STNB 2015 (29th year)
New developments on the Birch and Swinneton-Dyer conjecture
Presenters
Abstract
I will describe the circle of ideas appearing in recent joint works with Massimo Bertolini, Henri Darmon and Alan Lauder, where we construct $p$-adic families of cohomology classes associated to Beilinson-Flach elements and diagonal cycles, and show that their images under suitable regulator maps are related to special values of the $p$-adic $L$-function attached to the convolution of two (resp. three) Hida families of modular forms. This leads to significant progress on BSD for twists of elliptic curves by a large family of Artin representations, when the L-function vanishes to order 0, 1 and 2 at $s=1$.
We recover as particular cases the scenarios where the order of vanishing is 1 and classical Heegner points for imaginary quadratic fields or Darmon's Stark-Heegner points for real quadratic points are available. In the first case, we reprove and extend known results by completely different methods. In the second case, our methods give rise to global Selmer classes that are rational over the expected ring class fields unconditionally, while Darmon's construction at the turn of the century remains completely conjectural. Beyond these two classical settings, our construction is also available in other exciting settings that I will describe, including the case where the $L$-function vanishes with order 2.
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