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Quaternionic big Heegner points over totally real fields

Ignacio M. Jiménez

TL;DR

This paper extends Howard’s big Heegner point framework to towers over totally real fields, handling both totally definite (Gross curves) and indefinite (Shimura curves) quaternion algebras. It constructs compatible families of Heegner points and big Heegner points attached to Hida families of Hilbert modular forms, yielding a two-variable $p$-adic $L$-function in the definite setting and big Heegner classes in the indefinite setting. The work leverages Jacquet–Langlands to relate Hilbert and quaternionic objects, and establishes Euler-system-type compatibilities across horizontal, vertical, and Galois directions, culminating in an Iwasawa main conjecture analogue for the definite case and a Howard-style class construction for the indefinite case. These results advance interpolation, special-value formulas, and Iwasawa theory in the totally real, quaternionic context, with potential further connections to generalized Heegner cycles and higher-weight phenomena.

Abstract

In this work, we extend Howard's construction of compatible families of Heegner points to the setting of towers of Gross curves and Shimura curves over totally real fields. Following the strategy of Longo and Vigni, our approach simultaneously treats totally definite and indefinite quaternion algebras. We then extend their interpolation methods to define big Heegner points attached to families of Hilbert modular forms of parallel weight under the weak Heegner hypothesis. Applying this construction, we build in the definite setting a totally real analogue of Longo$-$Vigni's two-variable $p$-adic $L$-function, and in the indefinite setting, a system of big Heegner classes in the sense of Howard.

Quaternionic big Heegner points over totally real fields

TL;DR

This paper extends Howard’s big Heegner point framework to towers over totally real fields, handling both totally definite (Gross curves) and indefinite (Shimura curves) quaternion algebras. It constructs compatible families of Heegner points and big Heegner points attached to Hida families of Hilbert modular forms, yielding a two-variable -adic -function in the definite setting and big Heegner classes in the indefinite setting. The work leverages Jacquet–Langlands to relate Hilbert and quaternionic objects, and establishes Euler-system-type compatibilities across horizontal, vertical, and Galois directions, culminating in an Iwasawa main conjecture analogue for the definite case and a Howard-style class construction for the indefinite case. These results advance interpolation, special-value formulas, and Iwasawa theory in the totally real, quaternionic context, with potential further connections to generalized Heegner cycles and higher-weight phenomena.

Abstract

In this work, we extend Howard's construction of compatible families of Heegner points to the setting of towers of Gross curves and Shimura curves over totally real fields. Following the strategy of Longo and Vigni, our approach simultaneously treats totally definite and indefinite quaternion algebras. We then extend their interpolation methods to define big Heegner points attached to families of Hilbert modular forms of parallel weight under the weak Heegner hypothesis. Applying this construction, we build in the definite setting a totally real analogue of LongoVigni's two-variable -adic -function, and in the indefinite setting, a system of big Heegner classes in the sense of Howard.

Paper Structure

This paper contains 40 sections, 25 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Set-up
  4. Main results
  5. Totally definite setting
  6. Indefinite setting
  7. General notation
  8. Acknowledgments
  9. Review of Hilbert modular forms and Hida theory
  10. Notation
  11. Definition of Hilbert cuspform
  12. Hecke operators
  13. Diamond operators
  14. Operators $T_v$
  15. Galois representation attached to a Hilbert modular form
  16. ...and 25 more sections

Key Result

Theorem A

There is a family of points $\{\mathcal{P}_\mathfrak{c}\}_\mathfrak{c}\subset \bf{D}^\dag$, for $\mathfrak{c}$ conductor coprime with $\mathfrak{n}d_{K/F}$ and whose prime divisors are inert in $K$, such that and satisfying the following horizontal compatibility relations: for each $v\nmid \mathfrak{nc}p$ inert.

Theorems & Definitions (69)

  • Theorem A
  • Conjecture 1.1: Non-vanishing conjecture
  • Conjecture 1.2: Two-variable Iwasawa Main Conjecture
  • Theorem B
  • Definition 2.1: Hilbert cuspform
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • ...and 59 more