Plectic Heegner classes

TL;DR

This paper introduces plectic Heegner classes, a new collection of Galois cohomology classes that unify and extend previous constructions, offering finer arithmetic control for higher rank elliptic curves.

Contribution

It systematically constructs plectic Heegner classes using automorphic functions and Shimura curve uniformization, compatible with plectic Tate conjectures.

Findings

Plectic Heegner classes unify Heegner points and invariants.

Construction uses p-adic measures and automorphic functions.

Results enhance understanding of higher rank elliptic curve arithmetic.

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.