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Quaternionic families of Heegner points and $p$-adic $L$-functions

Matteo Longo, Paola Magrone, Eduardo Rocha Walchek

TL;DR

The paper extends Castella’s framework to indefinite quaternion algebras, proving an explicit reciprocity between a big analytic $p$-adic $L$-function and a big algebraic $p$-adic $L$-function arising from big Heegner points in a quaternionic Hida family. It builds a comprehensive infrastructure: quaternionic Shimura curves, CM points, Coleman integration, and Perrin–Riou theory, to interpolate $p$-adic $L$-values across weights and relate higher weight specializations to generalized Heegner cycles. The central result equates $\mathscr{L}_{\mathbb{I},\boldsymbol{\xi}}^{\mathrm{an}}$ and $\mathscr{L}_{\mathbb{I},\boldsymbol{\xi}}^{\mathrm{alg}}$, yielding non-torsion for quaternionic big Heegner points and enabling a weight-2 bridge to generalized Heegner cycles. The work also discusses current limitations in transferring GL$_2$-type Beilinson–Kato elements to the quaternionic setting, highlighting ongoing directions for aligning $p$-adic Hodge-theoretic pairings across settings.

Abstract

Following up a previous article of the authors which studies the interpolation of certain anticyclotomic $p$-adic $L$-functions associated to quaternionic modular forms in a Hida family, we extend the work of F. Castella on the interpolation and specialization of big Heegner points to the quaternionic setting. We prove an explicit reciprocity law relating the big $p$-adic $L$-function to the big Heegner points in this quaternionic setting.

Quaternionic families of Heegner points and $p$-adic $L$-functions

TL;DR

The paper extends Castella’s framework to indefinite quaternion algebras, proving an explicit reciprocity between a big analytic -adic -function and a big algebraic -adic -function arising from big Heegner points in a quaternionic Hida family. It builds a comprehensive infrastructure: quaternionic Shimura curves, CM points, Coleman integration, and Perrin–Riou theory, to interpolate -adic -values across weights and relate higher weight specializations to generalized Heegner cycles. The central result equates and , yielding non-torsion for quaternionic big Heegner points and enabling a weight-2 bridge to generalized Heegner cycles. The work also discusses current limitations in transferring GL-type Beilinson–Kato elements to the quaternionic setting, highlighting ongoing directions for aligning -adic Hodge-theoretic pairings across settings.

Abstract

Following up a previous article of the authors which studies the interpolation of certain anticyclotomic -adic -functions associated to quaternionic modular forms in a Hida family, we extend the work of F. Castella on the interpolation and specialization of big Heegner points to the quaternionic setting. We prove an explicit reciprocity law relating the big -adic -function to the big Heegner points in this quaternionic setting.

Paper Structure

This paper contains 29 sections, 12 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Quaternion algebras
  4. Shimura curves
  5. Moduli spaces
  6. Hida families
  7. Galois representations
  8. Modular forms
  9. Critical characters
  10. Big Galois representations
  11. Families of Hecke characters
  12. Twist of big Galois representations
  13. Eichler--Shimura cohomology
  14. The big Perrin-Riou map
  15. The algebraic $p$-adic $L$-function
  16. ...and 14 more sections

Key Result

Lemma 3.1

$\Psi\colon G_{K_\mathfrak{p}}\rightarrow\mathbb{I}^\times$ is unramified.

Theorems & Definitions (31)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Definition 4.2
  • Remark 5.1
  • Proposition 5.2
  • proof
  • Lemma 5.3
  • ...and 21 more