Derived Bockstein regulators and anticyclotomic $p$-adic Birch and Swinnerton-Dyer conjectures

TL;DR

This paper introduces derived Bockstein regulators, establishes a descent formalism, and applies it to connect various conjectures in p-adic number theory and Iwasawa theory, advancing understanding of Birch and Swinnerton-Dyer type conjectures.

Contribution

The paper develops derived Bockstein regulators and a descent formalism, linking multiple conjectures in p-adic and Iwasawa theory, and extending results on derivatives of Euler systems.

Findings

Proves a BSD-type conjecture for Heegner points from Perrin-Riou's main conjecture.

Derives a p-adic BSD conjecture from the Iwasawa-Greenberg main conjecture.

Extends conjectures on derivatives of Euler systems into a derived setting.

Abstract

We introduce "derived Bockstein regulators" by using an idea of Nekov\'a\v{r}. We establish a general descent formalism involving derived Bockstein regulators. We give three applications of this formalism. Firstly, we show that a conjecture of Birch and Swinnerton-Dyer type for Heegner points formulated by Bertolini and Darmon in 1996 follows from Perrin-Riou's Heegner point main conjecture up to a $p$-adic unit. Secondly, we show that a $p$-adic Birch and Swinnerton-Dyer conjecture for the Bertolini-Darmon-Prasanna $p$-adic $L$-function recently formulated by Agboola and Castella follows from the Iwasawa-Greenberg main conjecture up to a $p$-adic unit. Finally, we extend conjectures and results on derivatives of Euler systems for a general motive given by Kataoka and the present author into a natural derived setting.