Congruences of $p$-adic $L$-functions of modular forms at non-ordinary primes

TL;DR

This paper investigates congruences of $p$-adic $L$-functions for non-ordinary modular forms at primes where they are non-ordinary, establishing relations between their Iwasawa invariants and conditions for the main conjecture.

Contribution

It extends known results on $p$-adic $L$-function congruences to the non-ordinary setting, providing explicit formulas for Iwasawa invariants and conditions for the main conjecture.

Findings

Iwasawa invariants of $f$ and $g$ are related by explicit formulas.

The signed Iwasawa main conjecture holds for $f$ if and only if it holds for $g$ under certain conditions.

Results generalize previous ordinary case congruence results to non-ordinary primes.

Abstract

We present an analogue of Greenberg-Vatsal's and Emerton-Pollack-Weston's results on congruences of $p$-adic $L$-functions for $p$-non-ordinary cuspidal eigenforms $f$ and $g$ of equal weight that are $p$-congruent. In particular, we prove that the Iwasawa invariants of the analytic and algebraic signed $p$-adic $L$-functions of $f$ and $g$ are related by explicit formulae under appropriate hypotheses. We also show under the same assumptions that provided the algebraic and analytic $\mu$-invariants vanish, the signed Iwasawa main conjecture is true for $f$ if and only if it is true for $g$.