), an element v 0 ∈ Λ, and positive constants c 49 and c 50 such that where R u denotes the order End Hu (Z 4 ) of the quaternion algebra End Hu (Q 4 ). By Theorem 1.2, we can then fix positive constants c 51 and c 52 with the following property: for every u ∈ G(R), if H u = uH 0,R u −1 is defined over Q, then there exists a fundamental set for Γ ∩ H u (R) in H u (R) of the form where B u ⊂ Γ is a finite set such that Choosing a basis, we obtain Λ = Z d and we may refer to the height H(v) of any v ∈ Λ (defined as the maximum of the absolute values of the coordinates). For any v ∈ Λ R , we write G(v) = Stab G R ,ρ (v).
Proof. Let z ∈ π −1 (P ) ∩ F and let Y denote the smallest pre-special subvariety of H 2 containing z. Then π(Y ) is an E 2 or quaternionic curve and so Y = gH der 0 (R)x 0 = gH der 0 (R)g −1 · gx 0 = H der (R) · gx 0 , where g ∈ G(R), H is a Q-subgroup of G isomorphic to H B for some quaternion algebra B, as above, and H R = gH 0,R g −1 . Since H is the Mumford-Tate group of Y (that is, the smallest Q-subgroup of G containing x(S) for all x ∈ Y ), we have z(S) ⊂ H(R). By Proposition 5.1, we obtain v ∈ Aut ρ(G) (Λ R )v 0 such that ρ(g)v ∈ Λ and |v| ≤ c 49 |disc(R)| c 50 , where R = End H (L). Note that By Theorem 1.2 (with u = g), we obtain a finite set B g ⊂ Γ such that is a fundamental set for Γ H = Γ ∩ H(R) in H(R), and |ρ(b −1 g)v| ≤ c 51 |v| c 52 for all b ∈ B g . In particular, since z ∈ H(R)gx 0 , we can write z = γbsx 0 for some γ ∈ Γ H , b ∈ B g , and s ∈ CS. Hence, where we use the fact that γ ∈ H(R) = G(ρ(g)v)(R). Finally, from the above, we have 49 |disc(R)| c 50 c 52 and Therefore, since ρ(b −1 g)v ∈ Λ, we conclude that z and ρ(b −1 g)v satisfy the conditions of the proposition. Proof. Let Z be an E 2 or quaternionic curve satisfying ∆(Z) ≤ b and let P ∈ Z be a Hodge generic point on Z. Therefore, P ∈ Σ Quat ∪ Σ E 2 and, applying Proposition 6.3 and the first paragraph of its proof, we obtain v ∈ Λ satisfying H(v) ≤ c 51 c c 52 49 b c 50 c 52 such that Z is the image in A 2 of an orbit of G(v) der (R). As in the proof of Lemma 6.1, there is only one Shimura subdatum of (G, X) associated with G(v) and so the result follows. 6.5. Proof of Theorem 1.3 for quaternionic curves. Let C = π −1 (C) ∩ Fa set definable in the o-minimal structure R an,exp (see [KUY16] for more details). Let L be a finitely generated field of definition for C. Let P ∈ C ∩ Σ Quat . Varying over σ ∈ Aut(C/L), we obtain points σ(P ) ∈ C ∩ Σ Quat and, for each σ, we let z σ ∈ F ∩ π −1 (σ(P )) and we let v σ ∈ Aut ρ(G) (Λ R )ρ(G(R))v 0 ∩ Λ be the elements afforded to us by Proposition 6.3. That is, z σ (S) ⊂ G(v σ ) and H(v σ ) ≤ c 51 c c 52 49 |disc(R σ )| c 50 c 52 , where R σ denotes the ring End G(vσ) (L) ⊂ M 4 (Z). As above, |disc(R σ )| = ∆(σ(Z)). Note that we also have z σ ∈ C.
Let B ⊂ Λ C denote the Zariski closure of β 1 ([0, t 1 ]) ⊂ Λ R . By definable choice, there exists a complex algebraic setB ⊂ Aut ρ (Λ C )ρ(G(C)) of dimension at most 1 whose image under the algebraic map g → g · v 0 is B. Using the superscript ∨ to denote the compact dual of a hermitian symmetric domain, we obtain a complex algebraic setB × X ∨ 0 of dimension at most 2. Note thatB · H ∨ 2 = H ∨ 2 . Hence, B · X ∨ 0 ⊂ H ∨ 2 is algebraic of dimension at most 2. On the other hand,C is an irreducible complex analytic curve having an uncountable intersection withB · X ∨ 0 (in particular, it includes β 2 ([0, t 1 ]) because g −1 t z t ∈ X 0 and z t ∈C for all t ∈ [0, t 1 ]). Therefore,C is contained inB · X ∨ 0 , hence, so is H ∨ 2 . However, dim H ∨ 2 = 3, and we arrive at a contradiction.
6.6. Proof of Theorem 1.3 for E 2 curves. The proof is the same as in the previous section, working with Σ E 2 instead of Σ Quat . 6.7. Proof of Theorem 1.4. If C is an algebraic curve over a number field and A → C is an abelian scheme of relative dimension 2, we say that s ∈ C(Q) is a quaternionic point if the endomorphism algebra of the fiber A s is a quaternion algebra over Q not isomorphic to M 2 (Q). We claim that it suffices to prove the following theorem.
Theorem 6.5. Let C be an irreducible algebraic curve and let A → C be a principally polarised non-isotrivial abelian scheme of relative dimension 2 such that End(A η ) = Z, where η denotes a geometric generic point of C. Suppose that C and A are defined over a number field L and that there exist a curve C , a semiabelian scheme A → C and an open immersion ι : C → C , all defined over Q, such that A ∼ = ι * A and there is a point s 0 ∈ C (Q) \ C(Q) for which the fibre A s 0 is a torus.
Then there exist positive constants c 55 and c 56 such that, for any quaternionic point s ∈ C, # Aut(C/L) · s ≥ c 55 |disc(End(A s ))| c 56 .
To see that Theorem 6.5 implies Theorem 1.4, consider C as in Theorem 1.4. Then C is defined over a number field L and, furthermore, we can construct a curveC , a finite surjective morphism q :C → C, s 0 ∈C , and a semiabelian scheme A →C as in [DO,Proposition 9.4]. We can find a finite extensionL/L such thatC , q :C → C, s 0 and A are all defined overL. The abelian scheme A |C →C and the point s 0 ∈C (Q) satisfy the conditions of Theorem 6.5 and so, for any quaternionic points ∈C, # Aut(C/L) ·s ≥ c 55 |disc(End(A s ))| c 56 .
If s ∈ C ∩ Σ Quat , then we can find a quaternionic points ∈C such that q(s) = s, and since q is finite, # Aut(C/L) · s ≥ c 57 # Aut(C/L) ·s.
Let Z denote the unique special curve in A 2 containing s. Since s is a Hodge generic point of Z, the endomorphism ring of the associated abelian surface A s is isomorphic to the generic endomorphism ring of Z and so ∆(Z) = |disc(End(A s ))|. Since also A s = A s , we can combine the above inequalities to obtain # Aut(C/L) · s ≥ c 55 c 57 ∆(Z) c 56 , that is, Conjecture 6.2. Therefore, it remains to prove Theorem 6.5.
Proof of Theorem 6.5. After a finite extension, we may assume that C , A → C , Let s ∈ C be a quaternionic point. The image of s under the map C → A 2 induced by A → C is in the intersection between the image of C and a quaternionic curve. We deduce that s ∈ C(Q). Now End(A s ) ⊗ Q is a non-split quaternion algebra, so cannot inject into M 2 (Q). Hence, A s is exceptional in the sense of [DO,Section 8]. Therefore, by [ In order to deduce a bound for disc(End(A s )), we use the following theorem of Masser and Wüstholz. As remarked in [MW94] immediately following the statement of this theorem, one can replace End k (A) by End C (A) (the endomorphism ring which appears in the statement of Theorem 6.5) because one can find a finite extension K/k of degree bounded only in terms of n such that End K (A) = End C (A). Furthermore, as stated near the bottom of [MW94,p. 650], the constant c 58 (n, d, δ) is polynomial in d and δ, for a polynomial which depends only on n. Using also Lemma 5.6 to see that |disc(End(A), φ † )| = |disc(End(A))|, we conclude that there are constants c 60 , c 61 depending only on n such that, for all A as in the theorem, we have |disc(End(A))| ≤ c 60 max(δ, d, h F (A)) c 61 .
In our case, we have always n = 2 and δ = 1. So applying (17) together with the fact that h F (A s ) is polynomially bounded in terms of [L(s) : L] completes the proof of Theorem 6.5.