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The growth of Tate–Shafarevich groups in cyclic extensions
Published online by Cambridge University Press: 04 November 2022
Abstract
Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety
$A$ over a global field
$K$, the
$p$-Selmer group
$\mathrm {Sel}_{p}(A/L)$ grows unboundedly when
$L$ ranges over the
$(\mathbb {Z}/p\mathbb {Z})$-extensions of
$K$. Moreover, he raised a further problem: is
$\dim _{\mathbb {F}_{p}} \text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case
$p \neq \mathrm {char}\,K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential
$\text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.
Keywords
MSC classification
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- Research Article
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- © 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
References
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