일정: 2019년 8월 20~23일

장소: 고려대학교 아산이학관 525호 (위치)

강연자

이민 Min Lee (University of Bristol)

일정

8월 20일 4시~5시 15분 (강연 1)

8월 21일 4시~5시 15분 (강연 2)

8월 22일 10시 30분 ~ 11시 45분 (강연 3) 1시 30분 ~ 2시 45분 (강연 4)

8월 23일 10시 30분 ~ 11시 45분 (강연 5) 1시 30분 ~ 2시 45분 (강연 6)

강의 내용

The spectral theory of non-holomorphic automorphic forms began with H. Maass in the 1940s. A Maass form is a function on a hyperbolic surface which is also an eigenfunction of the Laplace-Beltrami operator. Although Maass discovered some examples by using Hecke L-functions, in general, the construction of explicit examples of Maass forms remains mysterious. Even the existence of such functions (except the examples discovered by Maass) was not clear.

In 1956, A. Selberg introduced his famous trace formula, now called the Selberg trace formula, which relates the spectrum of the Laplace operator on a hyperbolic surface to its geometry. By using his trace formula, Selberg obtained Weyl's law, which gives an asymptotic count for the number of Maass forms with Laplacian eigenvalues up to a given bound.

Let $\mathbb{H}$ be the Poincar\’e upper half plane and $\Gamma$ be a congruence subgroup of $SL_2(\mathbb{Z})$. The aim of this short course is to develop Selberg’s trace formulas for $\Gamma\backslash \mathbb{H}$ and study their applications.

참고 문헌

• Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997.
• Dorian Goldfeld, Automorphic forms and L-functions for the group GL(n,R).
• Dorian Goldfeld, Arthur’s truncated Eisenstein series for SL(2,Z) and the Riemann zeta function: a survey, Exploring the Riemann zeta function.
• Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory.
• Henryk Iwaniec, Spectral methods of automorphic forms.
• Elon Lindenstrauss and Akshay Venkatesh, Existence and Weyl’s law for spherical cusp forms.

강의 계획

Lecture 1 Introduction

Maass form : definition, Maass’ construction, Fourier expansion

Eisenstein series : definition, Fourier expansion, meromorphic continuation

Selberg trace formula motivation

Lecture 2 Spectral decomposition

The upper-half plane as a homogeneous space

-Iwasawa/Cartan decompositions, coordinates

Laplace operator and its eigenfunctions

Incomplete Eisenstein series

Cusp forms, Integral operator

Spectral decomposition

Selberg eigenvalue conjecture

-Vigneras’ proof for SL(2, \Z)

Lecture 3 Pre-trace formula

Spectral expansion of automorphic kernels

Lindenstrauss-Venkatesh operator and the existence of infinitely many

Maass cusp forms (if time allows)

Lecture 4 Selberg’s trace formula: Spectral side

Discrete spectrum

Arthur’s truncation (Maass-Selberg formula)

Continuous spectrum

Lecture 5 Selberg’s trace formula: Geometry side

The classification of conjugacy classes

Computing the trace formula for each conjugacy class

Lecture 6 Applications

Lattice points counting

Computational approaches