This week, the Forschungsinstitut für Mathematik is host to a conference to honor the 61st birthday of G. Wüstholz — of course, diophantine approximation, arithmetic geometry, and related areas were very much in focus. (I write “is” because, although this is most definitely Friday evening, there are still two talks scheduled tomorrow morning).
The programme was very enticing, so I attended most of the talks, despite not knowing much about some of the topics (e.g., Arakelov geometry). Among those I found especially interesting (partly because I was at least a bit more au courant) were the following:
(1) U. Zannier explained some recent work with D. Masser which can be described as trying to understand (and devise methods to study) intersections of “sparse” sets of arithmetic interest. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves
and the points
on Et (so there are two choices of the y-coordinates, which will not affect the question). What can one say about the set of parameters t for which both
are torsion points on the curve Et? It is not difficult to check that for either of the two points, there are infinitely many such parameters, forming “sparse” sets, and the results (or rather, the methods) of Masser and Zannier imply, in particular, that these two have at most a finite number of intersection points, i.e., that there are only finitely many t for which both are torsion points.
One may wonder why the question should be of any interest (I personally find it very nice), but Zannier emphasized that the new techniques they had to devise were quite significant and very likely to be useful in many contexts. These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). Zannier illustrated the link with a sketch of a new proof of the original Manin-Mumford conjecture (first proved by Raynaud; it states that if an algebraic curve defined over a number field is embedded in its Jacobian variety, then the curve only contains finitely many torsion points): using a transcendental parametrization, we can see the curve as a transcendental curve
for some lattice
whereas the torsion points are the elements of
and thus have rational coordinates. The intersection is thus, in principle, similar to the situation of Bombieri-Pila. Of course, much more work is required, and Zannier said that the extension to deal with Masser’s question require rather more subtle versions of the Bombieri-PIla ideas, including the very recent ones of Pila-Wilkie where — to add more fun to the mix — logic enters the game through the consideration of transcendental varieties with graphs definable in an o-minimal structure. (About which, despite looking at the book of van den Dries, I am still terribly ignorant).
(2) T. Shioda gave a very nice lecture — full of beautiful examples — on recent work of his concerning the determination and structure of the (finite) set of integral sections of an elliptic surface over the projective line (see his preprint); for rational elliptic surfaces, he explained a very beautiful general description involving commutative algebra. In particular, there are then at most 240 integral sections. For instance, there are exactly 240 polynomials
such that they are points on the rational elliptic surface with equation
indeed, those points, for the height pairing, can be identified with the 240 vectors of minimal length (squared) 2 on the famous E8 lattice.
(3) Y. Bilu explained his recently recovered-from-the-edge proof, with P. Parent, of the “split Cartan” case of the Serre uniformity question concerning the maximality of the Galois action on torsion points of prime order on elliptic curves over the rationals (see also this post for more background information– though as indicated, the first proof had a mistake, which was corrected in February–March this year, the overall strategy has remained the same). The preprint of Bilu and Parent is on arXiv.