Number Theory Working Seminar


Winter 2011

Purpose: The goal of this seminar is to understand Galois-theoretic aspects of maps between curves, starting with Belyi's theorem that on any curve defined over a number field there is a nonconstant rational function which has at most three branch points.

Meetings:
1/26/11Mike Zieve Belyi's theorem
2/2/11Mike Zieve Refinements of Belyi's theorem
2/9/11Alex Mueller Descent, I
2/16/11Sijun Liu abc implies effective Mordell
2/23/11Mike Zieve Descent, II
3/30/11Mike Zieve The converse of Belyi's theorem
4/6/11Zach Scherr Preimages of branch loci of Belyi maps
4/13/11Alex Mueller Dessins d'enfants
4/20/11Julian Rosen Geometric Galois actions
5/12/11Zach Scherr Belyi's theorem in positive characteristic


Participants: Hunter Brooks, Sijun Liu, Alex Mueller, Julian Rosen, Zach Scherr, Ben Weiss, Jeremy West, Mike Zieve



Summer 2010

Purpose: The goal of this seminar is to explain background material used in the research projects of Alex Carney, Ruthi Hortsch, and Ben Weiss. This includes topics from algebraic topology, Riemann surfaces, group theory, Galois theory, algebraic geometry, elliptic curves, height functions, etc.

Lecture notes: Here are some unpolished lecture notes. I (M.Z.) would welcome any feedback!

Meetings:
6/24/10Mike Zieve Riemann's existence theorem
6/28/10Alex Carney and Ruthi Hortsch Polynomials and group theory
7/15/10Mike Zieve Primitive and multiply transitive permutation groups
7/21/10Mike Zieve Generic maps between curves
8/10/10Mike Zieve Hurwitz spaces
8/18/10Mike Zieve Ramification and Puiseux series
8/26/10Mike Zieve Fibered products


Participants: Hunter Brooks, Alex Carney, Nic Ford, Ruthi Hortsch, Jeff Lagarias, Dylan Moreland, Alex Mueller, Julian Rosen, Zach Scherr, Ari Shnidman, Ben Weiss, Mike Zieve



Winter 2010 (b)

Purpose: The goal of this seminar is to introduce the various concepts and results which will be used in Matt Baker's Michigan Lectures in Number Theory.

Meetings:
4/2/10Alex Mueller and Julian Rosen The Berkovich projective line
4/9/10Sijun Liu and Zach Scherr Canonical heights and Bilu's equidistribution theorem
4/16/10Hunter Brooks and Ari Shnidman Green's functions and the Mandelbrot set


Participants: Hunter Brooks, Sijun Liu, Alex Mueller, Kartik Prasanna, Julian Rosen, Zach Scherr, Ari Shnidman, Ben Weiss, Mike Zieve



Winter 2010 (a)

Purpose: The first goal of this seminar is to understand the proof of the Ax-Katz theorem, which gives a lower bound on the power of p dividing the number of points on certain affine varieties over Fpn. The second goal is to understand Alex Mueller's improvement of the Ax-Katz bound, and to examine whether further improvements are possible.

Meetings: Alex Mueller led meetings on 3/22/10 and 3/29/10.

Participants: Sijun Liu, Alex Mueller, Julian Rosen, Zach Scherr, Mike Zieve



Fall 2009

Purpose: The goal of this seminar is to help students gain working knowledge of class field theory by using it to solve some concrete problems. As a bonus, it will likely lead to one or more publishable papers.

Meetings: Meetings involve a combination of student lectures on background material and group discussion of aspects of the problem. The topics and discussion leaders are:

11/12/09Mike Zieve Motivation and introduction to the problem
11/19/09Alex Mueller Decomposition of places in function field extensions
11/24/09Julian RosenThe Carlitz module
12/4/09
AllExamples


Topic: Find abelian extensions of Fq(x) having more degree-one places than any other known function fields of the same genus over the same constant field. More generally, analyze which combinations of conductor and splitting places yield the best function fields of this type. If there is time and interest, consider the same problems for towers of abelian extensions of function fields.

Participants: Alex Mueller, Hieu Ngo, Julian Rosen, Zach Scherr, Ari Shnidman, Sun Xinyun, Ben Weiss, Mike Zieve