In 2008, the Number Theory/Arithmetic Geometry Seminar meets, unless indicated otherwise, on Wednesday at 3:45 in room 304.

Other seminars in the area on related subjects include Emory and Clemson.

August 27: Dino Lorenzini  (UGA), The index of a variety over a discrete valuation field.

This is joint work with Qing Liu. I will explain a formula to compute the index of a variety over a discrete valuation field K using a regular model of the variety over the ring of integers of K. The proof uses a moving lemma which may be of independent interest.

September 3:  Daniel Krashen (UGA),  The index of a central simple algebra. 

 

September 10: Jonathan Hanke (UGA), Class Numbers and Mass Formulas for Quadratic Forms.

 

This talk is an introduction to class numbers of a quadratic forms, why they are important, and various techniques for understanding and computing them.  Our main approach will be to study them via the related local notion of the "mass" of a quadratic form.  This allows one to obtain asymptotic results about class numbers by careful local computations, and knowledge of special values of (twists of) Dedekind zeta functions. We will discuss how these techniques can be used to understand class numbers of individual quadratic forms, and computational techniques to enumerate all positive definite forms of small class number. These ideas can also be expressed more generally in the context of a linear algebraic group, and similar results should hold in that context as well.

September 17: Pete Clark  (UGA), Quadratic twists, modular curves and the Inverse Galois Problem.

We will discuss a geometric approach to (parts of the!) Inverse Galois Problem pioneered by K.-y. Shih, further developped by J.-P. Serre and applied by the speaker.  The problem is a meeting place for a cornucopia of number-theoretic themes and issues: modular curves (and possibly other Shimura varieties), Selmer groups of elliptic curves in quadratic twist families, twisting and arithmetic surfaces, class groups of quadratic fields, Galois groups...The emphasis here will be on identifying open problems and possible avenues of further work (e.g. thesis work).

September 24: Robert Rumely (UGA), Berkovich space and dynamics on Berkovich space.

This talk will begin with a description of the Berkovich projective line over a complete, algebraically closed nonarchimedean field, and the way that a rational function acts on the Berkovich projective line.  It will then compare the dynamics of a rational function on the  classical projective line over the complex numbers, and on the Berkovich projective line, focusing on the theory of periodic points and Fatou-Julia theory. 

 
October 1:   Robert Rumely  (UGA), Nonarchimedean potential theory and dynamical applications .

This talk will describe the Laplacian on the Berkovich projective line, and will use it to construct the ``canonical measure'' associated to a rational function of degree at least 2, which is analogous to the classical invariant measure constructed by Brolin, Lyubich, and Freire-Lopes-Mane.  It will then discuss equidistribution theorems relative to this measure, and use them to derive structural information about the Berkovich Julia set.

October 8:   Jim Stankewicz  (UGA), The possible torsion subgroups of CM elliptic curves.
 
A celebrated theorem of Mazur gives the complete list of 
possible torsion subgroups of elliptic curves over the rational numbers
and work of Kamienny, Kenku and Momose gives a complete list for
quadratic number fields. By implementing an algorithm given by Dr. Pete
Clark the UGA Number Theory VRG developed the tools to calculate the
lists for CM elliptic curves in degree up to 9. In this talk, expect to
see the lists and some other results which came about in the study of
this problem.

 

October 15:   Neil Lyall (UGA), Arithmetic Combinatorics .
 
I hope to give a very accessible introduction to the field of arithmetic combinatorics.
 

 

October 22:  Matthew Smith (UGA), Roth's Theorem on Arithmetic Progressions

 

I will give an overview of Roth's 1953 theorem that a subset of the natural numbers of positive upper density necessarily contains arithmetic progressions of length 3, or triples of the form {a,a+d,a+2d} for d nonzero, and how this result may be proved using a modification of the Hardy-Littlewood circle method.  If time permits, I will also give an overview of Roth's Theorem in the finite field setting.

 

October 29:  Mariah Hamel  (UGA), Roth's theorem on arithmetic progressions in random sets.

In this talk we will give a Fourier analytic proof of a version of Roth's theorem in random sets.

 
November 5: Pete Clark  (UGA), Introduction to Probabilistic Methods in Number Theory (Part I?)
 
I will argue that it is natural to analyze questions about the distribution of arithmetic quantities (e.g. prime numbers, number of points on varieties over finite fields, Hecke eigenvalues) using probabilistic reasoning.  Sometimes the established results of probability theory can be literally applied to obtain number-theoretic c onsequences.  More often though the situation is less cut-and-dried: since the "probability spaces" in question are usually countably infinite, the basic assumption of Kolmogorov's axiomatization modern probability theory -- namely the existence of a countably additive measure -- is often not appropriate.  We will focus on one particularly striking example, the idea  that on the "space" of positive integers the family of events "being divisible by p" (as p ranges over all primes) is "independent".  It is not hard to see that this cannot be literally true, but nevertheless it is "true enough" for it to be profitable to be familiar with some elements of the theory of independent, identically distributed random variables: there are some striking number theoretic applications here, up to and including a probabilistic interpretation of the Riemann hypothesis.
 
 

November 12:  Shahed Sharif  (Duke University),  Tate-Shafarevich and Brauer groups for split elliptic threefolds over C

 

Let Y and Z be two elliptic surfaces over a common base X. Consider the fiber product Y \times_X Z as an elliptic fibration over Z. Under certain hypotheses, we show how to compute the Tate-Shafarevich and Brauer groups of this threefold. This is joint work with Chad Schoen.

 
 
November 19: Michael Rosen ( Brown University), Class numbers not divisible by a fixed prime.

Let F be a finite field with q=p^f  elements and F(T) the rational function field. In 1999, H. Ichimura proved that there are infinitely many quadratic extensions K of F(T) whose class number is not divisible by 3. We will generalize this result in two directions. Let l be a prime number different from the characteristic and m an integer not divisible by l. We show that there are infinitely many extensions K of F(T) of degree m whose class number is not divisible by l. It is not known if there is an analogous theorem in the number field case.
This is joint work with Allison Pacelli and four of her undergraduate students.

 
December 10: Jennifer Paulhus (Kansas State University), Decomposing Jacobian varieties of curves.
 

Jacobian varieties of curves will sometimes factor into products of smaller dimensional abelian varieties.  Many questions can be asked about these  factors. How many isogenous elliptic curves can the Jacobian of a curve of a certain genus have as factors?  Do the factors share special arithmetic properties?  Are the factors of dimension at least 2 themselves Jacobian varieties?   We will first discuss how to find, given a curve, a decomposition of its Jacobian using the automorphism group, G, of the curve and idempotents in the group ring Q[G].  Then we will talk about partial answers to some of the questions above.

 

 

 

 

Archived seminars: 2006, 2007