On the structure of Selmer and Shafarevich-Tate groups of higher weight modular forms

TL;DR

This paper extends Kolyvagin's results on Shafarevich-Tate groups from elliptic curves to higher weight modular forms over imaginary quadratic fields, using Heegner cycles and Nekovár's theory.

Contribution

It proves the structure of Shafarevich-Tate groups for higher weight modular forms and improves bounds on their order, also analyzing Selmer groups in the Bloch--Kato sense.

Findings

Structure of Shafarevich-Tate groups is governed by cohomology classes from Heegner cycles.

Provides bounds on the order of Shafarevich-Tate groups.

Analyzes the structure of Selmer groups for modular forms.

Abstract

Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich-Tate groups of elliptic curves. In this paper we prove analogous results for ($p$-primary) Shafarevich-Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a "Heegner hypothesis". More precisely, we show that the structure of Shafarevich-Tate groups is controlled by cohomology classes built out of Nekov\'a\v{r}'s Heegner cycles on Kuga-Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the order of these groups. As a second contribution, we prove a result on the structure of ($p$-primary) Selmer groups of modular forms in the sense of Bloch--Kato.