On the Iwasawa invariants of BDP Selmer groups and BDP P-adic L-fucntions

TL;DR

This paper extends the understanding of Iwasawa invariants and p-adic L-functions from the cyclotomic to the anticyclotomic setting for certain modular forms over imaginary quadratic fields, revealing deep relations between Selmer groups and L-functions.

Contribution

It generalizes known results relating Iwasawa invariants and p-adic L-functions to the anticyclotomic context under specific hypotheses.

Findings

Relations between Iwasawa invariants of BDP Selmer groups for two forms.

Connections between BDP p-adic L-functions and Selmer groups in the anticyclotomic setting.

Extension of cyclotomic results to the anticyclotomic case.

Abstract

Let $p$ be an odd prime. Let $f_1$ and $f_2$ be weight-two Hecke eigen-cuspforms with isomorphic residual Galois representations at $p$. Greenberg--Vatsal and Emerton--Pollack--Weston showed that if $p$ is a good ordinary prime for the two forms, the Iwasawa invariants of their $p$-primary Selmer groups and $p$-adic $L$-functions over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, let $K$ be an imaginary quadratic field where $p$ splits. Suppose that the generalized Heegner hypothesis holds with respect to both $(f_1,K)$ and $(f_2,K)$. We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDP $p$-adic $L$-functions of $f_1$ and $f_2$.