### Math 8280: Topics in Number Theory (2017--2018)

#### Instructor:

Dr. Dihua Jiang
Office: VinH 224, Telephone: 625--7532, E-mail: jiang034@umn.edu
#### Lectures:

Lecture: 2:30--3:20pm, MWF at Vincent Hall 301 (Office Hours: by appointment)
#### Course Description:

This is a two-semester course for Automorphic L-functions and the Langlands Functoriality.

The Langlands functoriality conjecture is a profound core conjecture in the Langlands program, which

is also the name for the modern theory of automorphic forms. Automorphic L-functions are intrinsic invariants

attached to a given automorphic forms or representations, which also serve as bridges connecting the modern

theory of automorphic forms to number theory, algebraic geometry, representation theory and harmonic analysis

over certain locally compact topological spaces.

In this course, we will discuss the theory of automorphic L-functions from the point of view from

harmonic analysis, in particular, following the approach of the Rankin-Selberg method. Meanwhile, we

discuss as many known cases as possible and try to understand them from the guideline suggested by

Braverman and Kazhdan. The whole theory separates into the global part and the local part. Hence some part

of the representation theory of reductive groups over local fields (complex, real, p-adic) will be reviewed.

The local theory of the Rankin-Selberg method will be discussed with enough details.

In the global theory, we will discuss the roles of generalized Fourier transforms and Poisson summation formula

played in the study of automorphic L-functions. Meanwhile, we may also discuss the explicit construction of

certain types of the Langlands functorial transfer by automorphic integral transforms, via Fourier coefficients

automorphic forms.

Basic reference:

1) Moeglin and Waldspurger: Spectral Decomposition and Eisenstein Series. Cambridge University Press, 1995.

2) Arthur: The Endoscopy Classification of Representations: orthogonal and symplectic groups. AMS Colloquium 61, 2013.

3) Some relevant research papers.

4) TBA

#### Homework and Exams:

Homework Problems will be assigned, but no exams are required. Students may give reports to the class.