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ON THE $p$-ADIC VARIATION OF HEEGNER POINTS

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  18 February 2019

Francesc Castella
Francesc Castella*
Affiliation:
Mathematics Department, Princeton University, Fine Hall, Princeton, NJ 08544-1000, USA ([email protected])
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Abstract

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.

Keywords

Heegner pointsHida familiesp-adic L-functionsequivariant Birch–Swinnerton-Dyer conjecture

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 2127 - 2164
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Copyright
© Cambridge University Press 2019

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Footnotes

The author was supported in part by NSF grant DMS-1801385. This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682152).

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