Functiones et Approximatio Commentarii Mathematici

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 [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


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References

  • [1] K. Matsuno, An analogue of Kida's formula for the $p$-adic $L$-functions of modular elliptic curves, J. Number Theory 84(1) (2000), 80–92.
  • [2] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61.
  • [3] B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84(1) (1986), 1–48.
  • [4] W. Stein and C. Wuthrich, Algorithms for the arithmetic of elliptic curves using Iwasawa theory, Math. Comp. 82(283) (2013), 1757–1792.
  • [5] C. Wuthrich, On the integrality of modular symbols and Kato's Euler system for elliptic curves, Doc. Math. 19 (2014), 381–402.
  • [6] C. Wuthrich, The sub-leading coefficient of the $L$-function of an elliptic curve, in Publications mathématiques de Besançon. Algébre et théorie des nombres, 2016, volume 2016 of Publ. Math. Besançon Algébre Théorie Nr., pages 95–96, Presses Univ. Franche-Comtée, Besançon, 2017.