Stark points and Hida-Rankin p-adic L-function

TL;DR

This paper refines the elliptic Stark conjecture by determining an algebraic constant related to the Hida-Rankin p-adic L-function, providing deeper insight into the conjecture's validity in specific cases involving elliptic curves and Artin representations.

Contribution

It offers a refined result that identifies the algebraic constant in the elliptic Stark conjecture using Hida-Rankin p-adic L-functions, extending previous proofs to new settings.

Findings

Determined the algebraic constant in the conjecture for specific cases.

Connected local and global invariants of elliptic curves and quadratic fields.

Provided an alternative proof using Hida-Rankin p-adic L-functions.

Abstract

This article is devoted to the elliptic Stark conjecture formulated by Darmon, Lauder and Rotger [DLR], which proposes a formula for the transcendental part of a $p$-adic avatar of the leading term at $s=1$ of the Hasse-Weil-Artin $L$-series $L(E,\varrho_1\otimes \varrho_2,s)$ of an elliptic curve $E$ twisted by the tensor product $\varrho_1\otimes \varrho_2$ of two odd $2$-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a $2\times 2$ $p$-adic regulator involving the $p$-adic formal group logarithm of suitable Stark points on $E$. This conjecture was proved in [DLR] in the setting where $\varrho_1$ and $\varrho_2$ are induced from characters of the same imaginary quadratic field $K$. In this note we prove a refinement of this result, that was discovered experimentally in Remark 3.4 of [DLR] in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of [DLR] holds in a particular setting where the Hida-Rankin $p$-adic $L$-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both $E$ and $K$.