# NTS/Abstracts

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## Aug 28

 Robert Lemke Oliver The distribution of 2-Selmer groups of elliptic curves with two-torsion Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6. In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun.

## Sep 04

 Patrick Allen Unramified deformation rings Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari.

## Sep 11

 Melanie Matchett Wood The distribution of sandpile groups of random graphs *** The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian. An Erdős–Rényi random graph then gives some distribution of random abelian groups. We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution. In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture. *** This is officially a probability seminar, but will occur in the usual NTS room B105 at a slightly earlier time, 2:25 PM.

## Sep 18

 Takehiko Yasuda Distributions of rational points and number fields, and height zeta functions In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function.

## Sep 25

 Ramin Takloo-Bigash Counting orders in number fields and p-adic integrals In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech).

## Oct 02

 Pham Huu Tiep Nilpotent Hall and abelian Hall subgroups To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group.

## Oct 09

 Michael Woodbury An Adelic Kuznetsov Trace Formula for GL(4) An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications. In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4). I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem.

## Oct 16

 Robert Grizzard Small points and free abelian groups Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. We prove the failure of the converse to this statement by explicitly constructing counterexamples. This is joint work with Philipp Habegger and Lukas Pottmeyer.

## Oct 23

 Tonghai Yang A conjecture of Colmez In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable) height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula. In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, Φ) in terms of the log derivative at s=1 of some `Artin' L-function associated to the CM type Φ. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport.

## Oct 30

 Laura DeMarco Elliptic curves and complex dynamics I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields. On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P1.

## Nov 06

 Michael Magee Zero sets of Hecke polynomials on the sphere The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.

## Nov 13

 Yiwei She The Shafarevich conjecture for K3 surfaces Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture.

## Nov 20

 Simon Marshall Endoscopy and cohomology of unitary groups We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others. We will show how this can be used to prove asymptotics for the L2 Betti numbers of families of locally symmetric spaces.

## Dec 04

 Joel Specter Commuting Endomorphisms of the p-adic Unit Disk When can a pair of endomorphisms of $\mathbf{Z}_p[[X]]/\mathbf{Z}_p$ commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over $\mathbf{Z}_p$. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin.

## Dec 11

 Ila Varma The mean number of 3-torsion elements in ray class groups of quadratic fields In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over $\mathbb{Q}$. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava.

## Organizer contact information

Sean Rostami (srostami@math.wisc.edu)