The Joint Athens-Atlanta
Number Theory Seminar meets once a semester, usually on a Tuesday, with two
talks back to back, at 4:00 and at 5:15. Participants then go to dinner
together.
Spring 2012
Tuesday, April 10, 2012, at UGA, Room 323 in Boyd Graduate Studies Building
First talk at 4:00 by Max Lieblich (Univ. of
Washington)
Finiteness
of K3 surfaces and the Tate conjecture
Fix a finite field k. It is well known that there
are only finitely many smooth projective curves of a given genus over k. It
turns out that there are also only a finite number of abelian
varieties of a given dimension over k. What about other classes of varieties? I
will review the history of these results and describe joint work with Maulik and Snowden that links the finiteness of K3 surfaces
over k to the Tate conjecture for K3 surfaces over k. The key is a link between
certain lattices in the l-adic cohomology
of K3 surfaces and derived categories of sheaves on certain algebraic stacks. I
will not assume you know anything about any of this.
Second talk at 5:15 by Frank Calegari
(Northwestern Univ.)
Even
Galois Representations
What Galois representations "come" from
algebraic geometry? The Fontaine-Mazur conjecture gives a very precise
conjectural answer to this question. A simplified version of this conjecture in
the case of two dimensional representations says that "all nice
representations come from modular forms". Yet, by construction, all
representations coming from modular forms are "odd", that is, complex
conjugation acts by a 2x2 matrix of determinant -1. What happened to all the
even Galois representations?
Fall 2011
Wednesday, November 2, 2011, at Georgia Tech in Skiles room 005 (ground
floor).
First talk at 4:00 by Jared Weinstein (IAS)
Maximal
varieties over finite fields
This is joint work with Mitya
Boyarchenko. We construct a special hypersurface X over a finite field, which has the property
of "maximality", meaning that it has the
maximum number of rational points relative to its topology. Our variety
is derived from a certain unipotent algebraic group,
in an analogous manner as Deligne-Lusztig varieties
are derived from reductive algebraic groups. As a consequence, the cohomology of X can be shown to realize a piece of the
local Langlands correspondence for certain wild Weil
parameters of low conductor.
Second talk at 5:15 by David Brown (Emory)
Random Dieudonne modules and the Cohen-Lenstra
conjectures.
Knowledge of the distribution of class groups is
elusive -- it is not even known if there are infinitely many number fields with
trivial class group. Cohen and Lenstra noticed a
strange pattern --experimentally, the group $\mathbb{Z}/(9)$
appears more often than $\mathbb{Z{/(3) \times \mathbb{Z}/(3)$ as the 3-part of the classgroup
of a real quadratic field $\Q(\sqrt{d})$ - and
refined this observation into concise conjectures on the manner in which class
groups behave randomly. Their heuristic says roughly that $p$-parts of class
groups behave like random finite abelian $p$-groups,
rather than like random numbers; in particular, when counting one should weight
by the size of the automorphism group, which explains
why $\mathbb{Z}/(3) \times \mathbb{Z}/(3)$
appears much less often than $\mathbb{Z}/(9)$ (in
addition to many other experimental observations).
While proof of the Cohen-Lenstra conjectures remains
inaccessible, the function field analogue -- e.g., distribution of class groups
of quadratic extensions of $\mathbb{F}_p(t)$ -- is more tractable. Friedman and Washington
modeled the $\ell$-power part (with $\ell \neq p) of such class groups as random matrices and derived
heuristics which agree with experiment. Later, Achter
refined these heuristics, and many cases have been proved (Achter,
Ellenberg and Venkatesh).
When $\ell = p$, the $\ell$-power torsion of abelian
varieties, and thus the random matrix model, goes haywire. I will explain the
correct linear algebraic model -- Dieudone\'e
modules. Our main result is an analogue of the Cohen-Lenstra/Friedman-Washington
heuristics – a theorem about the distributions of class numbers of Dieudone\'e modules (and other invariants particular to
$\ell = p$). Finally, I'll present experimental evidence which mostly agrees
with our heuristics and explain the connection with rational points on
varieties.
Spring 2011
Tuesday, February 1, 2011
First talk at 4:00 by K. Soundararajan
(Stanford)
Moments of
zeta and L-functions
An important theme in number theory is to
understand the values taken by the Riemann zeta-function and related
L-functions. While much progress has been made, many of the basic questions
remain unanswered. I will discuss what is known about this question, explaining
in particular the work of Selberg, random matrix
theory and the moment conjectures of Keating and Snaith,
and recent progress towards estimating the moments of zeta and L-functions.
Second talk at 5:15 by Matthew Baker (Georgia Institute of Technology)
Complex
dynamics and adelic potential theory
I will discuss the following theorem: for any fixed
complex numbers a and b, the set of complex
numbers c for which both a and b both have finite orbit under iteration of
the map z -->z^2 + c is infinite if and only if a^2 = b^2. I will
explain the motivation for this result and give an outline of the proof.
The main arithmetic ingredient in the proof is an adelic
equidistribution theorem for preperiodic
points over product formula fields, with non-archimedean Berkovich spaces playing an essential role. This is joint
work with Laura DeMarco, relying on earlier
joint work with Robert Rumely.
Fall 2010
Tuesday, September 21, 2010
First talk at 4:00 by Ken Ono (Emory)
Mock modular
periods and L-values
Recent works have shed light
on the enigmatic mock theta functions of Ramanujan.
These strange power series are now known to be pieces of special
"harmonic" Maass forms. The speaker will
discuss recent joint work in the subject with regard to special values of
L-functions. This will include the study of values and derivatives of elliptic
curve L-functions, as well as general critical values of modular L-functions.
In addition, the speaker will derive new Eichler-Shimura
isomorphisms, and will derive new relations among the
"even" periods of modular L-functions. This is joint work with Jan Bruinier, Kathrin Bringmann, Zach
Kent, and Pavel Guerzhoy.
Second talk at 5:15 by Armand Brumer
(Fordham)
Abelian Surfaces
and Siegel Paramodular Forms
This expository talk will
survey recent progress on modularity of abelian
surfaces. After a brief review of the history, I'll describe work of Cris Poor and David Yuen on the modular side and Ken Kramer
and me on the arithmetic side.
Spring
2010
Tuesday, April 13, 2010
First talk at
4:00 by Venapally Suresh (Emory)
Degree three cohomology of function fields of
surfaces
Let
k be a global field or a local field. Class field theory says that every central division algebra over k is cyclic. Let l be a
prime not equal to the characteristic of k. If k contains a primitive l-th root of unity, then this leads to the fact that every
element in H^2(k, µ_l ) is a symbol. A natural
question is a higher dimensional analogue of this result: Let F be a function field in one variable over k which contains a
primitive l-th root of unity. Is every element in
H^3(F, µ_l ) a symbol? In this talk we answer
this question in affirmative for k a p-adic field or
a global field of positive characteristic. The main tool is a certain local
global principle for elements of H^3(F, µ_l ) in
terms of symbols in H^2(F µ_l ). We also show that this local-global
principle is equivalent to the vanishing of certain unramified
cohomology groups of 3-folds over finite fields.
Second talk at
5:15 by Antoine Chambert-Loir (IAS and University of Rennes)
Some applications of potential theory to number theoretical problems on
analytic curves
Slides available at http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/pdf/atlanta2010.pdf
Fall
2009
Tuesday, October 20, 2009
First talk at 4:00 by Doug Ulmer (GA Tech).
Constructing elliptic curves of
high rank over function fields
There are now several
constructions of elliptic curves of high rank over function fields, most
involving high-tech things like L- functions, cohomology,
and the Tate or BSD conjectures. I'll review some of this and then give a very
down-to-earth, low-tech construction of elliptic curves of high ranks over the
rational function field Fp(t).
Second talk at 5:15 by Jonathan Hanke (UGA).
Using Mass formulas to Enumerate Definite
Quadratic Forms of Class Number One
This talk will describe some
recent results using exact mass formulas to determine all definite quadratic
forms of small class number in n>=3 variables, particularly those of class
number one. The mass of a quadratic form connects the class number (i.e. number
of classes in the genus) of a quadratic form with the volume of its adelic stabilizer, and is explicitly computable in terms of
special values of zeta functions. Comparing this with known results about the
sizes of automorphism groups, one can make precise
statements about the growth of the class number, and in principle determine
those quadratic forms of small class number. We will describe some known
results about masses and class numbers (over number fields), then present some
new computational work over the rational numbers, and perhaps over some totally
real number fields.