A generalized Rubin formula for Hecke characters

TL;DR

This paper extends Rubin's theorem on Katz's $p$-adic $L$-functions to a broader class of self-dual algebraic Hecke characters using generalized Heegner cycles, broadening the scope of the original results.

Contribution

It generalizes Rubin's formula for $p$-adic $L$-values to more complex Hecke characters of higher infinity type, expanding the applicability of the theory.

Findings

Extended Rubin's theorem to characters of infinity type (1+ell, -ell)

Connected generalized Heegner cycles with broader class of Hecke characters

Provided new insights into the values of Katz's $p$-adic $L$-functions

Abstract

The goal of this paper is to generalize Rubin's theorem on values of Katz's $p$-adic $L$-function outside the range of interpolation from the case of Hecke characters of CM elliptic curves to more general self-dual algebraic Hecke characters. We follow the approach by Bertolini-Darmon-Prasanna, based on generalized Heegner cycles, which we extend from characters of imaginary quadratic fields of infinity type $(1,0)$ to characters of infinity type $(1+\ell,-\ell)$ for an integer $\ell\geq0$.