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 2017

Tuesday April 18, 2017, at Emory.

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

Galois action on homology of Fermat curves

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology.  Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve.  By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian.  Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

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.  


Fall 2016

Tuesday October 18, 2016, at UGA.

First talk at 4:00 by Florian Pop (University of Pennsylvania)

Local section conjectures and Artin-Schreier theorems

After a short introduction to (Grothendieck's) section conjecture (SC), I will explain how the classical Artin Schreier Thm and its p-adic analog imply the birational local SC; further, I will mention briefly local-global aspects of the birational SC. Finally, I will give an effective "minimalistic" p-adic Artin-Schreier Thm, which is similar in flavor to the classical Artin-Schreier Thm.

Second talk at 5:15 by Ben Bakker (UGA)

Recovering elliptic curves from their p-torsion

Given an elliptic curve E over the rationals Q, its p-torsion E[p] gives a 2-dimensional representation of the Galois group G_Q over F_p.  The Frey-Mazur conjecture asserts that for p>17, this representation is essentially a complete invariant:  E is determined up to isogeny by E[p].  In joint work with J. Tsimerman, we prove the analog of the Frey-Mazur conjecture over characteristic 0 function fields.  The proof uses the hyperbolic repulsion of special subvarieties in a modular surface to show that families of elliptic curves with many isogenous fibers have large volume.  We will also explain how these ideas relate to other uniformity conjectures about the size of monodromy representations.


Spring 2016

Thursday April 14, 2016, at Georgia Tech.

First Talk at 4:00 by Melanie Matchett-Wood (University of Wisconsin)

Nonabelian Cohen-Lenstra Heuristics and Function Field Theorems

The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and joint work with Boston proving cases of the non-abelian conjectures in the function field analog.

Second talk at 5:15 by Zhiwei Yun (Stanford University)

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).


Fall 2015

Tuesday November 17, 2015, at Emory.

First talk at 4:00 by John Duncan (Emory)

K3 Surfaces, Mock Modular Forms and the Conway Group

In their famous “Monstrous Moonshine” paper of 1979, Conway—Norton also described an association of modular functions to the automorphism group of the Leech lattice (a.k.a. Conway’s group). In analogy with the monstrous case, there is a distinguished vertex operator superalgebra that realizes these functions explicitly. More recently, it has come to light that this Conway moonshine module may be used to compute equivariant enumerative invariants of K3 surfaces. Conjecturally, all such invariants can be computed in this way. The construction attaches explicitly computable mock modular forms to automorphisms of K3 surfaces.

One expects the Brauer-Manin obstruction to control rational points on 1-parameter families of conics and quadrics over a number field when the base curve has genus 0. Results in this direction have recently been obtained as a consequence of progress inanalytic number theory. On the other hand, it is easy to construct a family of 2-dimensional quadrics over a curve with just one rational point over Q, which is a counterexample to the Hasse principle not detected by the \'etale Brauer-Manin obstruction. Conic bundles with similar properties exist over real quadratic fields, though most certainly not over Q.

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

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

One expects the Brauer-Manin obstruction to control rational points on 1-parameter families of conics and quadrics over a number field when the base curve has genus 0. Results in this direction have recently been obtained as a consequence of progress inanalytic number theory. On the other hand, it is easy to construct a family of 2-dimensional quadrics over a curve with just one rational point over Q, which is a counterexample to the Hasse principle not detected by the \'etale Brauer-Manin obstruction. Conic bundles with similar properties exist over real quadratic fields, though most certainly not over Q.


Spring 2015

Thursday April 9, 2015, at UGA, Room 304 in Boyd Graduate Studies Building.

First talk at 4:00 by Dick Gross (Harvard)

Pencils of quadrics

Quadric hypersurfaces, defined by homogeneous equations of degree 2, are the simplest projective varieties other than linear subspaces. In this talk I will review the theory of quadratic forms over a general field, and discuss the smooth intersection of two quadric hypersurfaces in projective space. The Fano scheme of maximal linear subspaces contained in this intersection is either finite or is a principal homogeneous space for the Jacobian of a hyperelliptic curve.  This gives an important tool for the arithmetic study of these 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.


Fall 2014

Tuesday, November 4, 2014, at Georgia Tech.

First talk at 4:00 by Arul Shankar (Harvard University)

Geometry-of-numbers methods over number fields

We discuss the necessary modifications required to apply Bhargava's geometry-of-numbers methods to representations over number fields. As an application, we derive upper bounds on the average rank of elliptic curves over any number field. This is joint work with Manjul Bhargava and Xiaoheng Jerry Wang.

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

Kolyvagin's conjecture on Heegner points

We recall a conjecture of Kolyvagin on Heegner points for elliptic curves of arbitrary analytic rank, and present some recent results on this conjecture for elliptic curves satisfying some technical conditions.


Spring 2014

Tuesday, February 25, 2014, at Emory.

First talk at 4:00 by Paul Pollack (UGA)

Solved and unsolved problems in elementary number theory

This will be a survey of certain easy-to-understand problems in elementary number theory about which "not enough" is known. We will start with a discussion of the infinitude of primes, then discuss the ancient concept of perfect numbers (and related notions), and then branch off into other realms as the spirit of Paul Erdös leads us.

Second talk at 5:15 by James Maynard (Université de Montréal)

Bounded gaps between primes

It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer $m$ there should be infinitely many sets of $m$ primes, with each set contained in an interval of size roughly $m\log{m}$. We will introduce a refinement of the `GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer $m$, and so there are infinitely many bounded length intervals containing $m$ primes.


Fall 2013

Tuesday, November 5, 2013, at UGA, Room 303 in Boyd Graduate Studies Building.

First talk at 4:00 by Joe Rabinoff (Georgia Tech)

Lifting covers of metrized complexes to covers of curves

Let K be a complete and algebraically closed non-Archimedean field and let X be a smooth K-curve.  Its Berkovich analytification X^an deformation retracts onto a metric graph Gamma, called a skeleton of X^an.  The collection of different skeleta of X are in natural bijective correspondence with the collection of semistable models of X over the valuation ring R of K.  We prove that, given a finite morphism of curves f: Y -> X, there exists a skeleton Gamma_X of X^an whose inverse image is a skeleton Gamma_Y of Y^an.  This can be seen as a "skeletal" simultaneous semistable reduction theorem, which can in fact be used to give new, more general proofs of foundational results of Liu, Liu-Lorenzini, and Coleman on simultaneous semistable reductions.  We then consider the following problem: given X, a skeleton Gamma_X, and a finite harmonic morphism of metric graphs Gamma' -> Gamma_X, can we find a curve Y and a finite morphism f: Y -> X such that f^{-1}(Gamma_X) is a skeleton of Y^an and is isomorphic to Gamma'?  In general the answer is no: one must enrich the skeleton with the structure of a metrized complex of curves.  In this context the answer is yes, and moreover the map Gamma' -> Gamma_X can be used to calculate the finitely many isomorphism classes of Y -> X, as well as their automorphisms.  We give an application to component groups of Jacobians, answering a question of Ribet.

Second talk at 5:15 by Kirsten Wickelgren (Georgia Tech)

Splitting varieties for triple Massey products in Galois cohomology

The Brauer-Severi variety a x^2 + b y^2 = z^2 has a rational point if and only if the cup product of cohomology classes associated to a and b vanish. The cup product is the order-2 Massey product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information carried in a differential graded algebra which can be lost on passing to the associated cohomology ring. For example, the cohomology of the Borromean rings is isomorphic to that of three unlinked circles, but non-trivial Massey products of elements of H^1 detect the more complicated structure of the Borromean rings. Analogues of this example exist in Galois cohomology due to work of Morishita, Vogel, and others. This talk will first introduce Massey products and some relationships with non-abelian cohomology. We will then show that b x^2 = (y_1^2 - ay_2^2 + c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 - 2 a y_2 y_4)^2 is a splitting variety for the triple Massey product <a,b,c>, and that this variety satisfies the Hasse principle. The method could produce splitting varieties for higher order Massey products. It follows that all triple Massey products over global fields vanish when they are defined. More generally, one can show this vanishing over any field of characteristic different from 2; Jan Minac and Nguyen Duy Tan, and independently Suresh Venapally, found an explicit rational point on X(a,b,c). Minac and Tan have other nice results in this direction.  This is joint work with Michael Hopkins.


Spring 2013

Tuesday, April 16, 2013, at Georgia Tech

First talk at 4:00 by Dick Gross (Harvard)

The arithmetic of hyperelliptic curves  

Hyperelliptic curves over Q have equations of the form y^2 = F(x), where F(x) is a polynomial with rational coefficients which has simple roots over the complex numbers. When the degree of F(x) is at least 5, the genus of the hyperelliptic curve is at least 2 and Faltings has proved that there are only finitely many rational solutions. In this talk, I will describe methods which Manjul Bhargava and I have developed to quantify this result, on average.

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

Arithmetic statistics over function fields

What is the probability that a random integer is squarefree? Prime? How many number fields of degree d are there with discriminant at most X? What does the class group of a random quadratic field look like? These questions, and many more like them, are part of the very active subject of arithmetic statistics. Many aspects of the subject are well-understood, but many more remain the subject of conjectures, by Cohen-Lenstra, Malle, Bhargava, Batyrev-Manin, and others. In this talk, I explain what arithmetic statistics looks like when we start from the  field Fq(x) of rational functions over a finite field instead of the field Q of rational numbers. The analogy between function  fields and number  fields has been a rich source of insights throughout the modern history of number theory. In this setting, the analogy reveals a surprising relationship between conjectures in number theory and conjectures in topology about stable cohomology of moduli spaces, especially spaces related to Artin's braid group. I will discuss some recent work in this area, in which new theorems about the topology of moduli spaces lead to proofs of arithmetic conjectures over function fields, and to new, topologically motivated questions about counting arithmetic objects.


Fall 2012

Thursday, October 25, 2012, at Emory.

First talk at 4:00 by Karl Rubin (UCI)

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 Jayce Getz (Duke)

An approach to nonsolvable base change for GL(2)

Motivated by Langlands' beyond endoscopy idea, the speaker will present a conjectural trace identity that is essentially equivalent to base change and descent of automorphic representations of GL(2) along a nonsolvable extension of fields.


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, at Emory

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, at UGA

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.