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Modularity theorems for abelian surfaces

Toby Gee

TL;DR

This work addresses the Hasse–Weil conjecture for genus two curves and abelian surfaces, and the modularity/potential modularity of these objects, by combining a 2--3 switch with a comprehensive p-adic modular-forms framework. The approach unites Sen theory, Hodge–Tate period maps, completed cohomology, and localisation to the partial flag variety, together with p-adic Eichler–Shimura decompositions and Taylor–Wiles/Calegari–Geraghty modularity lifting to transfer residual modularity to full modularity for $A/\mathbf{Q}$ under explicit hypotheses. The main contributions include potential modularity results for abelian surfaces over totally real fields, modularity for a positive proportion of abelian surfaces over $\mathbf{Q}$ under residually large-image assumptions, and a robust framework connecting $p$-adic and classical Siegel modular forms to automorphic $L$-functions. The work lays groundwork for extending modularity to broader totally real settings and for pursuing Serre-type conjectures for $\mathrm{GSp}_4$, aiming at a complete modularity picture for abelian surfaces.

Abstract

This is a brief account of my results with George Boxer, Frank Calegari and Vincent Pilloni on the (potential) modularity of abelian surfaces.

Modularity theorems for abelian surfaces

TL;DR

This work addresses the Hasse–Weil conjecture for genus two curves and abelian surfaces, and the modularity/potential modularity of these objects, by combining a 2--3 switch with a comprehensive p-adic modular-forms framework. The approach unites Sen theory, Hodge–Tate period maps, completed cohomology, and localisation to the partial flag variety, together with p-adic Eichler–Shimura decompositions and Taylor–Wiles/Calegari–Geraghty modularity lifting to transfer residual modularity to full modularity for under explicit hypotheses. The main contributions include potential modularity results for abelian surfaces over totally real fields, modularity for a positive proportion of abelian surfaces over under residually large-image assumptions, and a robust framework connecting -adic and classical Siegel modular forms to automorphic -functions. The work lays groundwork for extending modularity to broader totally real settings and for pursuing Serre-type conjectures for , aiming at a complete modularity picture for abelian surfaces.

Abstract

This is a brief account of my results with George Boxer, Frank Calegari and Vincent Pilloni on the (potential) modularity of abelian surfaces.

Paper Structure

This paper contains 14 sections, 14 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The 2--3 switch
  3. Galois representations
  4. Sen theory
  5. Ordinary Galois representations
  6. p-adic modular forms and classicality
  7. Modular forms and the Hodge--Tate period map
  8. Completed cohomology
  9. Localisation to the partial flag variety
  10. p-adic Eichler--Shimura
  11. The ordinary part
  12. Sen equals Cousin
  13. Modularity lifting
  14. Future directions

Key Result

Theorem 1.4

Let $X$ be either a genus two curve or an abelian surface over a totally real field $F$. Then $X$ is potentially modular, and Conjecture conj:serre holds for $X$.

Figures (1)

  • Figure 4.1: (Shifted) $M$-dominant Weyl chambers of weights $(k_1 ,k_2 )$. The $G$-dominant Weyl chamber is labelled by ${}^3w$. The red hearts are at $\kappa_{{}^3w}=\kappa_{{}^2w}=(1,1)$ and $\kappa_{{}^1w}=\kappa_{{}^0w}=(2,2)$ for our $\lambda=(1,1)$, while the blue squares represent the $\kappa_w$ for a typical regular weight.

Theorems & Definitions (30)

  • Conjecture 1.1: Hasse--Weil Conjecture, cf. Serre1969-1970, in particular Conj. C9
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proof 1: Proof of Theorem \ref{['first']}, given these steps
  • Remark 2.2
  • Lemma 2.3
  • Proof 2
  • ...and 20 more