## Functiones et Approximatio Commentarii Mathematici

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

Francesca Bianchi

#### 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

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

Digital Object Identifier
doi:10.7169/facm/1716

Mathematical Reviews number (MathSciNet)
MR3964261

Zentralblatt MATH identifier
07068532

#### 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

#### References

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