Mathematics 290b.2.10s
Topics in Number Theory
Arithmetic geometry
Spring Quarter 2010
Haruzo HIDA
Meeting Time: Fridays 3:30pm to 4:50pm at
MS5138
Office hours: From 12:00noon to 1:00pm (Fridays)
at my office: MS6308.
Texts: Some talk notes will be posted:
First Talk: Marc-Hubert Nicole (Luminy), Displays and deformation theory, 4/16/2010
Second Talk: Fabian Januszewski (Karlsruhe/UCLA), Generalized local and global Birch lemmas, 4/23/2010
Abstract:
In the construction of p-adic L-functions an explizit formula for the special values
of (twisted) L-functions might referred to as a Birch lemma for historical reasons.
I will present such an explicit formula for the Rankin-Selberg convolution
of cuspidal automorphic representations of $GL_n$ and $GL_{n-1}$ respectively.
This rather general formula can evenually be used to construct corresponding p-adic L-functions
over arbitrary number fields. If time permits I will give an outline of the method,
which relies on the Birch lemma and on a cohomological interpretation of this formula in terms of (relative) modular symbols.
Third Talk: Davide Reduzzi (UCLA), TBA, 4/30/2010
Fourth Talk: Miljan Brakocevic (UCLA), Anticyclotomic p-adic L-function of central critical Rankin-Selberg L-value, 5/7/2010
[Abstract] (pdf file)
Fifth Talk: Ashay Burungale (UCLA), Barsotti- Tate groups upto isogeny and F-isocrystals, 5/21/2010
Abstract:
The Dieudonne - Manin theorem classifies Barsotti - Tate
groups over an algebraically closed perfect field k of characteristic p > 0
interms of a semi-simple category of F- isocrystals over k. We sketch a
proof.
Sixth Talk:Patrick Allen (UCLA), 5/28/2010
Modularity of two dimensional Galois representations, 5/28/2010
Abstract:
The main tool for proving modularity of two dimensional p-adic Galois representations is the so called patching method
first introduced by Taylor and Wiles and later modified by Kisin.
This method is able to establish the modularity of a large class of p-adic Galois representations when p is an odd prime.
When p=2, a variant due to Khare and Wintenberger is needed, but requires the assumption that the residual image is non solvable.
I will give a brief introduction to these methods, and if time permits I will discuss a work in progress for proving modularity of
certain 2 dimensional 2-adic Galois representations whose residual image is dihedral.
Seventh Talk: Jacques Tilouine (Universite de Paris Nord/UCLA), Companion forms and differential equations, 6/4/2010
Abstract:
I plan to explain my proof of Gross' companion form theorem by using explicit calculations
on Gauss Manin and Ramanujan differentials.