Modular invariants for genus 3 hyperelliptic curves

Sorina Ionica (1) , Pinar Kilicer (2) , Kristin Lauter (3) , Elisa Lorenzo García (4) , Maike Massierer (5) , Adelina Manzateanu (6) , Christelle Vincent (7)

1 MIS - Modélisation, Information et Systèmes - UR UPJV 4290
2 OFFIS - Carl Von Ossietzky Universität Oldenburg = Carl von Ossietzky University of Oldenburg
3 Cryptography group
4 IRMAR - Institut de Recherche Mathématique de Rennes
5 USW - University of South Wales
6 University of Bristol [Bristol]
7 University of Vermont [Burlington]

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.

Domaines: Théorie des nombres [math.NT]

Sorina Ionica, Pinar Kilicer, Kristin Lauter, Elisa Lorenzo García, Maike Massierer, et al.. Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory, 2019, 5 (1), pp.article n°9. ⟨10.1007/s40993-018-0146-6⟩. ⟨hal-01845011v2⟩