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CM cycles on Kuga-Sato varieties over Shimura curves and Selmer groups. (English) Zbl 1445.11051

Summary: 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 [V. A. Kolyvagin, Prog. Math. 87, 435–483 (1990; Zbl 0742.14017)], as adapted by J. Nekovář [Invent. Math. 107, No. 1, 99–126 (1992; Zbl 0729.14004)] 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.

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R34 Galois cohomology
11F80 Galois representations
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