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

Abstract

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 $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $\chi$ is an anticyclotomic character of conductor $c{p}^n$ then $\chi (Ł)$ is equal (up to explicit constants) to $L(E,\chi ,1)$ or $L^{\prime }(E,\chi ,1)$.

17 pages, to appear in Comptes rendus - Mathématique

References in corpus (2)

• Explicit Gross-Zagier and Waldspurger Formulae
• p-adic interpolation of automorphic periods for GL(2)