Variation of Heegner points in Hida families

TL;DR

This paper constructs and studies a big Euler system of Heegner points varying in Hida families, extending nonvanishing results and proposing a conjecture linking these classes to Selmer group ranks.

Contribution

It develops a method to interpolate Heegner points in Hida families and constructs a big Euler system, extending previous results to all ordinary modular forms.

Findings

Big Heegner point Euler system constructed for Hida families.

Specializations of the Euler system are nontrivial for all forms in the family.

Proposes a nonvanishing conjecture implying Greenberg's conjecture on Selmer groups.

Abstract

Given a weight two modular form f with associated p-adic Galois representation V_f, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V_f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family, and construct an Euler system of big Heegner points for Hida's universal ordinary deformation of V_f. We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.