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 A over {mathbf {Q}} with operatorname{End}_{ {mathbf {C}}}A={mathbf {Z}}. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
Highlights
We show that any such ρ : GF → GSp4(F3) with suitable determinant and local conditions at places v|3 is equal to ρA,3 for infinitely many abelian surfaces A/F with EndC(A) = Z and with good ordinary reduction at v|3
L0 has a Galois-theoretic interpretation: the sum of the dimensions of the local deformation rings is l0 less than the corresponding dimension in the defect 0 case.) One key trick we employ in this paper is to reduce to situations where we only have to consider cohomology in at most two degrees, i.e. it suffices to work with complexes consisting of at most two terms
A comparison of this paper with [ACC+18]. — It follows from Theorem 1.1.3 that any elliptic curve E over a CM field K/F is potentially modular
Summary
This step relies upon an abstract multiplicity one result which we prove using the Taylor–Wiles method Exploiting these linear relations and using étale descent techniques, we first manage to construct a Klingen ordinary weight 2 modular form defined on the open subspace of the Hilbert–Siegel Shimura variety which has prank at least one at all v|p and carries a Klingen level structure. — It follows from Theorem 1.1.3 that any elliptic curve E over a CM field K/F is potentially modular ( consider the abelian surface given by Weil restriction of scalars of E from K to F)
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