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A generalized Rubin formula for Hecke characters

Matteo Longo, Stefano Vigni, Shilun Wang

TL;DR

The paper extends Rubin's Katz $p$-adic $L$-function results from CM elliptic curves to self-dual algebraic Hecke characters of imaginary quadratic fields, by adapting the Bertolini–Darmon–Prasanna framework of generalized Heegner cycles and a $p$-adic Gross–Zagier formula to infinity types $(1+ obreak ext{ell},- ext{ell})$ with $ ext{ell} obreak \ge0$. It builds a motivic framework using generalized Kuga–Sato motives, along with Deligne–Scholl and theta-factorized pieces, to realize the Hecke characters in both de Rham and étale cohomologies and to construct canonical cycle classes whose $p$-adic logarithms control Katz and BDP $p$-adic $L$-values. A central result is a generalized Rubin formula: under $ mathscr{L}_{p, rak c}( chi^ ext{*}) eq 0$, the $p$-adic $L$-value $ mathscr{L}_{p, rak c}( chi^ ext{*})$ is congruent to ${oldsymbol{ extOmega}_p( chi^*)}^{-1}igl( ext{log}_{ chi}(oldsymbol{z}_ chi)igr)^2$ modulo the appropriate coefficient field, where $oldsymbol{z}_ chi$ is a Bloch–Kato Selmer-class associated to $ chi$. The work ties this congruence to a $p$-adic Gross–Zagier framework, validates the conjectural links with the Bloch–Kato conjecture, and paves the way for understanding derivatives $L'( chi,1)$ in a $p$-adic context through generalized factorization and reciprocity laws. Overall, it broadens Rubin-type phenomena to a larger class of self-dual Hecke characters via a robust motivic and $p$-adic analytic apparatus with potential arithmetic applications.

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 $\ell\geq0$.

A generalized Rubin formula for Hecke characters

TL;DR

The paper extends Rubin's Katz -adic -function results from CM elliptic curves to self-dual algebraic Hecke characters of imaginary quadratic fields, by adapting the Bertolini–Darmon–Prasanna framework of generalized Heegner cycles and a -adic Gross–Zagier formula to infinity types with . It builds a motivic framework using generalized Kuga–Sato motives, along with Deligne–Scholl and theta-factorized pieces, to realize the Hecke characters in both de Rham and étale cohomologies and to construct canonical cycle classes whose -adic logarithms control Katz and BDP -adic -values. A central result is a generalized Rubin formula: under , the -adic -value is congruent to modulo the appropriate coefficient field, where is a Bloch–Kato Selmer-class associated to . The work ties this congruence to a -adic Gross–Zagier framework, validates the conjectural links with the Bloch–Kato conjecture, and paves the way for understanding derivatives in a -adic context through generalized factorization and reciprocity laws. Overall, it broadens Rubin-type phenomena to a larger class of self-dual Hecke characters via a robust motivic and -adic analytic apparatus with potential arithmetic applications.

Abstract

The goal of this paper is to generalize Rubin's theorem on values of Katz's -adic -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 to characters of infinity type for an integer .
Paper Structure (43 sections, 28 theorems, 190 equations)

This paper contains 43 sections, 28 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Motives of Hecke characters
  4. Algebraic Hecke characters
  5. Representations of Hecke characters
  6. Representations of CM abelian varieties
  7. Motives of Hecke characters
  8. Tate motives
  9. Motives of abelian varieties
  10. Artin motives
  11. Construction of motives of Hecke characters
  12. Standard factorization
  13. Deligne--Scholl motives
  14. Norm factorizations
  15. Theta factorizations
  16. ...and 28 more sections

Key Result

Proposition 2.7

The splitting $M(\theta_\psi)\times K\simeq M_{E_\psi}(\psi)\oplus M_{E_\psi}(\psi^*)$ holds in $\mathrm{Mot}_{E_\psi}(K)$.

Theorems & Definitions (75)

  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 65 more