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arXiv · 1611.02953

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

Francesca Bianchi
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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.

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Francesca Bianchi. 2016-11-09. Consequences of the functional equation of the $p$-adic $L$-function of an elliptic curve. https://doi.org/10.7169/facm%2F1716

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