| University of Georgia | Department of Mathematics |
Polynomials and Symmetry
Introductory VIGRE Research Group
Fall 2010
Leader: David Swinarski
About this group
In this Introductory VIGRE Research Group (IVRG) we will hunt for polynomial equations of algebraic curves which possess unusual symmetries.
Every finite group G arises as the automorphism group of some smooth compact Riemann surface S (which I will refer to below as a complex algebraic curve C). However, the property of #G being large compared to the genus g of C is special. A general curve of genus g>=2 has no nontrivial automorphisms at all. There is an upper bound for the number of automorphisms due to Hurwitz, #G <= 84(g-1), but most curves with automorphisms do not even come close to this. For instance, in genus 3, 84(g-1)=168, but only five curves have #G > 25.
Using Fuchsian group theory it is possible to set up computer searches for groups which are large with respect to the genus. Researchers have carried this out and published lists of pairs (G,g) such that there is a curve of genus g with automorphism group G, but we don't know equations of all these curves. In other words, there is no algorithm yet that, given the group, produces equations of the curve.
In some small genus cases g=3,4,5, the representation of the group on differentials has been published; in general, we can get this from the Chevalley-Weil formula or Eichler trace formula if necessary. Then, using a mix of invariant theory, representation theory, and algebraic geometry, we can sometimes get equations for the curve. See the Week 5 and 6 lectures for examples.
We will try to do this for all the superlarge automorphism groups of genus 4, 5, and 6. (Here, superlarge means #G > 12(g-1).) I have written a program in the computer package Magma to help us decompose G-actions on polynomial rings. We will have a few lectures geared to helping you understand the formula it implements. I also plan to implement the Chevalley-Weil formula for you. If these tricks aren't enough, we can also try to hunt for equations over finite fields and lift them.
Lecture notes and Magma tutorials Click on the links below for the notes from that week.
Magma code
Contact information
Office: 436 Boyd Graduate Studies
Cell phone: (917) 733-3016
e-mail: davids@math.uga.edu
Time and place
We will have two meeting times each week: Monday, 5-6:15pm or Tuesday, 4-5:15pm, both in 640 Boyd (the Laboratory for Experimental Mathematics). You need only attend one day per week.
Bibliography
Several papers are available to you in eLC. To respect copyrights, I won't post them here.
Lists of automorphism groups
Here is a link to Marston Conder's website. He has found and published all the automorphism groups of genus 2 through 101 curves C satisfying #Aut(C) > 4(g-1). Simply amazing.
For the full list, see his website. Here are abbreviated versions that have all the information we are likely to need: Genus 2 through 5, Genus 2 through 10 .
At the Science Library
The following books are on reserve for you at the Science Library: