Math 675 — Analytic theory of numbers

This is a first course in analytic number theory. The methods of analytic number theory are useful and applicable in many areas of mathematics. It will consider multiplicative number theory and properties of primes as a focus. It will emphasize methods. It will start with real analysis methods following Tenenbaum, arithmetic functions, prime number theory, hyperbola method, Euler-Maclaurin summation. For complex analysis methods and Dirichlet series it will follow Davenport and Montgomery. It will include basic theory of Riemann zeta function and Dirichlet L-functions to primes in arithmetic progressions. If time permits the course may include other topics, Selberg sieve, large sieve and a topic in probabilistic number theory.

Recent history:

• Fall 2013: Bob Vaughan
• Fall 2015: Jeff Lagarias
• Fall 2017: Wei Ho
• Fall 2019: Jeff Lagarias

Math 676 — Theory of algebraic numbers

This course covers basic algebraic number theory, including rings of integers, extensions of prime ideals, unique factorization of prime ideals, finiteness of the ideal class group, the structure of the unit group, decomposition and inertia groups in relation to prime splitting, absolute values, localizations, and completions. Applications will be given to diophantine equations, recursive sequences, and computing Galois groups of polynomials.

Recent history:

• Fall 2012: Mike Zieve
• Fall 2014: Mike Zieve
• Fall 2016: Kartik Prasanna | course page
• Fall 2018: Mike Zieve | course page
• Fall 2020: Mike Zieve

Math 677 — Diophantine problems

Recent history:

• Winter 2015: Mike Zieve

Math 678 — Modular forms

This course is a topics course, and so the subject changes each offering.

Recent history:

• Winter 2014: Kartik Prasanna
• Fall 2014: Jeff Lagarias
• Winter 2016: Andrew Snowden | Iwasawa theory | course page
• Fall 2017: Jeff Lagarias | Modular forms | course page
• Fall 2018: Kartik Prasanna | Motivic L-functions
• Fall 2019: Michał Zydor | Modular forms
• Fall 2020: Karol Koziol | Representation theory of p-adic groups

Math 679 — Elliptic curves

This course is a topics course, and so the subject changes each offering.

Recent history:

• Fall 2013: Andrew Snowden | Mazur's theorem | course page
• Winter 2017: Bhargav Bhatt | Perfectoid spaces | course page
• Winter 2018: Wei Ho | Arithmetic of surfaces
• Winter 2019: Tasho Kaletha | Automorphic forms
• Winter 2020: Serin Hong | p-adic Hodge theory

Math 775 — Topics in analytic number theory

Recent history:

• Winter 2014: Bob Vaughan
• Winter 2016: Hugh Montgomery
• Winter 2018: Hugh Montgomery
• Winter 2020: Wei Ho | Distributions of arithmetic invariants

Math 776 — Class field theory

Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century. It brings together, in a unified fashion, the quadratic and higher reciprocity laws of Gauss, Legendre et al, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.

This course is typically offered every other year in the winter term.

Recent history:

• Winter 2013: Jeff Lagarias | course page
• Winter 2015: Andrew Snowden | course page
• Winter 2017: Jeff Lagarias
• Winter 2019: Andrew Snowden | course page