Iwasawa theory of Heegner cycles, I. Rank over the Iwasawa algebra

TL;DR

This paper extends Iwasawa theory to higher-dimensional Heegner cycles associated with modular forms, demonstrating that a related Selmer group's dual has rank 1 over an anticyclotomic Iwasawa algebra.

Contribution

It generalizes previous results on Heegner points to higher-dimensional cycles in the context of modular forms, establishing new rank results over Iwasawa algebras.

Findings

Pontryagin dual of the Selmer group has rank 1

Extension of Iwasawa theory to higher-dimensional cycles

Results applicable to Galois cohomology of modular forms

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, the first in a series of two, 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 f of even weight >2. In this more general setting, the role of Heegner points is played by higher-dimensional Heegner cycles in the sense of Nekov\'a\v{r}. In particular, we prove that the Pontryagin dual of a certain Bloch-Kato Selmer group associated with f has rank 1 over a suitable anticyclotomic Iwasawa algebra.