# Finiteness of Tate-Shafarevich

Does anyone happen to know who conjectured the finiteness of the Tate-Shafarevich group?

We recall the conjecture. Let $E/K$ be an elliptic curve where $K$ is a number field. Then $Ш(E/K)$ is finite.

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For what it is worth, it does not seem to appear in the original Lang-Tate paper ("Principal Homogeneous Spaces over Abelian Varieties"), see Theorem 5 and the two preceding paragraphs on p. 681. I don't have easy access to Shafarevich's paper. –  B R May 9 '12 at 2:41
I'm going to assume that your $K$ is a number field; you haven't defined it. John Tate (On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, 9 (1964-1966), Exposé No. 306, 26 p. numdam.org/numdam-bin/fitem?id=SB_1964-1966__9__415_0) says that Another deep conjecture underlying [the Birch and Swinnerton-Dyer] Conjecture is that Ш is finite. He doesn't attribute this finiteness conjecture to anybody, so presumably he is the author. –  Chandan Singh Dalawat May 9 '12 at 3:12
You're right I forget to define $K$. I've fixed it in my edit. –  Eugene May 9 '12 at 3:14