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arXiv · 1801.01619

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

Daniel Kohen·Ariel Pacetti
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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 $L'(E,\chi,1)$.

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Daniel Kohen, Ariel Pacetti. 2018-01-05. Anticyclotomic $p$-adic $L$-functions for elliptic curves at some additive reduction primes. https://arxiv.org/abs/1801.01619

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