FA Seminar 2018

Field Arithmetic Seminar

We meet on Wednesdays in Schreiber 210 at 16:00.

Limits of the Diagonal Cartan Subgroup in SL(n,R) and SL(n, Q_p) (25/10/2017)

Arielle Leitner (Technion)

A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan

subgroup, C of SL(n) in the Chabauty topology. Over R, the group C is naturally associated to a projective n-1 simplex. We can compute the conjugacy limits of C by collapsing the n-1 simplex in different ways. In low dimensions, we enumerate all possible ways of doing this. In higher dimensions, we show there are infinitely many non-conjugate limits of C.

In the Q_p case, SL(n) has an associated p+1 regular affine building. (We'll give a gentle introduction to buildings in the talk). The group C stabilizes an apartment in this building, and limits are contained in the parabolic subgroups stabilizing the facets in the spherical building at infinity. There is a strong interplay between the conjugacy limit groups and the geometry of the building, which we exploit to extend some of the results above. The Q_p part is joint work in progress with Corina Ciobotaru.

Residual Girth of Groups and Algebras (01/11/2017)

Be'eri Greenfeld (Bar-Ilan University)

Residual girth of finitely generated (residually finite) groups measures how large are finite quotients into which the n-ball of the Cayley graph injects. While for a general group the residual girth may grow arbitrarily fast, bounds are known for special classes of groups (e.g. nilpotent, linear etc.) .

We study an analogous notion for algebras. We compare the residual girth of a group with that of its group algebra, showing they need not coincide. We prove that for special classes of algebras (e.g. representable algebras) the residual girth is asymptotically equivalent to the growth in the sense of the GK-dimension. Examples are constructed to show that without special assumptions on the algebra, its residual girth can be arbitrarily fast.

Intersection of finitely generated (Galois) groups (08/11/2017)

Mark Shusterman (Tel-Aviv University)

Howson's classical theorem says that the intersection of two finitely generated subgroups of a free group is finitely generated.

Hanna Neumann conjectured a bound on the number of generators of the intersection, that after many years of works, has been established independently by Friedman and Mineyev.

I will discuss the history of this problem, surveying the (elementary) proof techniques. I will then report on an extension of the results to Demushkin groups (a class of groups that is of great importance in Galois theory). My proof has to do with another classical problem called Kaplansky's zero-divisor conjecture.

No preliminaries are assumed beyond basic familiarity with the free group.

The talk is based on joint works with Andrei Jaikin-Zapirain, and Pavel Zalesskii.

Brauer classes supporting an involution (15/11/2017)

Uriya First (Haifa University)

The construction of the Brauer group of a field can be generalized to (commutative) rings, and more generally to schemes, by replacing central simple algebras with Azumaya algebras. As in the case of fields, the Brauer group is an important cohomological invariant of the scheme, appearing, for instance, in the Manin obstruction for rational points.
Many of the properties of central simple algebras generalize to Azumaya algebras, but sometimes modifications are needed. For example, Albert characterized the central simple algebras admitting an involution of the first kind as those whose Brauer class is 2-torsion. While this fails for Azumaya algebras over a ring R, Saltman showed that the 2-torsion classes in the Brauer group of R are precisely those containing some representative admitting an involution of the first kind. Knus, Parimala and Srinivas later gave a quantitative version of this statement: If A is an Azumaya algebra of over R such that its Brauer class is 2-torsion, then there exists an Azumaya algebra in the Brauer class of A that admits an involution and has degree 2*deg(A).

In this talk, we shall recall what are Azumaya algebras and how the Brauer group of a ring (or a scheme) is constructed. Then we will present a recent work with Asher Auel and Ben Williams where we use topological obstruction theory to show that the quantitative result of Knus, Parimala and Srinivas cannot be improved in general. Specifically, there are Azumaya algebras of degree 4 whose Brauer class if 2-torsion, but such that any algebra that is Brauer-equivalent to them and admits an involution has degree at least 8 = 2*4.

Spectral methods: from surfaces to graphs and back (22/11/2017)

Konstantin Golubev (Hebrew University)

A number of methods from algebraic graph theory were influenced by the spectral theory of Riemann surfaces. We pay it back, and take some classical results for graphs to the continuous setting. In particular, I will talk about colorings, average distance, and discrete random walks on surfaces. Based on joint works with E. DeCorte and A. Kamber.

Congruent modular forms (29/11/2017)

Devika Sharma (Weizmann Institute)

We introduce congruences in modular forms, and see how they lead to the notion of a Hida family. We also review the theory of deformations of Galois representations. We use these concepts to understand the (local) behavior of the p-adic Galois representation attached to a (p-ordinary) modular form.

Dirichlet's theorem in function fields (06/12/2017)

Arijit Ganguly (Tel-Aviv University)

I will talk about metric Diophantine approximation for function fields, specifically the problem of improving Dirichlet’s theorem in Diophantine approximation. This is a joint work with Anish Ghosh.

Finite growth order of Eisenstein series (13/12/2017)

Evgeny Musicantov (Tel-Aviv University)

Using an effective version of Bernstein’s continuation principle, combined with Bott-Chern interpretation of Nevanlinna characteristics function as the integrated ﬁrst Chern’s class of a certain hermitian holomorphic line bundle, we establish ﬁnite growth order of spherical Eisenstein series attached to SL2(Z) \ SL2(R).

Variance of sums of two squares in short intervals, and fractional divisor functions (20/12/2017)

Ofir Gorodetsky (Tel-Aviv University)

Landau's Theorem tells us that the density of positive integers up to X which are sums of two squares is proportional to 1/sqrt(ln(X)). A much harder problem is to understand the distribution of sums of two squares in the short interval [x,x+Delta(x)] where x is "random" (varies in [1,X]), Delta grows with x, and X goes to infinity.

Recent works of Smilansky and Freiberg-Kurlberg-Rosenzweig assumed unproven conjectures to show that the moments (of the aforementioned random variable), normalized suitably, converge to those of a Poisson-distributed variable. They treated only very short intervals (Delta grows like square root of ln(x)).

I will explain how one can obtain unconditional results in the analogous setting of the polynomial ring over a large finite field using monodromy calculations of Katz and an approach pioneered by Keating and Rudnick. Our methods give the variance of the distribution, and allow Delta(x) to grow like x^epsilon for any epsilon in (0,1).

This is joint work with Brad Rodgers.

Algebraic entropy in compactly covered locally compact groups (27/12/2017)

Menachem Shlossberg (University of Udine)

A topological group G is called compactly covered if each element of G is contained in some compact subgroup of G. We study the algebraic entropy in compactly covered locally compact groups (cclc for short) that satisfy certain cofinality conditions. Two important subclasses which we consider are:

1)Compactly covered locally compact abelian groups. This subclass contains for example all compact abelian groups, all LCA torsion groups and the p-adic numbers Q_p.

2)Discrete locally finite and normal groups. A group G is called locally finite and normal if each finite subset F is contained in a finite normal subgroup N of G. Among these groups are direct sums of finite groups, their subgroups, and quotient groups. The Hamiltonian groups, that is, the non-abelian groups in which every subgroup is normal are also locally finite and normal.

This is a joint work (in progress) with Anna Giordano Bruno and Daniele Toller.

Super-positivity of a family of L-functions (03/01/2018)

Bingrong Huang (Tel-Aviv University)

Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self-dual L-functions" which specifies that all derivatives of the completed L-function at the central value s=1/2 should be non-negative. The Riemann Hypothesis implies super-positivity for self-dual cuspidal automorphic L-functions on GL(n). This talk is based on recent joint work with Dorian Goldfeld where we prove, for the first time, that there are infinitely many L-functions associated to modular forms for SL(2,Z) each of which has the super-positivity property.

Maximal Pro-p Galois Groups and Kummer Theory (08/01/2018)

Ido Efrat (Ben-Gurion University)

The extrinsic primitive torsion problem (17/01/2018)

Patrick Hooper (City College New York)

We consider images of the free group F with two generators where images of all primitive elements have order k, for some positive integer k. We are interested in questions like when (in terms of k) the image must be finite, finitely generated, nilpotent, etc. Geometric motivation comes from understanding square tiled surfaces (punctured torus covers with a flat structure). I'll explain what we are able say about the answers to these questions, discuss the techniques we use, and mention some interesting open questions. This is joint work with Khalid Bou-Rabee.

Degenerate Whittaker Models for Cuspidal Representations of GL(kn,F)

(14/03/2018)

Zahi Hazan (Tel-Aviv University)

We present a generalization of a theorem by D. Prasad (2000), where he computed the twisted Jacquet module of an irreducible cuspidal representation of GL(2n,F), with respect to the "Shalika" character of the unipotent radical of the parabolic subgroup of type (n,n). Prasad computed this module as a GL(n,F)-module. We studied the twisted Jacquet module of an irreducible cuspidal representation of GL(kn,F), with respect to a "regular" character of the unipotent radical of the parabolic subgroup of type (n,n,...,n) (k times) and analyzed it as a GL(n,F)-module.

We have obtained a compact description of the GL(n,F)-representation obtained from the twisted Jacquet module above. Our description of the module involves a power of the Steinberg representation of GL(n,F), which in particular, sheds new light on Prasad's description. We use techniques from diverse disciplines: such as q-Hypergeometric series, combinatorics and linear algebra. In particular, the calculation of the dimension of the module leads to a deep problem on matrices and q-Hypergeometric identities.

This is a joint work with Ofir Gorodetsky.

Number of Points on Curves over Finite Fields

(21/03/2018)

Patrick Meisner (Tel-Aviv University)

Several authors have studied the probability that random curves over finite fields have a certain number of points. The families of curves most studied are those whose function field is an abelian Galois extension with sufficiently many roots of unity. We will present these results as well as a new work with Bary-Soroker on what happens when we remove the assumption on roots of unity.

Irreducibility of integral polynomials with a large gap

(11/04/2018)

Mark Shusterman (Tel-Aviv University)

Providing irreducibility criteria for integral polynomials is by now a classical topic. Yet, the irreducibility of "most" polynomials cannot be established using the existing techniques, and many problems remain open.

For example, establishing the irreducibility of random polynomials, and the irreducibility of various trinomials.

We will be interested in polynomials with only few nonzero coefficients, located "near the ends" (that is, a large gap in the middle). Focusing on a particular case, the family of polynomials X^{2k+1} -7X^2 + 1, we show how work of Schinzel (and generalizations by Bombieri-Zannier using unlikely intersections) imply irreducibility for large enough k. We then discuss work by Filaseta-Ford-Konyagin which makes this reasonably effective.

At last, we present our improved bounds that allow to obtain irreducibility for every k. We finish with some applications and conjectures.

This is a joint work with Will Sawin and Michael Stoll.

Restriction and Induction of phi-Gamma Modules

(25/04/2018)

Gal Porat (Hebrew University)

Let L be a non-archimedean local field of characteristic 0. We present a variant of the theory of (\varphi,\Gamma)-modules associated with Lubin-Tate groups, developed by Kisin and Ren, in which we replace the Lubin-Tate tower by the maximal abelian extension \Gamma=\mathrm{Gal}(L^{ab}/L). This variation allows us to compute the functors of induction and restriction for (\varphi,\Gamma)-modules, when the ground field L changes.

In the talk we will focus on the case of characteristic-p coefficients for simplicity.

No prior background will be assumed except for the basic theory of local fields.

This is a joint work with Ehud de Shalit.

Branching laws for non-generic representations

(30/05/2018)

Max Gurevich (National University of Singapore)

The celebrated Gan-Gross-Prasad conjectures aim to describe the branching behavior of representations of classical groups, i.e., the decomposition of irreducible representations when restricted to a lower rank subgroup.

These conjectures, whose global/automorphic version bear significance in number theory, have thus far been formulated and resolved for the generic case.

In this talk, I will present a newly formulated rule in the p-adic setting (again conjectured by G-G-P) for restriction of representations in non-generic Arthur packets of GL_n.

Progress towards the proof of the new rule takes the problem into the rapidly developing subject of quantum affine algebras. These techniques use a version of the Schur-Weyl duality for affine Hecke algebras, combined with new combinatorial information on parabolic induction extracted by Lapid-Minguez.

I will try to give an introductory point of view on some of the ingredients involved, for audience without prior background.