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Sorina Ionica, Pınar KılıÇer, Kristin Lauter, Elisa Lorenzo GarcíA, Adelina MÂnzĂŢeanu, Christelle Vincent, Determining the primes of bad reduction of cm curves of genus 3, The Quarterly Journal of Mathematics, Volume 75, Issue 1, March 2024, Pages 239–276, https://doi.org/10.1093/qmath/haae005
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Abstract
In this paper, we introduce a new problem called the Isogenous Embedding Problem (IEP) and relate the existence of solutions to this problem to the primes of bad reduction of complex multiplication curves of genus 3. More precisely, the absence of solutions to IEP implies potentially good reduction. We propose an algorithm for computing the solutions to the IEP and run the algorithm through different families of curves. Using this algorithm, we were able to prove the reduction type of some particular curves at certain primes that were open cases in R. Lercier, Q. Liu, E. Lorenzo García and C. Ritzenthaler, Reduction type of smooth plane quartics, Algebra Number Theory 15 no. 6 2021, 1429–1468.