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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
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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 cyclotomic extension, Shafarevich-Tate group, Bloch-Kato Selmer group, modular form, non-ordinary prime, p-adic Hodge theory
MSC Classifications: 11R18, 11F11, 11R23, 11F85 show english descriptions Cyclotomic extensions
Holomorphic modular forms of integral weight
Iwasawa theory
$p$-adic theory, local fields [See also 14G20, 22E50]
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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