Math 204C - Number Theory

Course description: This is the third in a string of three courses, which is an introduction to algebraic and analytic number theory. Part A treated the basics of number fields (their rings of integers, failure of unique factorization, class numbers, the Dirichlet unit theorem, splitting of primes, cyclotomic fields, and more). Part B focused on local fields and local properties of number fields (completions of number fields, finite extensions of local fields, ramification, different and discriminant, decomposition and inertia groups, basics of local class field theory). Part C will focus on zeta functions. The exact topics and order of presentation will be decided in consultation with the audience; the topics will most likely be a subset of the following.

The Riemann zeta function and the prime number theorem
Dirichlet L-functions and primes in arithmetic progressions
Dedekind zeta functions, class number formulas
Artin L-functions, Chebotarev density theorem
Adelic interpretations of zeta functions ("Tate's thesis")
The Riemann hypothesis and its consequences
Stepanov's proof of the Riemann hypothesis for function fields
Zeta functions for higher-dimensional schemes, the Weil conjectures
Computational techniques for zeta functions
External applications (cryptography, combinatorial geometry, etc.)

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours Tu 1-2 (note change), or by appointment, in APM 7202.

Lectures: MWF 1-1:50, in APM 7421; Tu 10-11, in APM 6402. See below for modifications to the schedule.

Textbook: No specific text. See lecture notes (and references) on the topics list.

Prerequisites: Math 204A or Math204B or equivalent with instructor's permission. I will be flexible about this, adapting the presentation based on the audience.

Homework: Informal. I will assign some exercises, but I do not expect to collect or grade them.

Final exam: None.

Grading: Let me know if you need a meaningful grade for this course.

Announcements:

First lecture: Monday, March 30. Last lecture: Friday, May 22.
No lectures: May 6 or May 8.
On April 14, we will meet in APM 5218 instead of APM 6402.

Topics covered so far (with notes):

March 30: overview, the Riemann zeta function (notes).
April 1: the prime number theorem (notes from 2007)
April 3: Dirichlet L-functions (notes from 2007, more notes from 2007)
April 6: prime number theorem in arithmetic progressions (notes from 2007)
April 7: Dedekind zeta functions (notes)
April 8: class number formula for Dedekind zeta functions (Janusz, Algebraic Number Fields, 4.2)
April 10: zeta functions for function fields (notes)
April 13: demonstration of L-Functions and Modular Forms Database (link to LMFDB).
April 14: Artin L-functions (notes).
April 15: more on function fields (notes).
April 17: RH for function fields, part 1 (notes, not yet finished; also Lorenzini, Chapter X)
April 20: RH for function fields, part 2
April 21: Tate's thesis: local multiplicative and additive characters (Cassels-Fröhlich XV.2.1-2)
April 22: Tate's thesis: local zeta functions (Cassels-Fröhlich XV.2.3-4)
April 24: Tate's thesis: sign of the functional equation; restricted direct products (Cassels-Fröhlich XV.2.5, 3.1-2)
April 27: Tate's thesis: measures on restricted direct products, additive lattice property (Cassels-Fröhlich XV.3.3, 4.1)
April 28: Tate's thesis: Poisson summation formula, multiplicative lattice property, global zeta functions (Cassels-Fröhlich XV.4.2-4.4)
April 29: Tate's thesis: proof of the main theorem, application to classical zeta functions (Cassels-Fröhlich XV.4.4-5)
May 1: Statement of the Weil conjectures (notes to follow)
May 4: Comparison between ell-adic (etale) and p-adic (rigid) Weil cohomologies
May 5: Survey of algorithms for computing zeta functions over finite fields
May 6: Guest lecture by René Schoof on computing zeta functions of elliptic curves over finite fields ("Schoof's algorithm")
May 8: NO MEETING
May 10: p-adic method for computing zeta functions of hyperelliptic curves
May 11: continuation
May 12: Explicit formulas relating primes to zeta zeroes (Davenport, Multiplicative Number Theory 17)
May 14: Statement of the Bombieri-Vinogradov theorem (notes from 2007)
May 17: Hardy-Littlewood prime k-tuples conjecture (notes from 2007), Selberg sieve (notes from 2007)
May 18: Short gaps between primes after Maynard (Maynard's preprint)
May 19: continuation
May 20: continuation
May 22: Dwork's proof of the rationality of the zeta function (Dwork-Gerotto-Sullivan, Introduction to G-Functions Chapter 2)