Kolyvagin classes versus non-cristalline diagonal classes

TL;DR

This paper establishes a formula connecting Kolyvagin classes, derived from Heegner points on elliptic curves with multiplicative reduction, to $p$-adic iterated integrals linked to triple-product $L$-functions of certain eigenforms.

Contribution

It provides a new explicit formula relating Kolyvagin classes to $p$-adic integrals in the context of elliptic curves and triple-product $L$-functions, extending previous work on Heegner points.

Findings

Derived a formula linking Kolyvagin classes and $p$-adic integrals.

Connected triple-product $L$-functions with arithmetic classes.

Extended the theory of Heegner points in the multiplicative reduction case.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve having multiplicative reduction at a prime $p$. Let $(g,h)$ be a pair of eigenforms of weight $1$ arising as the theta series of an imaginary quadratic field $K$, and assume that the triple-product $L$-function $L(f,g,h,s)$ is self-dual and does not vanish at the central critical point $s=1$. The main result of this article is a formula expressing the $p$-adic iterated integrals introduced in [DLR] to the Kolyvagin classes associated by Bertolini and Darmon to a system of Heegner points on $E$.