**Instructor**: Jeff Thunder**Office**: WH 362/320**Phone**: 753-6725**e-mail**: jthunder@niu.edu**Office Hours**: Monday, Wednesday, Friday, 11:00-12:00 p.m. or by appointment.

The textbook for the course is *Number Theory in Function Fields*
by Michael Rosen. We will attempt to cover most of the material
from the first eight chapters, then move on to other stuff as time allows.

Grades for section 1 will be based on homework and the final exam. The weights for these are 70% and 30%, respectively.

Homework will be assigned somewhat irregularly; I will usually have one assignment per week. I'll announce assignments (and due dates) in class and also post them to this webpage. You are free to work with other students on the homework; in fact, this is encouraged. Sloppy and/or illegible work will be returned back with no credit! Your homework is something of which you should be proud (notice how I didn't end with a preposition there). Expect to spend lots of time on it.

- Week #1: Do at least 8 of the 16 exercises (your choice!) from Chapter 1, Section 1. This is due Wednesday, Jan. 23.
- Week #3: Do exercises 6-9 from Chapter 2 and read Chapters 3 and 4 on your own. This is due Monday, February 4.
- Week #4: Do the exercises in the handout on adeles (below). This is due Monday, February 11.
- Week #6: Do the exercises in the second handout on adeles (below). This is due Friday, February 22.
- Week #8: Do the exercises in the second handout on the Riemann-Roch Theorem. This is due the Wednesday after spring break, March 20.

- Here is the first section of the textbook (including the exercises) for those of you who don't have the text yet.
- This handout about places of number fields and function fields is recycled from last year's algebraic number theory course.
- Here is a brief writeup of the adele ring of a number field and function field.
- For those who have never seen it before, here is a fairly detailed writeup from a past MATH 430 class where the real numbers are constructed from the field of rational numbers via Cauchy sequences. The same method works almost word-for-word to topologically complete any field with an attached non-trivial absolute value.
- Last spring in MATH 681 we discussed the product formula for number fields and function fields.
- This handout goes into more depth on the topology of the adele ring and also has a few exercises.
- The Riemann-Roch Theorem is a fundamental result which has many consequences, just a few of which are addressed in this handout.
- Here is a proof of the Riemann-Roch Theorem together with an extension and an application, plus six homework exercises.
- This handout discusses the "Riemann Hypothesis" for function fields, along with some necessary background on the zeta function and L-function.
- The proof we give for the "Riemann Hypothesis" is rather lengthy; this discusses some linear algebra with fields and a useful application to estimating the number of places of degree 1.
- And this uses some group theory and Galois theory to finish off the proof of the Hasse-Weil Theorem (a.k.a. the "Riemann Hypothesis" for curves over a finite field).
- Finally, as a last hurrah we have a function field version of the classical Polya-Vinogradov Inequality.

Last update: May 1, 2019