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

TL;DR

This paper constructs an anticyclotomic $p$-adic $L$-function for elliptic curves with additive reduction at $p$, establishing its interpolation properties relating to special values of complex $L$-functions.

Contribution

It introduces a new $p$-adic $L$-function for elliptic curves with additive reduction, extending the theory to cases with semistable reduction over abelian extensions.

Findings

Defines the anticyclotomic $p$-adic $L$-function $ extbackslash L$

Proves $ extbackslash L$ satisfies expected interpolation properties

Relates $ extbackslash L$ values to $L(E, extbackslash chi,1)$ and its derivatives

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