From: Dave Rusin Date: Tue, 14 Jul 1998 13:21:35 -0500 (CDT) To: [Permission pending] Subject: Presenting surfaces as elliptic surfaces Last year I wrote you with some questions regarding a family of elliptic curves and you were kind enough to give me some pointers which proved very helpful. I am again looking at number theoretic questions which lead me to an algebraic surface on which I seek rational points, and I am led to ask about some structural questions regarding these surfaces. I didn't see what I needed to know in your books but thought you might suggest the most appropriate references. I am under the general impression that nothing can be said about the rational points on a _surface_ of low degree, which is nearly as satisfactory as the situation for an elliptic _curve_. But elliptic surfaces lend themselves to a better analysis, so my question is, how can we recognize an elliptic surface when we have one? Let me explain in the context of present my problem, although I'm interested in the general case as well. One description of the surface I want to study is as the zero locus of y^2 = x ( x+1 ) ( x + w^2) z^2 =-x (-x+1 ) (-x + w^2) w^2 = t^2 + 1. The last equation describes a conic, which we may parameterize. For each such w, the other two equations describe an intersection of elliptic surfaces, whose intersection is a curve of genus > 1. There are at most finitely many points for each such w; the question is whether there are any. Here's one approach. The elliptic curve described in the first equation has plenty of 2-torsion and so (thanks in part to the work on last year's problem) it's usually straightforward to show that for some w there are no points on the curve at all; and it's fairly easy to generate the points on the curve, at least to within finite index. For each one we may see if the corresponding z in the second equation is rational (which it never seems to be). So, after not too much computation, we deduce that there is no solution of the first two equations with x, y, and z anywhere short of gargantuan -- for this w. We can then try more values of w in turn, but this is hardly satisfying, since for no single w are our results ever conclusive. An alternative approach would hold x fixed and view the first two equations as describing an intersection of two conics, which is again an elliptic curve, and in fact again has plenty of 2-torsion. So again it's quite easy to check that certain small values of x are impossible, and for others to list all the possible solutions (y,z,w). We must then compare against the last equation; as before, this effectively replaces the elliptic curve with one of higher genus, so there are at most finitely many rational points; while we can rule out the existence of solutions of modest height by direct computations on the elliptic curve it covers, this is no more satisfying than the method of the previous paragraph. Now, these suggestions to hold one variable fixed at a time simply present the surface as a covering over P^1 whose fibres are curves of genus greater than 1. Is it possible that the surface may also be presented as an elliptic surface? It seems that whether the base space were of genus 0, 1, or greater, as long as the _fibres_ were elliptic curves one could make a much more fruitful search for points or (more likely) develop a proof that there are none. I'm inclined to think the surface cannot be presented as an elliptic surface. From a topologist's perspective I would argue that a complex surface can be presented as a fibration F -> S -> B in at most one way unless S is a product. (The fundamental group of S , computed from the homotopy long exact sequence, would give a nontrivial extension of pi_1(B) by pi_1(F), from which, I think, the genera of B and F may be recovered.) On the other hand, the algebraic and topological categories are hardly isomorphic, especially when as in this case there are singularities in the varieties, so I don't really know whether the surface is elliptic or not. So these are the questions I'm hoping are already known: 1) How might a person determine if an algebraic surface is elliptic? 2) Has there been any success describing the set of rational points on any surfaces of low degree which are not elliptic? Any pointers would be much appreciated. Thanks! dave ============================================================================== From: Dave Rusin Date: Wed, 5 Aug 1998 23:07:31 -0500 (CDT) To: [Permission pending] Subject: Re: Presenting surfaces as elliptic surfaces I recently wrote asking for some information about the structure of algebraic surfaces. I haven't dealt with those problems since, but when trying another angle I realized that my main question was actually rather foolish. I had written >From a topologist's perspective I would argue that a complex surface can be >presented as a fibration F -> S -> B in at most one way unless S is a >product. [...] On the other hand, the >algebraic and topological categories are hardly isomorphic It's a good thing I qualified the comments this way because my geometric model is pretty wide of the mark. The affine surface S described by x+y^2+z^3=0 projects to the first coordinate with fibres F which are of genus 1 (except at x=0) but the projections to the other coordinates have fibres of genus 0. One can put this observation in a positive light: a surface which is known to admit maps to curves of high genus with fibres of high genus might also map onto curves of lower genus, with fibres also of lower genus. So there's still room for amateurish fiddling :-) The other questions I asked > 1) How might a person determine if an algebraic surface is elliptic? > 2) Has there been any success describing the set of rational > points on any surfaces of low degree which are not elliptic? are still beyond me. I suppose algebraists must have a construct Map(V,W) classifying maps between varieties, just as topologists do. I appreciate your having given my previous message your attention. dave