MATH 780 Spring 2019 Section 1
- Instructor: Jeff Thunder
- Office: WH 362/320
- Phone: 753-6725
- Office Hours: Monday, Wednesday, Friday, 11:00-12:00 p.m. or
Text and Syllabus
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
- 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.
Disability Resource Center Statement
If you need an accommodation for this class, please contact the Disability Resource Center as soon as possible. The DRC coordinates accommodations for students with disabilities. It is located on the 4th floor of the Health Services Building, and can be reached at 815-753-1303 or firstname.lastname@example.org.
Also, please contact me privately as soon as possible so we can discuss your accommodations. Please note that you will not be required to disclose your disability, only your accommodations. The sooner you let me know your needs, the sooner I can assist you in achieving your learning goals in this course.
Homer does math!
Yes indeed, there's plenty of math humor to be found in the
look and see!
Last update: May 1, 2019