![Stable homology of braid groups with symplectic coefficients](/media/cache/video_light/uploads/video/2024-05-07_Petersen.mp4-02e4b37b08b4d31a5bc8706d66c76471-video-339dfc29f5d7136e6a7bcf8ea9ae0a67.jpg)
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Stable homology of braid groups with symplectic coefficients
By Dan Petersen
Appears in collection : Galois Representations, Automorphic Forms and L-Functions / Représentations galoisiennes, formes automorphes et leurs fonctions L
I will discuss recent joint work with Sarah Zerbes in which we use Euler systems and reciprocity laws for GSp(4) to study the analytic rank 0 case of the Birch--Swinnerton-Dyer conjecture for abelian surfaces. Via restriction of scalars, this also includes the BSD conjecture for analytic rank 0 elliptic curves over imaginary quadratic fields. Our main result is a conditional proof of the conjecture subject to an assumption about the local geometry of the GSp4 eigenvariety at non-regular-weight points. I will explain how this conjecture arises and some motivation for why it seems plausible that it should hold.