NTS Fall 2013/Abstracts

September 5

Guillermo Mantilla-Soler (EPFL)

Title: The spinor genus of the integral trace and local arithmetic equivalence

Abstract: In this talk I'll explain my recent results on the spinor genus of the integral trace form of a number field. I'll show how from them one can decide in terms of finitely many ramification invariants, and under some restrictions, whether or not a pair of number fields have isometric integral trace forms. Inspired by the work of R. Perlis on number fields with the same zeta function I'll define the notion of local arithmetic equivalence, and I'll show that under certain hypothesis this equivalence determines the local root numbers of the number field, and the isometry class of integral trace form.

September 12

Simon Marshall (Northwestern)

Title: Endoscopy and cohomology growth on U(3)

Abstract: I will use the endoscopic classification of automorphic forms on U(3) to determine the asymptotic cohomology growth of families of complex-hyperbolic 2-manifolds.

September 19

Valerio Toledano Laredo (Northeastern)

Title: From Yangians to quantum loop algebras via abelian difference equations

Abstract: For a semisimple Lie algebra g, the quantum loop algebra and the Yangian are deformations of the loop algebra g[z, z − 1] and the current algebra g[u], respectively. These infinite-dimensional quantum groups share many common features, though a precise explanation of these similarities has been missing so far. In this talk, I will explain how to construct a functor between the finite-dimensional representation categories of these two Hopf algebras which accounts for all known similarities between them. The functor is transcendental in nature, and is obtained from the discrete monodromy of an abelian difference equation canonically associated to the Yangian. This talk is based on a joint work with Sachin Gautam.

September 26

Haluk Şengün (Warwick/ICERM)

Title: Torsion homology of Bianchi groups and arithmetic

Abstract: Bianchi groups are groups of the form SL(2, R) where R is the ring of integers of an imaginary quadratic field. They form an important class of arithmetic Kleinian groups and moreover they hold a key role for the development of the Langlands program for GL(2) beyond totally real fields. In this talk, I will discuss several interesting questions related to the torsion in the homology of Bianchi groups. I will especially focus on the recent results on the asymptotic behavior of the size of torsion, and the reciprocity and functoriality (in the sense of the Langlands program) aspects of the torsion. Joint work with N. Bergeron and A. Venkatesh on the cycle complexity of arithmetic manifolds will be discussed at the end. The discussion will be illustrated with many numerical examples.

October 3

Andrew Bridy (Madison)

Title: The Artin–Mazur zeta function of a Lattes map in positive characteristic

Abstract: The Artin–Mazur zeta function of a dynamical system is a generating function that captures information about its periodic points. In characteristic zero, the zeta function of a rational map from P1 to P1 is known to always be a rational function. In positive characteristic, the situation is much less clear. Lattes maps are rational maps on P1 that are finite quotients of endomorphisms of elliptic curves, and they have many interesting dynamical properties related to the geometry and arithmetic of elliptic curves. I show that the zeta function of a separable Lattes map in positive characteristic is a transcendental function.

October 10

Bogdan Petrenko (Eastern Illinois University)

Title: Generating an algebra from the probabilistic standpoint

Abstract: Let A be a ring whose additive group is free Abelian of finite rank. The topic of this talk is the following question: what is the probability that several random elements of A generate it as a ring? After making this question precise, I will show that it has an interesting answer which can be interpreted as a local-global principle. Some applications will be discussed. This talk will be based on my joint work with Rostyslav Kravchenko (University of Chicago) and Marcin Mazur (Binghamton University).

October 17

Anthony Várilly-Alvarado (Rice)

Title: Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups

Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X --> P^1 such that A is "vertical" for this map. This implies, for example, that if there is a cohomological obstruction to the existence of a point on X, then there is a genus-one fibration X --> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of "seeing" a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X. Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray.

October 24

Paul Garrett (Minnesota)

Title: Spectra of pseudo-Laplacians on spaces of automorphic forms

Abstract: Faddeev–Pavlov and Lax–Phillips observed that certain restrictions of the Laplacian to parts of automorphic continuous spectrum have discrete spectrum. Colin de Verdiere used this to prove meromorphic continuation of Eisenstein series, and proposed ways to exploit this idea to construct self-adjoint operators with spectra related to zeros of L-functions. We show that simple forms of this construction produce at most very sparse spectra, due to incompatibility with pair correlations for zeros. Ways around some of the obstacles are sketched. (Joint with E. Bombieri.)

October 31

Jerry Wang (Princeton)

Title: Pencils of quadrics and the arithmetic of hyperelliptic curves

Abstract: In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over Q of genus g have no points over any odd degree extension of Q. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser–Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages.

November 7

who? (where?)

Title: tba

Abstract: tba

November 12

Evan Dummit (Madison)

Title: Counting extensions of number fields of given degree, bounded (rho)-discriminant, and specified Galois closure

Abstract: A very basic question in algebraic number theory is: how many number fields are there? A natural way to order the fields of a fixed degree n is by discriminant, and classical results of Minkowski then assure us that there are only finitely many fields with a given discriminant. We are also often interested in counting number fields, or relative extensions, with other properties, such as having a particular Galois closure. A folk conjecture sometimes attributed to Linnik states that the number of extensions of degree n and absolute discriminant less than X is on the order of X. A great deal of recent and ongoing work has been focused towards achieving upper bounds on counts of this nature (quite successfully, in degree 5 and lower), but there is comparatively little known in higher degrees, for relative extensions, or for sufficiently complicated Galois closures: the primary results are those of Schmidt and Ellenberg-Venkatesh. I will discuss these results and my thesis work, in which I generalize several of their results and introduce another counting metric, the "rho-discriminant".

November 21

Michael Lipnowski (Duke)

Title: tba

Abstract: tba

November 26

Daniel Kane (Stanford)

Title: Diffuse decompositions of polynomials

Abstract: We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q1, ..., qm so that the joint probability density function of q1(G), ..., qm(G) is close to being bounded. This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the qi. We then present some applications of this result.

December 5

Jennifer Park (MIT)

Title: tba

Abstract: tba

December 12

Vivek Shende (Berkeley)

Title: Equidistribution on the space of rank two vector bundles over the projective line

Abstract: I will discuss how the algebraic geometry of hyperelliptic curves gives an approach to a function field analogue of the 'mixing conjecture' of Michel and Venkatesh. (For a rather longer abstract, see the arxiv posting of the same name as the talk). This talk presents joint work with Jacob Tsimerman.

Organizer contact information

Robert Harron

Sean Rostami

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