## Math 847: Algebraic Curves and Varieties over Finite Fields

taught by Melanie Matchett Wood

## Time and Place:

MWF 9:55-10:45am, Van Vleck B129
#### There will be no class on Mon Sep 5 (Labor Day).

## Office Hours:

Thursday 3:40-5 (after number theory seminar), Van Vleck 315, or by appointment (email me)

## Course Content:

The emphasis will be on algebraic curves over finite fields, and the methods will include both number theoretic perspectives through the number theory of their function fields and geometric perspectives. The course will quickly "review" necessary background in algebraic number theory and algebraic geometry, when needed. Hence the prerequisites are algebraic number theory and some algebraic geometry, especially of algebraic curves, OR a willingness to learn either/both of these along with the course. For students who do not have background in algebraic geometry, the course content should provide both motivation and examples for this. For students who do not have background in number theory, a knowledge of algebraic curves will help them quickly learn the number thory background they need. The course will start with foundational topics on curves over finite fields, such as the study of their points and zeta functions. In the second half of the semester, we will cover current research topics in the area.
There will be no textbook, thoough references will be suggested as necessary.

## Expectations/Grading:

Each student will be required to (and graded on) providing latexed notes
for three days of lecture. Each day of lecture notes must also contain an (additional) example written by the student of something from that class.
Students will sign-up in class for which days they will be responsible for notes, and the notes will be due via email to the instructor by the start of class time one week from the lecture.
Please use the following latex template for your notes.
Here is the sign-up for who is taking notes what day.

## Tentative Course Plan:

(Note: the topics do not correspond to lectures, some lectures will include multiple topics and some topics will take multiple lectures.)
### Foundational Topics

*Finite fields
*Plane curves over an arbitrary field

*Field valued points on plane curves

*Base field extension

*Zeta functions of curves over finite fields

*Statement of the Weil conjectures and resulting bounds on number of points on curves

*Analogy of number theory and affine curves

*Points and topology of affine curves

*Residue fields, localization, and completion

*Integral closure, normality, and smoothness

*Gluing affine curves

*Class groups, line bundles, divisors, and linear systems

*Curves in higher dimensional projective spaces

*Riemann-Roch

*Sheaf cohomology

*Genus and the Riemann-Hurwitz formula

### Current Research Topics

*Bertini theorems over finite fields
*Average number of points on curves over a fixed finite field

*Applications of the Lefschetz trace formula to cohomology of moduli spaces