Viernes 5 de Diciembre, 14:30-16:40, sala 2-1, IMA de la PUCV
Expositor: Bas Edixhoven (U. Leiden)
Título: Modular forms, Galois representations, computations, sums of squares and Gauss’s theorem on sums of 3 squares.
Resumen: The first half of the lecture will consist of an overview of joint work with Couveignes, de Jong, Merkl, and of work of Bruin, on the computation of Galois representations attached to modular forms. A nice application of this is the fast computation of the number of solutions of x_1^2+….+x_d^2 = n for even d. The second half of the lecture concerns the case d=3. Gauss has shown for example that for a positive square free integer n that is 1 modulo 4 the number of solutions in integers of x^2+y^2+z^2=n equals 12 times the class number of the ring Z[t]/(t^2+n). Gauss’s proof is long. The aim of the talk is to give a short proof, using the action of the group scheme SO(3). This proof shows in fact that the class group in question acts freely and transitively of the set of solutions. These results on Gauss’s theorem is work in progress of Albert Gunawan, PhD student with me and Qing Liu. Basic knowledge of sheaves on Spec(Z) with its Zariski topology suffices for understanding this second half. Reference for the first half: http://www.math.u-bordeaux1.fr/~jcouveig/book.htm