Wach modules and critical slope p-adic L-functions

TL;DR

This paper analyzes the structure of critical slope p-adic L-functions for ordinary modular forms, providing a decomposition into explicit measures and distributions, and studying their zeros in relation to local properties at p.

Contribution

It introduces a novel decomposition of critical slope p-adic L-functions into bounded measures and explicit distributions based on local properties, advancing understanding of their zeros.

Findings

Decomposition of p-adic L-functions into measures and distributions.

Results on the zeros of the p-adic L-function.

Consistency with observed behaviors in examples by Pollack and Stevens.

Abstract

We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form, using the methods of our earlier work with Lei. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens.