Modularity in genus 2

3:00pm, Wednesday 23rd Sep 2026

Abstract

Given an elliptic curve E over the rational numbers, the modularity theorem shows that its L-function L (E, s) can equally well be realized as an L-function L (f, s) associated to a modular form of weight 2 for GL_{2, QQ} with rational coefficients. This association gives rise to a bijection between the set of isogeny classes of elliptic curves E over QQ of a given conductor N and the set of normalized newforms f with respect to Gamma_0 (N). This talk discusses how these results generalize (at least conjecturally) to curves of genus 2.

Given such a genus 2 curve X over a number field F, we start by exploring the information contained in the L-function L (J, s) of its Jacobian J = Jac (X). Things get considerably stranger, and in particular the above bijection may break down because of the presence of false elliptic curves, so that we have to state carefully where to look for a modular form f for GL_{2, F} associated to J, if indeed any such form exists at all. If the endomorphism ring of J is a quadratic number field M (so that J is of so-called GL_2-type), then a conjecture by Ribet claims the existence of a modular form f for GL_{2, F} with coefficient field M such that L (J, s) = L (f, s). In general, it turns out that there always exists a quadratic extension K of the base field F of X such that J is of GL_2-type over K. A consideration involving restrictions of scalars then shows that we have L (J, s) = L (f, s), where f is a modular form for GL_{2, K} associated to the base extension of J to K; moreover, the property that said base extension actually comes from a curve over K (namely X) is reflected in a peculiar property of the modular form f that we call Galois alignment.

We illustrate these constructions by means of concrete examples. This is joint work with Andrew Booker, Andrew Sutherland, John Voight, and Dan Yasaki.