CM cycles on Kuga-Sato varieties over Shimura curves and Selmer groups

TL;DR

This paper constructs an Euler system of CM cycles on Kuga-Sato varieties over Shimura curves for higher weight modular forms and uses it to bound Selmer groups and prove finiteness results for Shafarevich-Tate groups.

Contribution

It introduces a new system of CM cycles on Kuga-Sato varieties over Shimura curves for higher weight forms, extending Kolyvagin's method to this setting.

Findings

Bound the size of Selmer groups associated to modular forms and quadratic fields.

Prove finiteness of the primary part of the Shafarevich-Tate group under certain conditions.

Construct an Euler system of CM cycles with compatibility properties.

Abstract

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a suitable cohomology class does not vanish.