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Abelian surfaces over totally real fields are potentially modular
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  • Open Access
  • Published: 29 November 2021

Abelian surfaces over totally real fields are potentially modular

  • George Boxer1,
  • Frank Calegari1,
  • Toby Gee2 &
  • …
  • Vincent Pilloni3 

Publications mathématiques de l'IHÉS volume 134, pages 153–501 (2021)Cite this article

  • 1197 Accesses

  • 5 Citations

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Abstract

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces \(A\) over \({\mathbf {Q}}\) with \(\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}\). We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

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  1. The University of Chicago, 5734 S University Ave, Chicago, IL, 60637, USA

    George Boxer & Frank Calegari

  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Toby Gee

  3. Unité de Mathématiques pures et appliquées, Ecole normale supérieure de Lyon, 46 allée d’Italie, 69 364, Lyon Cedex 07, France

    Vincent Pilloni

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G.B. was supported in part by NSF postdoctoral fellowship DMS-1503047. F.C. was supported in part by NSF Grants DMS-1404620, DMS-1701703, and DMS-2001097. T.G. was supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, ERC Starting Grant 306326, and a Royal Society Wolfson Research Merit Award. V.P was supported in part by the ANR-14-CE25-0002-01 Percolator, and the ERC-2018-COG-818856-HiCoShiVa.

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Boxer, G., Calegari, F., Gee, T. et al. Abelian surfaces over totally real fields are potentially modular. Publ.math.IHES 134, 153–501 (2021). https://doi.org/10.1007/s10240-021-00128-2

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  • Received: 25 December 2018

  • Revised: 13 September 2021

  • Accepted: 27 October 2021

  • Published: 29 November 2021

  • Issue Date: December 2021

  • DOI: https://doi.org/10.1007/s10240-021-00128-2

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