## 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. – BR May 9 2012 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/…) 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 2012 at 3:12
You're right I forget to define $K$. I've fixed it in my edit. – Eugene May 9 2012 at 3:14

In Cassels's 1962 ICM paper (available here), he says the following: "Indeed, Tate and Šafarevič have, I believe, independently conjectured(5) that Ш itself is always finite", and the (5) is a footnote stating: "In his lecture, Tate denied paternity but adopted the conjecture. In conversation during the Congress Šafarevič expressed strong doubts."

So, maybe no one knows!

EDIT: As an added bonus, I found the following quote in Cassels's review of Silverman's book: "Without doubt the reviewer's most lasting contribution to the theory is the introduction of the Cyrillic letter Ш ("sha") to denote this group, a usage which has become universal."

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Cassels has the same footnote ("This is the author's most lasting contribution to the subject") on p. 109 of his Lectures on Elliptic Curves. – KConrad May 9 2012 at 5:22
Actually, proving the order a square when finite would seem to be a candidate. But he likes his jokes. – Charles Matthews May 9 2012 at 11:08
Perhaps he views the conceptual leap to the Cyrillic alphabet as a real game changer. – Rob Harron May 9 2012 at 13:33
Tate told me that, when he first started thinking about elliptic curves, he assumed that Ш would be finite and that it would be easy to prove it, in analogy with the class group. He then realized that matters were much more difficult. – Felipe Voloch May 9 2012 at 21:27