Kolyvagin systems and Iwasawa theory of generalized Heegner cycles

TL;DR

This paper extends Iwasawa theory to the Galois cohomology of p-adic representations associated with modular forms, using higher-dimensional Heegner cycles to generalize classical Heegner point results.

Contribution

It introduces new results on Iwasawa theory for Galois cohomology of modular form representations, replacing abelian varieties with higher-dimensional Heegner cycles.

Findings

Extension of Iwasawa theory to higher-dimensional Heegner cycles

Comparison with Fouquet's deformation-theoretic results in Hida families

Generalization of classical Heegner point results to modular forms of higher weight

Abstract

Iwasawa theory of Heegner points on abelian varieties of GL_2 type has been studied by, among others, Mazur, Perrin-Riou, Bertolini and Howard. The purpose of this paper is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne's p-adic representation attached to a modular form of even weight >2. In this setting, the role of Heegner points is played by higher-dimensional Heegner-type cycles that have been recently defined by Bertolini, Darmon and Prasanna. Our results should be compared with those obtained, via deformation-theoretic techniques, by Fouquet in the context of Hida families of modular forms.