Some explicit elliptic modular surfaces

TL;DR

This paper studies specific algebraic surfaces constructed from hyperelliptic curves, demonstrating they are elliptic modular surfaces that serve as universal families for certain elliptic curve moduli.

Contribution

It proves that Schreieder's surfaces are elliptic modular surfaces, establishing their role as universal families of elliptic curves in the moduli space.

Findings

Surfaces are smooth models of quotients of hyperelliptic curve products.

They are shown to be elliptic modular surfaces in the sense of Shioda.

These surfaces act as universal families for explicit elliptic curve moduli.

Abstract

We consider algebraic surfaces, recently constructed by Schreieder, that are smooth models of the quotient of the self-product of a complex hyperelliptic curve by a $(\mathbb{Z}/3^c\mathbb{Z})$-action. We show that these surfaces are elliptic modular surfaces in the sense of Shioda, meaning in particular that they are universal families of explicit moduli of elliptic curves.