# Journal of the Ramanujan Mathematical Society

Volume 38, Issue 2, June 2023 pp. 177–182.

On the order of Tate-Shafarevich groups over finite Galois extensions

**Authors**:
Hoseog Yu

**Author institution:**
Department of Mathematics, Sejong University, 209 Neungdong-ro Gwangjin-gu, 05006, Seoul, Korea

**Summary: **
Let A be an abelian variety defined over a number field K. Let L be a finite Galois extension of K with
Galois group G and let Ш (A/K) and Ш(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and
of A over L. Define A{φ} to be an abelian variety defined over K derived from nontrivial group representations of G.
Assuming Ш(A/L) is finite, we compute [Ш(A/L)]/[Ш(A/K)], where [X] is the order of a finite abelian group X.

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