Course Descriptions

16:640:571 - Number Theory I

Fall 2024

Matthew Young

Course Description:

This course will cover the analytic theory of L-functions with a focus on families and moments.  Topics to be covered include:

Definitions of L-functions
Constructions of L-functions (Dirichlet characters, modular forms, Rankin-Selberg, etc.)
Approximate functional equation and the explicit formula
Poincare series and the Petersson formula
Families of L-functions
Bounds and asymptotics for moments of L-functions
Additional topics as time permits

Text:

Analytic Number Theory, by Iwaniec and Kowalski

Prerequisite:

Standard undergraduate math curriculum should be sufficient

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Fall 2023

Michael Woodbury

Course Description:

ll cover various topics in the theory of modular forms. Potential topics include computational aspects of modular forms and representation theoretical approaches to automorphic forms on GL(n).

Text:

TBD

Prerequisite:

None

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Fall 2022

Jerrold Tunnell

Course Description:

This will be a introductory course in Algebraic Number Theory. The subject matter of the course should be useful to students in areas of algebra and discrete mathematics, which often have a number theoretic component to problems, as well as students in number theory and algebraic geometry. Basic properties of number fields (field extensions of the rational numbers of finite degree) will be introduced -- rings of integers, ideal classes, units groups, zeta functions, reciprocity laws,adele ring, group of ideles. The relation of these invariants to Diophantine problems of finding rational solutions to collections of polynomial equations will be developed. Examples such as quadratic and cyclotomic fields will be studied. Galois extensions of number fields, Chebotarev density theorem and L-functions will be introduced.

Topics will include:

1. Number fields, lattices and rings of integers

2. Dedekind domains and their ideals and modules

3. Ideal class groups and Class number

4. Zeta functions of number fields

5. Quadratic fields and binary forms

6. Cyclotomic fields and Gauss sums

7. Diophantine problems and algebraic number theory

8. Algorithms in number theory

9. Adeles and Ideles of number fields

10.Chebotarev Density Theorem

Text:

no text

Prerequisite:

graduate algebra and complex analysis courses

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Fall 2021

Stephen Miller

Subtitle:

Automorphic representations

Course Description:

This course will cover the theory of automorphic representations for SL(2,R) from the analytic point of view, stressing Maass forms and their L-functions. Time permitting, we may explore the analogous p-adic theory due to Jacquet-Langlands.

Text:

Automorphic Forms for SL(2,R), by Armand Borel

Prerequisite:

First year core courses

 

Schedule of Sections:

 

Previous Semesters