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Non-Archimedean plectic Jacobians

Michele Fornea, Lennart Gehrmann

Abstract

Plectic Stark-Heegner points were recently introduced to explore the arithmetic of higher rank elliptic curves: the concept was inspired by Nekovář and Scholl's plectic philosophy, while the construction is based on Bertolini and Darmon's groundbreaking use of the $p$-adic uniformization of Shimura curves to study the Birch-Swinnerton-Dyer conjecture. In this note we give a geometric interpretation of plectic Heegner points using the non-Archimedean uniformization of higher-dimensional quaternionic Shimura varieties. To this end, we define and study a plectic Jacobian functor from a category of Mumford varieties to topological groups extending the classical Jacobian functor on Mumford curves.

Non-Archimedean plectic Jacobians

Abstract

Plectic Stark-Heegner points were recently introduced to explore the arithmetic of higher rank elliptic curves: the concept was inspired by Nekovář and Scholl's plectic philosophy, while the construction is based on Bertolini and Darmon's groundbreaking use of the -adic uniformization of Shimura curves to study the Birch-Swinnerton-Dyer conjecture. In this note we give a geometric interpretation of plectic Heegner points using the non-Archimedean uniformization of higher-dimensional quaternionic Shimura varieties. To this end, we define and study a plectic Jacobian functor from a category of Mumford varieties to topological groups extending the classical Jacobian functor on Mumford curves.
Paper Structure (15 sections, 29 theorems, 113 equations)

This paper contains 15 sections, 29 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Plectic subgroups
  3. Limit points
  4. Plectic groups
  5. Partial intersection with compact open subgroups
  6. Associated polysimplicial complex
  7. Plectic Jacobians
  8. Mumford varieties and zero cycles
  9. Integration theory
  10. Universal measures and multiplicative integrals
  11. Plectic Jacobians and the Abel--Jacobi map
  12. Hecke correspondences
  13. Modularity
  14. Quaternionic Shimura varieties
  15. Geometric modularity

Key Result

Proposition 2.3

Let $\Gamma\subseteq \mathop{\mathrm{PGL}}\nolimits_2(F_S)$ a subgroup with $\mathcal{L}_\Gamma^S=\emptyset$. Then $\Gamma$ is finite.

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 59 more