Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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
1  
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
add comment

1 Answer 1

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."

share|improve this answer
    
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 '12 at 5:22
1  
Actually, proving the order a square when finite would seem to be a candidate. But he likes his jokes. –  Charles Matthews May 9 '12 at 11:08
6  
Perhaps he views the conceptual leap to the Cyrillic alphabet as a real game changer. –  Rob Harron May 9 '12 at 13:33
9  
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 '12 at 21:27
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.