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We show that abelian surfaces (and consequently curves of genus 2) overtotally real fields are potentially modular. As a consequence, we obtain theexpected meromorphic continuation and functional equations of their Hasse--Weilzeta functions. We furthermore show the modularity of infinitely many abeliansurfaces A over Q with End_C(A)=Z. We also deduce modularity and potentialmodularity results for genus one curves over (not necessarily CM) quadraticextensions of totally real fields.

Abelian Surfaces over totally real fields are Potentially Modular