Abelian surfaces over totally real fields are potentially modular

Boxer, George; Calegari, Frank; Gee, Toby; Pilloni, Vincent

doi:10.1007/s10240-021-00128-2

Article

Abelian surfaces over totally real fields are potentially modular

Boxer, George ; Calegari, Frank ; Gee, Toby ; Pilloni, Vincent

Publications Mathématiques de l'IHÉS, Tome 134 (2021), pp. 153-501

Résumé

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 $𝐐$ with ${End}_{𝐂} A = 𝐙$ . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

Reçu le : 2018-12-25
Accepté le : 2021-10-27
Première publication : 2021-11-29
Publié le : 2021-12-28

MR Zbl

DOI : 10.1007/s10240-021-00128-2

@article{PMIHES_2021__134__153_0,
     author = {Boxer, George and Calegari, Frank and Gee, Toby and Pilloni, Vincent},
     title = {Abelian surfaces over totally real fields are potentially modular},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {153--501},
     year = {2021},
     publisher = {Springer International Publishing},
     address = {Cham},
     volume = {134},
     doi = {10.1007/s10240-021-00128-2},
     mrnumber = {4349242},
     zbl = {1522.11045},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-021-00128-2/}
}

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Boxer, George; Calegari, Frank; Gee, Toby; Pilloni, Vincent. Abelian surfaces over totally real fields are potentially modular. Publications Mathématiques de l'IHÉS, Tome 134 (2021), pp. 153-501. doi: 10.1007/s10240-021-00128-2

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