Open Access
June 2019 Consequences of the functional equation of the $p$-adic $L$-function of an elliptic curve
Francesca Bianchi
Funct. Approx. Comment. Math. 60(2): 227-236 (June 2019). DOI: 10.7169/facm/1716


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 [6], 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.


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Francesca Bianchi. "Consequences of the functional equation of the $p$-adic $L$-function of an elliptic curve." Funct. Approx. Comment. Math. 60 (2) 227 - 236, June 2019.


Published: June 2019
First available in Project Euclid: 28 March 2018

zbMATH: 07068532
MathSciNet: MR3964261
Digital Object Identifier: 10.7169/facm/1716

Primary: 11G05 , 11G40
Secondary: 11R23

Keywords: $p$-adic $L$-functions , Elliptic curves

Rights: Copyright © 2019 Adam Mickiewicz University

Vol.60 • No. 2 • June 2019
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