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

Abstract

Abstract Given a modular form f {{f}} of even weight larger than two and an imaginary quadratic field K {{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 {{f}} enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to f {{f}} and K {{K}} and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.