Modular invariants for genus 3 hyperelliptic curves

TL;DR

This paper establishes a link between invariants of binary octics and Siegel modular forms of genus 3, enabling explicit computations of hyperelliptic curves with complex multiplication by analyzing modular functions' values.

Contribution

It proves an analogue of a theorem connecting binary octic invariants to genus 3 Siegel modular forms and demonstrates how to compute hyperelliptic curves with CM using these modular functions.

Findings

Modular functions on hyperelliptic locus have denominators with primes of bad reduction.

Explicit computations illustrate the theoretical results.

The work facilitates effective models of hyperelliptic curves with CM.

Abstract

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.