Abelian surfaces over totally real fields are potentially modular. (English) Zbl 1522.11045
Summary: 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 \(\mathbb{Q}\) with \(\mathrm{End}_{{\mathbb{C}}}A={\mathbb{Z}}\). We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
MSC:
| 11F80 | Galois representations |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14K15 | Arithmetic ground fields for abelian varieties |
| 11G10 | Abelian varieties of dimension \(> 1\) |
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