Higher modularity of elliptic curves over function fields

TL;DR

This paper introduces a new concept of higher modularity for elliptic curves over function fields, proving that certain nonisotrivial elliptic curves with degree 4 conductors are 2-modular, using properties of K3 surfaces.

Contribution

It defines higher modularity for elliptic curves over function fields and proves that elliptic curves with degree 4 conductors are 2-modular, linking to K3 surface properties.

Findings

Nonisotrivial elliptic curves over F_q(t) with degree 4 conductor are 2-modular.

A K3 surface admits a finite morphism to a Kummer surface iff its Picard lattice matches.

The proof leverages properties of K3 surfaces and their Picard lattices.

Abstract

We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as an elliptic surface. The (known) case $r=1$ is analogous to the notion of modularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if $E/\mathbf{F}_q(t)$ is a nonisotrivial elliptic curve whose conductor has degree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3 surfaces. Along the way we prove a result of independent interest: A K3 surface admits a finite morphism to a Kummer surface attached to a product of elliptic curves if and only if its Picard lattice is rationally isometric to the Picard lattice of such a Kummer surface.