Higher modularity of elliptic curves over function fields

Logan, Adam; Weinstein, Jared

doi:10.2140/ant.2026.20.801

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Higher modularity of elliptic curves over function fields

Adam Logan and Jared Weinstein

Interlude: Masato Kuwata

Vol. 20 (2026), No. 4, 801–860

DOI: 10.2140/ant.2026.20.801

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 $ℚ$ . Our main theorem is that if $E ∕ 𝔽_{q} (t)$ is a nonisotrivial elliptic curve with tame fibers 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.

Keywords

number theory, elliptic curve, function field, modularity, K3 surface, Tate conjecture

Mathematical Subject Classification

Primary: 11G05, 14J28

Milestones

Received: 20 December 2022

Revised: 10 February 2025

Accepted: 2 April 2025

Published: 30 April 2026

Authors

Adam Logan
School of Mathematics and Statistics Carleton University Ottawa, ON Canada	Department of Mathematics and Statistics McGill University Montreal, QC Canada

Jared Weinstein
Department of Mathematics and Statistics Boston University Boston, MA United States

Masato Kuwata
Department of Mathematics, Faculty of Science and Engineering Chuo University Hachioji, Tokyo Japan

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Vol. 20, No. 4, 2026