Number theory
Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes
Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 973-983.

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/Qp 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 cpn, 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/Qp 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 cpn, alors χ(L) est égal (à des constantes explicites près) à L(E,χ,1) ou L(E,χ,1).

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Accepted:
Published online:
DOI: 10.1016/j.crma.2018.09.005
Kohen, Daniel 1; Pacetti, Ariel 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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Kohen, Daniel; Pacetti, Ariel. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.09.005/

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