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Plectic Heegner classes

Michele Fornea

Abstract

We introduce a new collection of partially global Galois cohomology classes subsuming both plectic Heegner points and mock plectic invariants. The former are recovered as localizations of plectic Heegner classes, while the latter arise as eigenspace projections with respect to a "partial Frobenius"-action. By overcoming some limitations of previous constructions, plectic Heegner classes are expected to provide finer control over the arithmetic of higher rank elliptic curves. We are able to perform our construction via a systematic use of certain automorphic functions whose coefficients are p-adic measures valued in Galois cohomology. As we produce these functions through the uniformization of Shimura curves -- rather than higher dimensional quaternionic Shimura varieties -- our results are compatible with a plectic refinement of Tate's conjectures.

Plectic Heegner classes

Abstract

We introduce a new collection of partially global Galois cohomology classes subsuming both plectic Heegner points and mock plectic invariants. The former are recovered as localizations of plectic Heegner classes, while the latter arise as eigenspace projections with respect to a "partial Frobenius"-action. By overcoming some limitations of previous constructions, plectic Heegner classes are expected to provide finer control over the arithmetic of higher rank elliptic curves. We are able to perform our construction via a systematic use of certain automorphic functions whose coefficients are p-adic measures valued in Galois cohomology. As we produce these functions through the uniformization of Shimura curves -- rather than higher dimensional quaternionic Shimura varieties -- our results are compatible with a plectic refinement of Tate's conjectures.

Paper Structure

This paper contains 57 sections, 60 theorems, 436 equations.

Table of Contents

  1. Introduction
  2. Plectic Heegner classes
  3. Relation to mock plectic invariants.
  4. Relation to plectic Heegner points.
  5. Conjectures
  6. Overview of the construction
  7. Automorphic functions.
  8. Plectic Abel--Jacobi map.
  9. Preliminaries
  10. $p$-adic Measures and Distributions.
  11. Shimura curves
  12. CM points.
  13. Hecke correspondences.
  14. Towers of Shimura curves
  15. Directed systems.
  16. ...and 42 more sections

Key Result

Theorem A

Under Assumptions assum: square-free, global classes assumptions, we construct a "plectic Heegner class" belonging to a tensor product of local and global Galois cohomology groups.

Theorems & Definitions (162)

  • Definition 1.2
  • Remark 1.3
  • Theorem A
  • Remark 1.6
  • Theorem B
  • Theorem C
  • Remark 1.8
  • Conjecture 1.9
  • Conjecture 1.10
  • Remark 1.11
  • ...and 152 more