CMC special weeks on number theory





November 12-22, 2019                                      KIAS 8101

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Jaeho Haan

Title : The basic of p-adic group representation

Abstract : This is a preliminary lecture for Prof. Lapid’s main lecture. We will review the basic theory of representation of GL_n(Q_p).




Yeansu Kim


Abstract: The classification of discrete series is one important subject with numerous ap- plication in the harmonic analysis and in the theory of automorphic forms. The classification consists of two steps, whose first step is the classification of strongly positive representations. With Ivan Matic’ (University of Osijek, Croatia), we obtain the first step, i.e., classification of strongly positive representations of a quasi-split similitude unitary groups GU(n, n) defined with respect to a quadratic extension local non-archimedean local fields. This is second result of the project "classification of discrete series of all similitude classical groups". If time permits, we are going to briefly explain the application of the results, which is in progress.




Erez Lapid

Title: Some aspects of irreducibility of parabolic induction for the general linear group

Abstract: The representation theory of the general linear group over a non-archimedean local field was inaugurated by Bernstein-Zelevinsky in their seminal work in the 1970s. The classification of irreducible representations is particularly elegant. It is largely combinatorial and reflects a beautiful geometric picture. In the 1990s it was realized that the category of representations of GL_n(F) is related on the one hand to category O for Lie algebras of type A by the Arakawa-Suzuki functors and on the other hand to finite-dimensional representations of quantum affine algebras of type A by Chari-Pressley's quantum Schur-Weyl duality. More recently, the works of Kang-Kashiwara-Kim-Oh and Hernandez-Leclerc give a new perspective on representations of quantum affine algebras (and more).
I will introduce some of these ideas and discuss some recent work, mostly joint with Alberto Minguez and Maxim Gurevich, on parabolic induction.




Olivier Fouquet

Title: Congruences, Euler systems and Taylor-Wiles systems
Abstract: The Equivariant Tamagawa Number Conjectures of Block-Kato and Kato are a far-reaching system of conjectures predicting the special values of $L$-functions of motives and their variations in $p$-adic families or under the action of a supplementary algebra. We discuss the formulation of the Equivariant Tamagawa Number Conjectures for modular forms with coefficients in universal deformation rings and Hecke algebras. We show in particular that these conjectures are compatible with change of level. Using an adaptation of the method of Taylor-Wiles system, we deduce that the Iwasawa Main Conjecture in the cyclotomic direction is true for modular forms in many cases.




Shunsuke Yamana

Title: Seesaw dual pairs and theta correspondence I

        Seesaw dual pairs and theta correspondence II

Abstract: TBA




Zhengyu Mao

Title: Analogue of the Ichino–Ikeda conjecture for Whittaker coefficients

Abstract: We report on the work with E. Lapid about a conjecture relating Whittaker-Fourier coefficients of cusp forms to special values of $L-$functions. We discuss the format of the conjecture and the proof in cases of general linear groups and metaplectic groups.




Takenori Kataoka

Title: Equivariant Iwasawa theory and Euler systems

Abstract: We will discuss the equivariant refinements of Iwasawa theory, mainly for elliptic curves. As the main result, under suitable hypotheses, we prove one divisibility of the equivariant main conjecture for elliptic curves, via Euler system argument. For the proof, we will reinterpret a recent result of Burns-Sakamoto-Sano on Stark systems, from the view point of derived categories.




Tadashi Ochiai

Title: Iwasawa Main Conjecture for Galois deformation spaces and glueing of Euler systems I
        Iwasawa Main Conjecture for Galois deformation spaces and glueing of Euler systems II


Abstract: We discuss Iwasawa Main Conjecture for elliptic modular forms and for p-adic families of modular forms. The first goal is to review the Iwasawa Main Conjecture the precise statement and the known result. We will also state a generalization of Iwasawa Main Conjecture to families of modular forms such as Hida families and Coleman families.
The second goal is to explain the Euler system approach to Iwasawa Main conjecture for these generalized Iwasawa Main Conjectures.
Especially we focus on how to glue Euler systems and construct Euler systems on Hida families and Coleman families.




Kenichi Namikawa

Title: Recent developments on Iwasawa theory for Asai representations

Abstract: Twisted tensor products of representations of the absolute Galois group over quadratic fields are called Asai representations. In this talk, I will survey recent developments on Iwasawa main conjecture for Asai representations of degree 4. In particular, I introduce a congruence method by theta correspondences, and I construct p-adic L-functions for Asai representations.

This talk is partially based on joint works with Ming-Lun Hsieh.




Chan-Ho Kim

Title: On the quantitative variation of congruence ideals of modular forms

Abstract: We discuss a quantitative version of Ribet's level lowering theorem for modular forms of higher weight. This is joint work with Kazuto Ota.





Yongxiong Li




Hwajong Yoo

Title : The rational torsion subgroup of J_0(N)

Abstract : In this talk, we try to compute the rational torsion subgroup of the modular Jacobian J_0(N). Let N be a positive integer. For a prime p whose square does not divide 12N, we determine the structure of the p-primary part of the rational torsion subgroup of J_0(N) by proving a generalized version of the conjecture of Ogg.