From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Question about Elliptic Curves Date: 3 Oct 1999 01:11:58 GMT Newsgroups: sci.math Keywords: Shafarevich group Sha . In article , Allan Trojan wrote: > >In Husemoeller's book "Elliptic Curves" page 13 >he states >"As for cubics, there is no known method for determining, > in a finite number of steps, > whether there is a rational point on a given cubic curve. > This very important question is still open." > > Can anyone tell me what is the present status of this problem. Still important, still open. Here's the basic difficulty: it's pretty easy to determine whether or not there are p-adic solutions to an equation of the form f(x,y)=0 when f is a polynomial. (There are only finitely many primes at which there might not be p-adic points on any given elliptic curve.) If there are no p-adic points for even one p, then there can't be any rational points, either. On the other hand, if there are p-adic solutions for every p, what then? There _is_ a theorem (the "local-to-global principle", various forms of which are attributed to many authors) which states that there IS a rational solution in this case, but it applies only to quadratic equations. There are examples in which a cubic curve has p-adic points for every p but no rational point: Selmer found the simple example 3 x^3 + 4 y^3 + 5 = 0. There is a group "Sha" (that's the Cyrillic letter which looks like III ) which measures the gap between rational solvability and everywhere-local solvability (named after Shafarevich). The problem is that there's no algorithm known which can compute Sha for a curve, in terms of the initial data. Even the best software can only start poking around looking for rational points and, if none are found, quit with the declaration that Sha may be nontrivial. (Well, that's a little harsh -- there are some other tools which might apply in various circumstances. But the general case is still open.) dave For more on elliptic curves: index/14H52.html