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Smoothness of eigenvarieties and BSD for abelian surfaces

By David Loeffler

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.

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Citation data

  • DOI 10.24350/CIRM.V.19935303
  • Cite this video Loeffler, David (21/06/2022). Smoothness of eigenvarieties and BSD for abelian surfaces. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19935303
  • URL https://dx.doi.org/10.24350/CIRM.V.19935303



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