On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms

Masoero, Daniele

doi:10.1090/tran/7407

On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms
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Trans. Amer. Math. Soc. 371 (2019), 8381-8404 Request permission

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 Nekovář’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.

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