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The growth of Tate–Shafarevich groups in cyclic extensions

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields Abelian varieties and schemes

Published online by Cambridge University Press:  04 November 2022

Yi Ouyang  and
Jianfeng Xie
Yi Ouyang
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China [email protected] Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
Jianfeng Xie
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China [email protected]
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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

abelian varietiesSelmer groupsTate–Shafarevich groupsfine Tate–Shafarevich groupstwist of abelian varieties

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$ 14K15: Arithmetic ground fields
Secondary: 11G05: Elliptic curves over global fields 11R34: Galois cohomology 14K05: Algebraic theory
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 10 , October 2022 , pp. 2014 - 2032
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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