프로그램

제목: The distribution of Selmer ranks of Jacobians of hyperelliptic curves

연사: 유명준

초록: In algebraic number theory, the theory of elliptic curves plays a very important role. For example, Andrew Wiles proved Fermat's last theorem by establishing the modularity of semistable elliptic curves over the rational field. In the first half of the talk, we discuss the basic theory of elliptic curves. Selmer groups and quadratic twists will be introduced as tools to study elliptic curves. We will also talk about generalizations of the results on elliptic curves to the family of hyperelliptic curves.

제목: On the descent: with more focus on Galois descent and examples

연사: 조성문

초록: Descent theory has its origin on the relation between Fundamental groups on topological spaces and the automorphism groups on their covering spaces.

Grothendieck and others made fundamental works of such theory in the case of algebraic schemes. Galois descent is one special form of such theory, having lots of applications to modern number theory.

The goal of this lecture series is to make graduate/undergraduate students feel comfortable using Galois descent. I will provide some interesting examples on algebraic torus, elliptic curves, characters, and so on. If time allows, then we will study more general formulation, fppf descent. The best reference, written in English, is Section 6 of Neron Models by BLR, although this book is notoriously difficult to follow.

제목: Introduction of Integral $p$-adic Hodge Theory

연사: 천현민

초록: Thanks to Colmez-Fontaine, there exists an equivalence of categories between semi-stable representations and admissible filtered (phi, N)-modules. There is also an equivalence of categories between Galois stable lattices of semi-stable representations and strongly divisible modules (under certain assumption), where strongly divisible modules can be regarded as lattices of admissible filtered (phi, N)-modules. Moreover, there are Breuil modules, that correspond to the mod-p reduction of strongly divisible modules, and these Breuil modules correspond to the mod-p reduction of semi-stable representations. In this talk we will introduce all of these categories as well as morphisms between these categories.

제목: Galois-stable lattice of semi-stable Galois representation and examples

연사: 박정효

초록: Given a semi-stable Galois representation $\rho$, one tries to find its Galois-stable lattice. On the other hand, Colmez-Fontaine theorem says that there is a categorical equivalence between the category of semi-stable representations and the category of (weakly) admissible filtered $(\phi, N)$-modules. Anti-equivalence between the category of Galois-stable lattices and strongly divisible modules is given by Tong Liu. With these results, Galois-stable lattice of a given semi-stable representation can be described by computing strongly divisible modules associated to the filtered $(\phi, N)$-module. In this talk, I give a simple example of finding strongly divisible modules when $\rho: G_{Q_{p}} \to GL_{3}(E)$ is semi-stable, with Hodge-Tate weights (0, 1, 2) and rank N = 2.

제목: A short introduction to the trace formula

연사: 강태엽

초록: In a long series of papers from 1974 to 2003, James Arthur developed the trace formula of a reductive group over a global field as a generalization of the Selberg trace formula. The trace formula is an identity between two kinds of interpretation of a trace : the geometric terms given by the conjugacy classes, and the spectral terms given by the induced representations. In this talk, I will briefly explain how the trace formula is constructed.

제목: Orbital integrals for gln and smooth integral models

연사: 이유찬

초록: The orbital integral is a significant tool to handle the stable trace formula, directly related to the Langlands functoriality conjecture. There are a lot of attempts to calculate the orbital integral; using Bruhat-Tits building or extending Langlands' idea.

In this talk, I will provide the entirely new method to calculate the orbital integrals through the geometric techinique; smoothening method; and the explicit value of them, when n=2 and n=3.