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equivariant birch–swinnerton-dyer conjecture

Journal of the Institute of Mathematics of Jussieu (1)

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ON THE $p$-ADIC VARIATION OF HEEGNER POINTS
Part of
- Arithmetic algebraic geometry
Francesc Castella
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 6 / November 2020

Published online by Cambridge University Press:

18 February 2019, pp. 2127-2164

Print publication:

November 2020
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In this paper, we prove an ‘explicit reciprocity law’ relating Howard’s system of big Heegner points to a two-variable $p$-adic $L$-function (constructed here) interpolating the $p$-adic Rankin $L$-series of Bertolini–Darmon–Prasanna in Hida families. As applications, we obtain a direct relation between classical Heegner cycles and the higher weight specializations of big Heegner points, refining earlier work of the author, and prove the vanishing of Selmer groups of CM elliptic curves twisted by 2-dimensional Artin representations in cases predicted by the equivariant Birch and Swinnerton-Dyer conjecture.