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# On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms over Cyclotomic Extensions

Published:2016-12-06
Printed: Aug 2017
• Antonio Lei,
Département de mathématiques et de statistique , Pavillon Alexandre-Vachon, Université Laval , Québec, QC, Canada G1V 0A6
• David Loeffler,
Mathematics Institute, Zeeman Building, University of Warwick , Coventry CV4 7AL, UK
• Sarah Livia Zerbes,
Department of Mathematics, University College London , Gower Street, London WC1E 6BT, UK
 Format: LaTeX MathJax PDF

## Abstract

We study the asymptotic behaviour of the Bloch--Kato--Shafarevich--Tate group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
 Keywords: cyclotomic extension, Shafarevich-Tate group, Bloch-Kato Selmer group, modular form, non-ordinary prime, p-adic Hodge theory
 MSC Classifications: 11R18 - Cyclotomic extensions 11F11 - Holomorphic modular forms of integral weight 11R23 - Iwasawa theory 11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]

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