Abstract
LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L)G]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.
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Yu, H. On Tate-Shafarevich groups over galois extensions. Isr. J. Math. 141, 211–220 (2004). https://doi.org/10.1007/BF02772219
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DOI: https://doi.org/10.1007/BF02772219