July 21, 2024

Thematic Program on Automorphic Forms

Graduate Courses - Winter Semester 2003

Courses Offered :

January 21- April 29, 2003
Tuesdays, 1:30 pm- 3:00 pm
Course on Automorphic Functions

Instructors: H. Kim
NOTE: Last class rescheduled from Tuesday, April 20 to Thursday, May 1 at 1:30

January 21- May 1, 2003
Thursday, 10:30 am- 12:00 pm
Course on Symmetric Power L-Functions And Applications To Analytic Number Theory

Instructor: R. Murty
NOTE: Last class is on April 17, 2003

January 21 - April 29, 2003 (Tuesday 10:30 am - 12:00 pm)
Course on L-functions, Converse Theorems, and Functoriality for GL(n)
Instructor: Jim Cogdell (Oklahoma State)
NOTE: Last class rescheduled from Tuesday, April 20 to Thursday, May 1 at 10:30

Course Descriptions:
Course on Automorphic l-Functions

January 21 - April 29, 2003
Course on Automorphic Functions
Tuesdays, 1:30 pm- 3:00 pm
Instructor: H. Kim

This is a joint course with Ram Murty. The goal of this course is to give a proof of functoriality of symmetric cube and symmetric fourth of cusp form on GL(2), and their applications in number theory. We hope to cover the following
topics (mostly without proofs):

1. semi-simple algebraic groups and their properties (sketch)
2. cuspidal representations
3. induced representations
4. Eisenstein series
5. constant terms and intertwining operators
6. L-groups
7. L-functions in the constant terms
8. meromorphic continuation of L-functions (Langlands)
9. generic representations and their Whittaker models
10. local coefficients and local L-functions
11. non-constant terms and functional equations (Shahidi)
12. holomorphy and boundedness in vertical strips of L-functions
13. converse theorem of Cogdell and Piatetski-Shapiro
14. functoriality of symmetric cube
15. functoriality of symmetric fourth

January 23 - May 1, 2003
Course on Symmetric Power L-Functions And Applications To Analytic Number Theory
Thursdays, 10:30 am- 12:00 pm
Instructor: R. Murty

We will begin with a general overview of the Langlands program and then discuss the Sato-Tate conjecture, the Ramanujan conjecture, the Selberg eigenvalue conjecture, Artin's holomorphy conjecture and Langlands reciprocity conjecture. We will emphasize how the Langlands program proposes to solve each of these conjectures. We then apply the recent work of Kim and Shahidi on symmetric power L-functions to these conjectures as well as cognate questions in analytic number theory.

Course on L-functions, Converse Theorems, and Functoriality for GL(n)

January 21 - April 29, 2003 -- Tuesday 10:30 am - 12:00 pm
Instructor: Jim Cogdell (Oklahoma State)

The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the zeta-function. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL(2,Z) by Hecke and his school. Much of our current theory is a direct outgrowth of Hecke's. L-functions of automorphic representations were first developed by Jacquet and Langlands for GL(2). Their approach followed Hecke combined with the local-global techniques of Tate's thesis. The theory for GL(n) was then developed along the same lines in a long series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Shalika. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come from a modular form, the so called Converse Theorem of Hecke. In the context of automorphic representations, the Converse Theorem for GL(2) was developed by Jacquet and Langlands, extended and significantly strengthened to GL(3) by Jacquet, Piatetski-Shapiro, and Shalika, and then extended to GL(n).

In these lectures we hope to present a synopsis of this work and in doing so present the paradigm for the analysis of general automorphic L-functions via integral representations. We will begin with the classical theory of Hecke and then a description of its translation into automorphic representations of GL(2) by Jacquet and Langlands. We will then turn to the theory of automorphic representations of GL(n), particularly cuspidal representations. We will first develop the Fourier expansion of a cusp form and present results on Whittaker models since these are essential for defining Eulerian integrals.We will then develop integral representations for L-functions for GL(n) × GL(m) which have nice analytic properties (meromorphic continuation, boundedness in vertical strips, functional equations) and have Eulerian factorization into products of local integrals.

We next turn to the local theory of L-functions for GL(n), in both the archimedean and non-archimedean local contexts, which comes out of the Euler factors of the global integrals. We finally combine the global Eulerian integrals with the definition and analysis of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties.

We will then turn to the various Converse Theorems for GL(n). We will begin with the simple inversion of the integral representation. Then we will show how to proceed from this to the proof of the basic Converse Theorems, those requiring twists by cuspidal representations of GL(m) with m at most n-1. We will then discuss how one can reduce the twisting to m at most n-2. Finally we will consider what is conjecturally true about the amount of twisting necesssary for a Converse Theorem.

We will end with a description of the applications of these Converse Theorems to new cases of Langlands Functoriality. We will discuss both the basic paradigm for using the Converse Theorem to establish liftings to GL(n) and the specifics of the lifts from the split classical groups SO(2n+1), SO(2n), and Sp(2n) to the appropriate GL(N).

Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.

Financial Assistance

As part of the Affiliation agreement with some Canadian Universities, graduate students are eligible to apply for financial assistance to attend graduate courses. Interested graduate students must forward a letter of application with a letter of recommendation from their supervisor.
Two types of support are available:

  • Students outside the greater Toronto area may apply for travel support. Please submit a proposed budget outlining expected costs if public transit is involved, otherwise a mileage rate is used to reimburse travel costs. We recommend that groups coming from one university travel together, or arrange for car pooling (or car rental if applicable).

  • Students outside the commuting distance of Toronto may submit an application for a term fellowship. Support is offered up to $1000 per month. Send an application letter, curriculum vitae and letter of reference from a thesis advisor to the Director, Attn.: Course Registration, The Fields Institute, 222 College Street, Toronto, Ontario, M5T 3J1.

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