On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms
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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.References
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Additional Information
- Daniele Masoero
- Affiliation: Departimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
- Address at time of publication: Dipartimento di Matematica, Universitá degli studi di Milano, via C. Saldini 50, 20133 Milano, Italy
- Email: daniele.masoero@unimi.it
- Received by editor(s): May 9, 2017
- Received by editor(s) in revised form: September 14, 2017
- Published electronically: February 22, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8381-8404
- MSC (2010): Primary 11F11, 14C25
- DOI: https://doi.org/10.1090/tran/7407
- MathSciNet review: 3955550