Trivial zeros of p-adic L-functions at near central points

TL;DR

This paper establishes a formula relating derivatives of p-adic L-functions at near central points for elliptic modular forms, extending previous work to include potentially crystalline reduction cases.

Contribution

It introduces a Mazur-Tate-Teitelbaum style formula for derivatives of p-adic L-functions at near central points, utilizing a new $ ext{L}$-invariant and covering more cases.

Findings

Proves a new formula for derivatives of p-adic L-functions

Extends results to potentially crystalline reduction cases

Provides tools for understanding p-adic L-functions at near central points

Abstract

Using the $\scr L$-invariant constructed in our previous paper we prove a Mazur-Tate-Teitelbaum style formula for derivatives of p-adic L-functions of elliptic modular forms at near central points. In the second version of the paper the case of potentially crystalline reduction is also covered.