Author:
Boxer George,Calegari Frank,Gee Toby,Pilloni Vincent
Abstract
AbstractWe 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 $A$
A
over ${\mathbf {Q}}$
Q
with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$
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.
Publisher
Springer Science and Business Media LLC
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