1 results
The growth of Tate–Shafarevich groups in cyclic extensions
- Part of
-
- Journal:
- Compositio Mathematica / Volume 158 / Issue 10 / October 2022
- Published online by Cambridge University Press:
- 04 November 2022, pp. 2014-2032
- Print publication:
- October 2022
-
- Article
- Export citation
-
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.
