Variation of Heegner points in Hida families

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.

Cited by in corpus (8)

• On the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves
• On the $p$-adic variation of Heegner points
• The universal $p$-adic Gross-Zagier formula
• Heegner points in Coleman families
• Kolyvagin systems and Iwasawa theory of generalized Heegner cycles
• Interpolation of Generalized Heegner Cycles in Coleman Families
• Big Heegner point Kolyvagin system for a family of modular forms
• On anticyclotomic Euler and Kolyvagin systems