Eseen 2010 1st day, Algebraic Geometry and Arithmetic

Yesterday was the 1st day of the conference on algebraic geometry and arithmetic in Essen, Germany. The internet connection (WiFi) was not established until now, so that I read the e-mail for the first time in Germany at the conference room.

Before the lecture, I had to look for how to reach the yellow building. While I was thinking about a public street (and the tram station) around the campus, it was found out that the lecture room is not at the math department, bur rather at the center of Duisburg-Essen University.

The morning talks were delivered by Vistoli and Chenever. Before that the organizer delivered a speech on the passing away of Prof. Viehweg. Vistoli was on the "essential dimension" of a curve and the genericity theorem. After the definition of Merkuriev, the theorem of Brousnan-Retchtustein-Vistov holds for any ground filed with characteristic 0 (genus 0: 2, genus 1: +infty, genus 2: 5, genus >= 3, 3g - 3). After the theorem on smooth connected Deligne-Mumford stack of finite type over a field, the genericity theorem (Beosnan-Reichstein-Vistoli) holds. After thinking about the Artin stack and stable curves with n points, one conjectured that the automorphism group G should be "extremely reductive". Then we switched the case for d & n (especially for whether 3 divides d).

After the first talk, I could see Prof. Beilinson and talked about my research projects and his recent works with Gaitsogry. I reported my current updates at the topological methods (Jacob Lurie et.al.) in mathematical physics at Komaba compus and IPMU of University of Tokyo.

Chenever was on the analytic number theoretical methods to the modularity problem of complex multiplication. Mazur's results for p > 13 (p not 631) for Ramanujan delta function were stated, and the theorem of Gowea-Mazur was that the quasi-modular points are Zariski dense. After the theorem on the first cohomology on generic situation, its kernel is isomorphic to 2nd cohomology.

The lunch was at the shopping mall, and I took a Thai-Reise Spaghetti, which was very hot and I feel refreshed. After the lunch, we went to a cafe and we saw a carnival with bright contume.

The afternoon session was by Ngo and Brosnan. Ngo talked about the Langlands program. It started from some classical results about Godement, Jacquet, Authur on the L-funciton. The endoscopy and analytic continuation were analyzed on the pole. After the Beilinson-Drinfeld Grassmanian (n=1 case and geometric Langlands), he showed the intersection cohomology of perverse sheaf (Hecke) on the Grassmanian. For the adelization of the trace formula, he utilized the Lie group and its Haar measure, Tamagawa measure on the coset. Then the Poisson resummation formula enables us to make the summation absolutely convergent.