Timeline for Which primes can divide orders of Tate-Shafarevich groups?
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10 events
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| Mar 23, 2011 at 6:28 | comment | added | Pete L. Clark | BTW, asking for the finiteness of the Shafarevich-Tate group seems to complicate the question, in my opinion. You can rephrase by asking whether the Shafarevich-Tate group has an element of order $p$, as I did in my answer below. (And indeed, in the cases where I and other people have constructed various "large" Shafarevich-Tate groups, they may indeed be infinite for all we know. I mean, conjecturally not, of course, but it seems absolutely hopeless to prove that.) | |
| Mar 23, 2011 at 6:20 | answer | added | Pete L. Clark | timeline score: 10 | |
| Mar 23, 2011 at 6:09 | comment | added | Denis Chaperon de Lauzières | It's Delaunay, not Delauney... | |
| Mar 23, 2011 at 0:02 | comment | added | David Hansen | Kevin--There's only 44 papers listed by mathscinet with "Tate-Shafarevich" in their title, and only four of them predate Rubin's work. :) | |
| Mar 22, 2011 at 20:25 | comment | added | Remke Kloosterman | However, for every prime number $p$ and every integer $k$ there is an abelian variety $A/\mathbb{Q}$ such that the $p$-torsion in the Tate-Shafarevich group has $\mathbb{F}_p$-dimension at least $k$. In this construction the dimension of $A$ grows with $p$. | |
| Mar 22, 2011 at 19:35 | comment | added | Kevin Buzzard | David---when the first edition of Silverman came out, there wasn't a single elliptic curve for which Sha was known to be finite! Facts about Sha are hard to come by... | |
| Mar 22, 2011 at 19:06 | comment | added | David Hansen | Wow! I somehow figured that this "must be known", but I guess I figured wrong... Thanks for the great references. | |
| Mar 22, 2011 at 19:01 | comment | added | Tim Dokchitser | I believe that even if you replace ${\mathbb Q}$ by any fixed number field, this is still not known. (The papers by Matsuno and Alex Bartel (arxiv.org/abs/0805.1231) study Sha and Selmer growth in small extensions $K/{\mathbb Q}$, but the degree of $K$ grows with $p$.) | |
| Mar 22, 2011 at 18:45 | comment | added | Franz Lemmermeyer | K. Matsuno, elliptic curves with large Tate-Shafarevich groups (google will give you a pdf file) wrote in 2006 that the answer was not known. | |
| Mar 22, 2011 at 18:29 | history | asked | David Hansen | CC BY-SA 2.5 |