On the non-vanishing of generalized Kato classes for elliptic curves of rank $2$

TL;DR

This paper proves the first cases of a conjecture relating to the non-vanishing of generalized Kato classes for elliptic curves of rank 2, linking it to the structure of their $p$-adic Selmer groups.

Contribution

It introduces a novel formula connecting the leading term of an anticyclotomic $p$-adic $L$-function with derived $p$-adic heights and regulators, advancing understanding of elliptic curves of rank 2.

Findings

Non-vanishing of generalized Kato classes implies 2-dimensional $p$-adic Selmer group.

Established cases of Darmon--Rotger conjecture for rank 2 elliptic curves.

Derived a new formula for the leading term of an anticyclotomic $p$-adic $L$-function.

Abstract

We prove the first cases of a conjecture by Darmon--Rotger on the non-vanishing of generalized Kato classes attached to elliptic curves $E$ over $\mathbf{Q}$ of rank $2$. Our method also shows that the non-vanishing of generalized Kato classes implies that the $p$-adic Selmer group of $E$ is $2$-dimensional. The main novelty in the proof is a formula for the leading term at the trivial character of an anticyclotomic $p$-adic $L$-function attached to $E$ in terms of the derived $p$-adic height of generalized Kato classes and an enhanced $p$-adic regulator.