Eseen 2010 4th day, Algebraic Geometry and Arithmetic

The 4th day (February 19) morning was by Takuro Mochizuki, Venjakob, Fantechi. Mochizuki was on the recent development on the irregular singularity after Sabbah. The motivation came from \mathbb{Q}-holonomic D-module \mathcal{M} / X. Regular singularity case has Riemann-Hilbert correspondence for f_{\ast}, f_!, f^{\ast}, f^!, \otimes, RHom, \psi_g (nearby cycle), \phi_g (vanishing cycle). [Saito-Terasoma: \mathcal{M} regular singular meromorphic bundle, Beilinson-Bloch-Deligne-Esnault: regular, dim X = 1]. Then, rather than naive hope for "pre-Betti structure", he worked on an extra condition of "canonical Betti strucutre" for functoriality of "Stokes structure" (Stokes filration rather than Deligne filtration) after gluing of \mathbb{C}-perverse sheaves.

Venjakov was on the non-commutative Iwasawa main conjecture for elliptic curves with complex multiplication (CM). The definition of Iwasawa was on p-adic zeta function \zeta_p up to unit, and L_p for elliptic curve. The CM case was E \ \mathbb{Q} with \mathcal{O}_K = End (E).

Fantechi (SISSA) talked about "Twists or logs as you please", Degeneration form for Gromov-Witten invariants. She defined the Gromov-Witten invariants, which were numerical invariants of smooth projective variety over \mathbb{C}. Rather than computing symplectic invariants of Li-Ruan (and relative version by Ionel-Parker), here came the algebraic geometry of Jun Li (2001). After the talks, I went to the student restaurant and a cafe, and explained to participants about the recent understanding of mirror symmetry conjecture on Gromov-Witten potential and generating function of correlation function of cohomological field theory (and Frobenius structure on the Gauss-Manin integrable connection).

Afternoon talk was only by Prof. Hida on analytic number theory of 'big' Hecke algebra. As is often the case with automorphic forms, he introduced the space of cusp forms S_{k+1} with weight k+1. He explained something important for definition of analytic L-function by Greenberg's definition written in his books.