Consequences of the functional equation of the $p$-adic $L$-function of an elliptic curve

TL;DR

This paper investigates the relationship between the first two coefficients of the $p$-adic $L$-function of an elliptic curve around $s=1$, revealing a formula involving the conductor and extending to base changes.

Contribution

It establishes a new relation between the initial coefficients of the $p$-adic $L$-function and the conductor, generalizing previous results and exploring implications for base changes.

Findings

First two coefficients are related by a conductor-involving formula

Extension of results to base-change over abelian number fields

Implications for the structure of $p$-adic $L$-functions

Abstract

We prove that the first two coefficients in the series expansion around $s=1$ of the $p$-adic $L$-function of an elliptic curve over $\mathbb{Q}$ are related by a formula involving the conductor of the curve. This is analogous to a recent result of Wuthrich for the classical $L$-function, which makes use of the functional equation. We present a few other consequences for the $p$-adic $L$-function and a generalisation to the base-change to an abelian number field.