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 $cp^n$ then $\chi(Ł)$ is equal (up to explicit constants) to $L(E,\chi,1)$ or $L'(E,\chi,1)$.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.01619
 Bibcode:
 2018arXiv180101619K
 Keywords:

 Mathematics  Number Theory;
 Primary: 11G05;
 Secondary: 11G40
 EPrint:
 17 pages, to appear in Comptes rendus  Math\'ematique