Factorisation of two-variable p-adic L-functions

TL;DR

This paper extends the factorization of two-variable p-adic L-functions associated with modular forms to cases where the form is non-ordinary at p, using Sprung's logarithmic matrix for the splitting.

Contribution

It generalizes Pollack's plus and minus splitting to non-ordinary cases where a_p ≠ 0, utilizing Sprung's logarithmic matrix.

Findings

Splitting of p-adic L-functions can be extended beyond the a_p=0 case.

Uses Sprung's logarithmic matrix for the generalization.

Provides a broader framework for analyzing p-adic L-functions.

Abstract

Let $f$ be a modular form which is non-ordinary at $p$. Kim and Loeffler have recently constructed two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, they showed that, as in the one-variable case, Pollack's plus and minus splitting applies to these new objects. In this short note, we show that such a splitting can be generalised to the case where $a_p\ne0$ using Sprung's logarithmic matrix.