Abelian surfaces over totally real fields are potentially modular
George Boxer, Frank Calegari, Toby Gee, Vincent Pilloni
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝐐</mml:mi> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo form="prefix">End</mml:mo> <mml:mi>𝐂</mml:mi> </mml:msub> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>𝐙</mml:mi> </mml:mrow> </mml:math> . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.