Anticyclotomic <i>p</i>-adic <i>L</i>-functions for elliptic curves at some additive reduction primes

Daniel Kohen; Ariel Pacetti

doi:10.1016/j.crma.2018.09.005

Number theory

Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes

Presented by: the Editorial Board

Daniel Kohen ¹; Ariel Pacetti ²

¹ Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, Conicet, Argentina
² FAMAF-CIEM, Universidad Nacional de Córdoba, C.P:5000, Córdoba, Argentina

Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 973-983.

Abstract
Résumé

Let E be a rational elliptic curve and let p be an odd prime of additive reduction. Let K be an imaginary quadratic field and fix a positive integer c prime to the conductor of E. The main goal of the present article is to define an anticyclotomic p-adic L-function $L$ attached to $E / K$ when $E / Q_{p}$ attains semistable reduction over an abelian extension. We prove that $L$ satisfies the expected interpolation properties; namely, we show that if χ is an anticyclotomic character of conductor $c p^{n}$ , then $χ (L)$ is equal (up to explicit constants) to $L (E, χ, 1)$ or $L^{'} (E, χ, 1)$ .

Soit E une courbe elliptique rationnelle et p un premier impair de réduction additive. Soit K un corps quadratique imaginaire et c un entier positif, premier au conducteur de E. Le but de cette Note est de définir une fonction L p-adique, anti-cyclotomique, notée $L$ , attachée à $E / K$ lorsque $E / Q_{p}$ atteint la réduction semi-stable sur une extension abélienne. Nous montrons que $L$ satisfait les propriétés d'interpolation escomptées. Précisément, nous montrons que, si χ est un caractère anti-cyclotomique de conducteur $c p^{n}$ , alors $χ (L)$ est égal (à des constantes explicites près) à $L (E, χ, 1)$ ou $L^{'} (E, χ, 1)$ .

Received: 2018-04-27
Accepted: 2018-09-11
Published online: 2018-09-27

DOI: 10.1016/j.crma.2018.09.005

Author's affiliations:

Daniel Kohen ¹; Ariel Pacetti ²

¹ Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, Conicet, Argentina
² FAMAF-CIEM, Universidad Nacional de Córdoba, C.P:5000, Córdoba, Argentina

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     author = {Daniel Kohen and Ariel Pacetti},
     title = {Anticyclotomic \protect\emph{p}-adic {\protect\emph{L}-functions} for elliptic curves at some additive reduction primes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {973--983},
     publisher = {Elsevier},
     volume = {356},
     number = {10},
     year = {2018},
     doi = {10.1016/j.crma.2018.09.005},
     language = {en},
}

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Daniel Kohen; Ariel Pacetti. Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes. Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 973-983. doi : 10.1016/j.crma.2018.09.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.09.005/

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