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.

*Cohen-Lenstra heuristics in the Presence of Roots of Unity*

The class group is a natural abelian group one can associated to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet to general number fields. However, their model was observed by Malle to have issues when the base field contains roots of unity. We explain that in this setting there are naturally defined additional invariants on the class group, and based on this we propose a refined model in the number field setting rooted in random matrix theory. Our conjecture is based on keeping track not only of the underlying group structure, but also certain natural pairings one can define in the presence of roots of unity. Specifically, if the base field contains roots of unity, we keep track of the class group G together with a naturally defined homomorphism G*[n] --> G from the n-torsion of the Pontryagin dual of G to G. Using methods of Ellenberg-Venkatesh-Westerland, we can prove some of our conjecture in the function field setting.

*Gonality** and the strong uniform boundedness
conjecture for periodic points*

*Periods, motivic Gamma functions, and
Hodge structures*

Golyshev and Zagier found an interesting new source of periods
associated to (eventually inhomogeneous) solutions generated by the Frobenius method for Picard Fuchs equations in the
neighborhood of singular points with maximum unipotent
monodromy. I will explain how this works, and how one can associate "motivic Gamma functions" and
generalized Beilinson style variations of mixed Hodge
structure to these solutions. This is joint work with M. Vlasenko.

*Galois
action on homology of Fermat curves*

*Weakly
commensurable Zariski-dense subgroups of semi-simple groups and isospectral locally symmetric spaces *

I will discuss the notion of weak
commensurability of Zariski-dense subgroups of
semi-simple groups. This notion was introduced in my
joint work with Andrei Rapinchuk (Publ. Math. IHES 109(2009),
113-184), where we determined when two Zariski-dense S-arithmetic subgroups of
absolutely almost simple algebraic groups over a field of characteristic zero
can be weakly commensurable. These results enabled us to prove that in many
situations isospectral locally symmetric spaces of
simple real algebraic groups are necessarily commensurable. This settled the
famous question "Can one hear the shape of a drum?" of Mark Kac for these spaces. The arguments use algebraic and
transcendental number theory.

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under the assumption that the relevant objects are everywhere unramified. Our formula relates the self-intersection number of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

*Local-to-global
principle for rational points on conic and quadric bundles over curves*

*When is an
error term not really an error term? *

The classical case of Iwasawa
theory has to do with how quickly the p-parts of the ideal class groups of
number fields grow in certain towers of number fields. I will discuss the
connection of these growth rates to Chern classes,
and how the "error terms" in various formulas have to do with higher Chern classes. I will then describe a result linking second
Chern classes in Iwasawa
theory over imaginary quadratic fields to pairs of p-adic
L-functions. This is joint work
with F. Bleher, R. Greenberg, M. Kakde,
G. Pappas, R Sharifi and M. Taylor.

*Kolyvagin's** conjecture on Heegner
points*

Second talk at 5:15 by Jordan Ellenberg (Wisconsin)

I will discuss some recent conjectures and results on the distribution of Mordell-Weil ranks and Selmer ranks of elliptic curves. After some general background, I will specialize to families of quadratic twists, and describe some recent results in detail.

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.

** 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.

**First talk at
4:00 by ****Venapally**** Suresh (Emory)**

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)**

Slides available at http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/pdf/atlanta2010.pdf

**Fall
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). **