Please visit the new page at https://research.franklin.uga.edu/agant/joint-athens-atlanta-number-theory-seminar

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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.

TBA

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### Second talk at 5:15 by Jacob Tsimerman (U. Toronto)

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.

### First talk at 4:00 by Bjorn Poonen (MIT)

Gonality and the strong uniform boundedness conjecture for periodic points

### Second talk at 5:15 by Spencer Bloch (U. Chicago)

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.

### First talk at 4:00 by Rachel Pries (Colorado State)

Galois action on homology of Fermat curves

### Second talk at 5:15 by Gopal Prasad (U. Michigan)

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.

### Intersection numbers and higher derivatives of L-functions for function fields

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).

### Second talk at 5:15 by Alexei Skorobogatov (Imperial College)

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

### Second talk at 5:15 by Ted Chinburg (University of Pennsylvania)

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.

### Second talk at 5:15 by Wei Zhang (Columbia University)

Kolyvagin's conjecture on Heegner points

### Ranks of elliptic curves

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.

### 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.

### 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)

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).