Order of Ш (Sha) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:19:19Z http://mathoverflow.net/feeds/question/95555 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95555/order-of-sha Order of Ш (Sha) Srilakshmi 2012-04-30T06:08:26Z 2012-04-30T12:50:24Z <p>To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not yet known.</p> <p>Is there any example of an elliptic curve of rank 2 such that $p$-primary components of Ш are trivial for $p$ outside a finite set of primes?. In particular, $Ш(E/\mathbb{Q})[p]$ is trivial for $p$ $\neq$ 2, 3, 5, 7.</p> http://mathoverflow.net/questions/95555/order-of-sha/95562#95562 Answer by Alex Bartel for Order of Ш (Sha) Alex Bartel 2012-04-30T08:20:18Z 2012-04-30T09:59:50Z <p>No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically by performing $p^n$-descent for higher and higher $n$, until the upper bound on the rank of $Ш(E/\mathbb{Q})[p^n]$ stabilises. Of course, we cannot prove a priori that this would ever happen, but in practice, if you knew finiteness of $p$-primary parts of sha outside a finite set of primes, you would run your computer to do $p^n$-descent for the remaining primes, until you establish finiteness for this finite set, too.</p>