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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Daniele Masoero
Journal: Trans. Amer. Math. Soc. 371 (2019), 8381-8404
MSC (2010): Primary 11F11, 14C25
DOI: https://doi.org/10.1090/tran/7407
Published electronically: February 22, 2019
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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.


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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

DOI: https://doi.org/10.1090/tran/7407
Keywords: Modular forms, Shafarevich--Tate groups, Selmer groups, Heegner cycles
Received by editor(s): May 9, 2017
Received by editor(s) in revised form: September 14, 2017
Published electronically: February 22, 2019
Article copyright: © Copyright 2019 American Mathematical Society