Math 7315 (Number Theory in Function Fields), Fall 2021



Course Information
Instructor Class Times Office Hours
Evan Dummit
edummit at northeastern dot edu 		MR 6:00pm-7:30pm
Ryder 159 		W 3:00pm-5:00pm
via Zoom or in Lake 571
This course will provide a modern introduction to number theory in the setting of function fields. Here is the course syllabus.
Course Motivation: Classical elementary number theory consists of studying the properties of arithmetic in Q and its associated ring of integers Z (e.g., prime factorization, Fermat's little theorem, quadratic reciprocity, the prime number theorem, etc.). Algebraic and analytic number theory then expand this discussion to algebraic extensions K of Q and their associated rings of integers OK (e.g., unique factorization into prime ideals, Dirichlet's unit theorem, zeta functions and L-functions, etc.).

The primary aim of this course is to discuss the analogous stories of elementary number theory in the setting of the function field Fq[t] and of algebraic and analytic number theory in finite extensions of Fq[t]. There are a great many analogies between number theory over algebraic number fields and over algebraic function fields, and our goal is to elucidate (as much as we can) the connections between the two. Much of this material is rarely treated in graduate-level courses, despite its simplicity and its importance to modern number theory and algebraic geometry.
Course Topics: This course is billed as "From Fermat's Last Theorem to the Riemann Hypothesis" and this billing is meant quite literally: on the first day of class we will prove Fermat's Last Theorem, and on the last day of class we will prove the Riemann hypothesis. (Of course, these will be done in the function field setting!)
We will begin with a treatment of elementary number theory in Fq[t], studying primes and factorization, zeta functions, the power reciprocity law, and the analogue of Dirichlet's theorem on primes in arithmetic progression. We will then discuss algebraic function fields, valuations and primes, the dictionary between function fields and algebraic curves, differentials and divisors, Riemann-Roch, zeta functions, Galois theory of function fields, class groups, elliptic curves and elliptic function fields, cyclotomic function fields, L-functions, S-units, and other related topics as interest dictates. We will finish up with a treatment of the Weil conjectures for curves, culminating in a proof of the abc conjecture for function fields along with a proof of the Riemann hypothesis.
Textbook: I will generally follow Rosen's "Number Theory in Function Fields", but may supplement the discussion with various papers and material from other books, depending on student interest. (If you have difficulty obtaining a copy of the textbook, please contact the instructor.)
Background: There are no formal prerequisites, but students should have comfort with algebra and number theory at the level of Math 4527 or 5111 or 5112. I will freely refer to some results from elementary number theory, commutative algebra, algebraic geometry, Galois theory, and complex analysis, but the goal is to make the course as self-contained as possible.
Grades: Grades will be based on attendance and participation (70%), and on occasional homework assignments (30%), which will be collaborative and may involve roundtable discussions of the solutions. Depending on student and instructor interest, there may also be student presentations on course-related topics during the semester.


Lecture Notes
Current version of the lecture notes (through Lecture 24 on 12/9):
Notes, Lectures 1-24


Homework Assignments
Homework #1. Please prepare to present problems from this assignment in class on Thu Oct 7th.

Homework #2. Please prepare to present problems from this assignment in class on Mon Nov 8th.

Homework #3. Please prepare to present problems from this assignment in class on Thu Dec 9th. [Note: The second part of the problem from 0.1.5 is false as stated; try giving a counterexample and seeing if you can add additional hypotheses to make it true.]