On anticyclotomic variants of the $p$-adic Birch and Swinnerton-Dyer conjecture

TL;DR

This paper formulates and partially proves $p$-adic Birch and Swinnerton-Dyer conjectures for elliptic curves at ordinary primes, linking Iwasawa theory with $p$-adic $L$-functions and Heegner distributions.

Contribution

It introduces new $p$-adic BSD conjectures for elliptic curves and proves key inequalities and leading coefficient formulas using Iwasawa theory.

Findings

Proved one inequality of the conjecture under mild hypotheses.

Established the leading coefficient formula up to a $p$-adic unit.

Related results to conjectures by Bertolini-Darmon from 1996.

Abstract

We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves $E/\mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove under mild hypotheses one of the inequalities predicted by the rank part of our conjectures, as well as the predicted leading coefficient formula up to a $p$-adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in 1996 for certain Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.