Non-commutative p-adic L-functions for supersingular primes

TL;DR

This paper proposes a conjecture for the existence of plus and minus p-adic L-functions for supersingular elliptic curves over certain non-abelian extensions, extending methods from p-ordinary cases.

Contribution

It introduces a new conjecture and constructs non-commutative p-adic L-functions for supersingular elliptic curves in specific non-abelian extensions, adapting existing methods.

Findings

Formulated a conjecture on p-adic L-functions for supersingular primes.

Constructed such functions under certain technical conditions.

Extended methods from p-ordinary to supersingular cases.

Abstract

Let E/Q be an elliptic curve with good supersingular reduction at p with a_p(E)=0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the Zp-cyclotomic extension of a finite Galois extension of Q where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.