## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 60, Number 2 (2019), 227-236.

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

#### 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 [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.

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 60, Number 2 (2019), 227-236.

**Dates**

First available in Project Euclid: 28 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1522202461

**Digital Object Identifier**

doi:10.7169/facm/1716

**Mathematical Reviews number (MathSciNet)**

MR3964261

**Zentralblatt MATH identifier**

07068532

**Subjects**

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 11G05: Elliptic curves over global fields [See also 14H52]

Secondary: 11R23: Iwasawa theory

**Keywords**

elliptic curves $p$-adic $L$-functions

#### Citation

Bianchi, Francesca. Consequences of the functional equation of the $p$-adic $L$-function of an elliptic curve. Funct. Approx. Comment. Math. 60 (2019), no. 2, 227--236. doi:10.7169/facm/1716. https://projecteuclid.org/euclid.facm/1522202461