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Field Arithmetic Seminar

We meet on Wednesdays in  Kaplun 118 at 16:00.

2-coverable Groups and Intersective Polynomials (02/11/2016)

Joachim Konig (Technion)

We present new theoretical results on the existence of intersective polynomials (that is, integral polynomials that have a root mod every n, but do not have a rational root) with certain prescribed Galois groups, namely the projective and affine linear groups PGL_2(â„“) and AGL_2(â„“), as well as the affine symplectic groups AGSp_4(â„“):=(F_â„“)^4â‹ŠGSp_4(â„“). For further families of affine groups, existence results are proven conditional on the existence of certain tamely ramified Galois extensions of the rationals. We also compute explicit families of intersective polynomials with certain non-solvable Galois groups.

Semi-free subgroups of profinite surface groups (16/11/2016)

Matan Ginzburg (Tel-Aviv University)

We prove that every infinite index normal subgroup of a profinite surface group is contained in a normal semi-free profinite subgroup. This generalizes a result of Bary-Soroker, Zalesskii and Stevenson, in a direction suggested by their work. Time permitting, we will discuss work in progress on some generalizations and analogues for Hilbertian fields.

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This is a joint work with Mark Shusterman.

Elementary solution to primes of the form x^2+ny^2 (23/11/2016)

Ohad Avnery (Open University)

Using class field theory, Cox gave a characterization of the primes p represented by the quadratic form x^2+ny^2. The characterization has to do with an irreducible integral polynomial f_n and its roots modulo p.

 

We give a new proof of this result, inspired by explicit class field theory and formulas for x and y. For us, the polynomial f_n is just the minimal polynomial of the j-invariant of a suitably chosen elliptic curve.

 

The proof also gives us an explicit formula for x in the representation of p as x^2 + ny^2, as an exponential sum over F_p (or equivalently, as the normalized number of points on an elliptic curve E over F_p). This simplifies and generalizes formulas of Gauss, Eisenstein, and others.
 

This is a joint work with Mark Shusterman.

Explicit formulas for the numbers of prime polynomials in short intervals (30/11/2016)

Ofir Gorodetsky (Tel-Aviv University)

Gauss was the first to determine the number of prime polynomials in F_q[T] of degree n, establishing the Prime Polynomial Theorem. A natural extension of this problem is to count prime polynomials with first L coefficients prescribed. We describe the values of q and L for which it is possible to give a closed form formula for this counting question, by relating it to supersingularity of certain curves. This extends works of Carlitz and Kuzmin.

Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case (07/12/2016)

Gianluca Paolini (Hebrew University)

We give a quick overview of homogeneous model theory, and then see how to apply this machinery to the case of right-angled Coxeter groups. Central to this is the notion of strong rigidity of such groups, which will be explored.

Asymptotically Uniform Period Distribution for Binary Expansions of Rational Numbers (14/12/2016)

Gal Porat (Hebrew University)

The binary expansion of a rational number q is periodic, and if we choose at random k consecutive digits from a period, we see that q induces a probability measure on binary strings of length k. We show that there exists a density one sequence of primes, such that the sequence of measures induced by their reciprocals converges to the uniform measure for every k.

 

This result and some others are obtained by basic methods of Fourier analysis and analytic number theory, using results of Pappalardi and Kurlberg.

 

This is a joint work with Guy Kapon carried out under the supervision of Ofir David and Uri Shapira.

Approximation of groups, characterizations of sofic groups, and equations over groups (21/12/2016)

Lev Glebsky (Universidad Autonoma de San Luis Potosi)

Recently, a family of groups called sofic (coming from the word 'sofi' in Hebrew) has been defined. These groups possess many nice properties, and various widely open problems have an answer for them.

Amazingly, not a single example of a non-sofic group is known. I will present a couple of new characterizations of soficity, and discuss some applications to equations and approximations over groups.

Rational functions with algebraic constraints (28/12/2016)

Elad Levi (Hebrew University)

A polynomial P(x,y) over an algebraically closed field k has an algebraic constraint if the set

{(P(a,b),(P(a′,b′),P(a′,b),P(a,b′)|a,a′,b,b′∈k}

does not have the maximal Zariski-dimension. Tao proved that if P has an algebraic constraint then it can be decomposed: there exists Q,F,G∈k[x] such that P(x,y)=Q(F(x)+G(y)), or P(x,y)=Q(F(x)⋅G(y)). We will discuss the generalisation of this result to rational functions with 3-variables and show the connection to a problem raised by Hrushovski and Zilber regarding 3-dimensional indiscernible arrays.

Howson's Theorem for (pro-p) Surface groups (11/01/2017)

Mark Shusterman (Tel-Aviv University)

A classical result of Howson says that the intersection of two finitely generated subgroups of a free group is finitely generated.

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This result has been generalized to surface groups by Greenberg,  to limit groups by Dahmani, and established in many other cases as well. Furthermore, the problem of bounding the number of generators of the intersection received much attention, culminating in the groundbreaking works of Friedman, Mineyev, and Jaikin-Zapirain.


I will prove Howson's theorem for various pro-p groups: pro-p completions of surface groups, maximal pro-p extensions of p-adic fields, maximal pro-p quotients of etale fundamental groups, Sylow subgroups of the absolute Galois group of a global field, and other pro-p groups of arithmetic origin.

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We will then see that the proof gives bounds on the number of generators of the intersection in discrete groups such as Fuchsian groups, thus improving upon the bounds of Soma in certain cases.
 

This is a joint work with Pavel Zalesskii.

Rank gradient of sequences of subgroups in a direct product

(18/01/2017)

Tzvika Shemtov (Tel-Aviv University)

The Reidemeister-Schreier process shows that the number of generators of any finite index subgroup U of a finitely generated group G is at most (d(G)-1)[G : U] + 1. We define the rank gradient of a sequence G_i of finite index subgroups of a finitely generated group G to be


inf_i (d(G_i) - 1)/[G : G_i]

 

thus measuring the strictness of the aforementioned inequality.

We will prove that excluding few trivial cases,

the rank gradient of sequences in direct products is zero.

 

This is a joint work with Nikolay Nikolov and Mark Shusterman.

Linked Fields of Characteristic 2 and their u-Invariant

(25/01/2017)

Adam Chapman (Tel-Hai College)

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field  of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time.

We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker.

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You can find the paper at: https://arxiv.org/abs/1701.01367

Grothendieck-Teichmüeller theory from a topological viewpoint

(15/03/2017)

Pierre Lochak (Universite Paris Pierre et Marie Curie)

I will explain the main features of what has come to be known as Grothendieck-Teichmueller theory, insisting on the technical similarity and differences between the different versions. I will also introduce a new approach which stresses profinite graphs and simplices sets rather than groups. Finally I will try to keep the prerequisites minimal.

Free Subalgebras, Graded Algebras and Nilpotent Elements

(22/03/2017)

Be'eri Greenfeld (Bar-Ilan University)

The famous Koethe conjecture asserts that the sum of two nil left ideals is always nil. This still open problem, which is sometimes considered the central open problem in ring theory has attracted many researchers and inspired a flurry of results toward a better understanding of its validity.

 

Its most popular equivalent formulation nowadays is, that the polynomial ring R[x] over a nil ring R is equal to its own Jacobson radical.

The observation that R[x] is naturally graded, and every homogeneous element is nilpotent (i.e. R[x] is "graded nil") motivated L. Small and E. Zelmanov to ask ('06) whether a graded nil algebra is always equal to its Jaocbson radical.

This was disproved by A. Smoktunowicz a few years ago, and should be mentioned together with another result by Smoktunowicz, disproving a conjecture of L. Makar-Limanov: she proved that there exists a nil ring R such that after tensoring with central variables (specifically: R[x_1,...,x_6]) it contains a free subalgebra. Such ring can exist only over countable base fields.

 

In this talk we present a new construction, which provides a monomial, graded nilpotent ring (a stronger property than graded nil) which contains a free subalgebra. Our methods involve combinatorics of infinite words, and gluing together sequences of letters which arise from appropriate morphisms of free monoids.

In particular, this resolves Small-Zelmanov's question and can be thought of as a continuation of Smoktunowicz's counterexample to Makar-Limanov's conjecture (since in our construction the base field can be arbitrary).

 

We also construct finitely generated graded Golod-Shafarevich algebras in which all homogeneous elements are nilpotent of bounded index, and prove that such phenomenon cannot appear in monomial algebras.

 

The talk is based on a joint work with Jason P. Bell.

Groups without finite parametric sets over Q

(29/03/2017)

Francois Legrand (Technion)

Given a finite group G, we consider in this talk "parametric sets'' (over the field Q of rational numbers), that is, sets S of (regular) Galois extensions of Q(T) with Galois group G whose specializations provide all the Galois extensions of Q with Galois group G. This is related to the Beckmann-Black Problem (that asks whether the strategy of solving the Inverse Galois Problem by specializations is optimal) which can be formulated as follows: does a given finite group G have a parametric set over Q?
We show that many finite groups G do not have a finite parametric set over Q.

 

This is a joint work with Joachim König.

Galois groups of local fields, Lie algebras and ramification

(19/04/2017)

Victor Abrashkin (Durham University)

Rel leaves of the Arnoux-Yoccoz surfaces

(26/04/2017)

Barak Weiss (Tel-Aviv University)

In a joint work with Hooper, we solved a geometrical problem regarding the existence of a dense leaf in a foliation of the moduli space of holomorphic one-forms called the "rel foliation". One of the inputs to the proof is an algebraic result of Bary-Soroker, Shusterman and Zannier about a certain sequence of number fields. The reduction to the algebraic question relies on the breakthroughs of Eskin and Mirzakhani, for which Mirzakhani was awarded a Fields medal in 2014.

 

I will give a quick introduction to the geometry and sketch some of the ideas of the proof, including a (not completely formal) statement of the Eskin-Mirzakhani theorem and related results of Mirzakhani and Wright.

 

The talk is intended for a wide audience.

Quasi-formations of finite groups

(03/05/2017)

Sela Fried (Tel-Aviv University)

We define quasi-formations that generalize formations of finite groups and show that for every quasi-formation C there exists an (up to an isomorphism) unique pro-C group of at most countable rank with the embedding property, that has C as its class of finite images. This group is an analogue of a free pro-C group.

 

We will apply our theory to the absolute Galois group of the maximal pro-solvable extension of Q under the assumption that Shafarevich conjecture is valid, namely that the absolute Galois group of the maximal abelian extension of Q is the free profinite group on a countable set. 

 

The content of the lecture is part of my dissertation written under the supervision of Dan Haran.

Algebraic geometry - topological methods in the classification of surfaces

(10/05/2017)

Meirav Topol Amram (Shamoon College)

I will give a review of the subject. I will present the steps of the classification of surfaces, using very nice methods and techniques, such as: degeneration of surfaces, braid monodromy, calculations of fundamental groups and Coxeter groups. We will see interesting examples of classification of known and significant surfaces, such as Hirzebruch surfaces.

​Subgroup growth in Heisenberg groups over number rings

(17/05/2017)

Michael Schein (Bar-Ilan University)

Let G be a finitely generated group, and let a_n be the number of normal subgroups of G of index n, which is always finite.  The zeta function Z(s) = \sum a_n n^{-s} counts the finite index subgroups of G and has been an object of active study for the past 25 years.  The zeta function splits into an Euler product of local factors, and in some cases these factors possess a striking symmetry (a functional equation).  It is an interesting and deep problem to explain this symmetry in terms of the algebraic properties of G.

 

We consider the special case of a Heisenberg group over a number ring.  Let K be a number field with ring of integers O.  The Heisenberg group H(O) consists of upper triangular matrices with entries in O and ones on the diagonal.  We consider the local zeta factors of the group H(O).  If p is either unramified or non-split in K, we have explicit formulae for these factors, expressed as a sum of terms parametrized by Dyck words of length 2[K:Q].  The local zeta factor at any prime p appears to satisfy a functional equation that depends on the ramification of p in K.  This is joint work with Christopher Voll.

The Congruence Subgroup Problem for Free Metabelian Groups

(24/05/2017)

David el-Chai Ben-Ezra (Hebrew University)

On certain finiteness questions in the arithmetic of Galois representations

(07/06/2017)

Gabor Wiese (Universite du Luxembourg)

​Let p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary and unramified outside p. In contrast, there are infinitely many such with Galois group inside GL_2(Z_p^bar), even if one restricts to ones coming from modular forms (this restriction is believed to be local at p). It is tempting to ask what happens "in between" F_p^bar and Z_p^bar, i.e. whether there is still finiteness modulo fixed prime powers.
In the talk, I will motivate and explain a conjecture made with Ian Kiming and Nadim Rustom stating that the set of such Galois extensions `modulo p^m' (a proper definition will be given in the talk) coming from modular forms is finite. I will present partial results and a relation of the finiteness conjecture to a strong question by Kevin Buzzard.
The talk is based on joint work with Ian Kiming and Nadim Rustom.

Jordan's Theorem on primitive permutation groups

(14/06/2017)

Bar Ben-Yair (Tel-Aviv University)

I will present an elementary proof of a 1872 theorem on finite permutation groups by Jordan.

 

Jordan's theorem says that the only primitive permutation groups on n letters containing a cycle of prime length p with n-p ≥ 3 are the alternating group A_n and the full symmetric group S_n.

The Neukirch-Uchida Theorem

(21/06/2017)

Lidor Eldabbah (Tel-Aviv University)

I will present a proof of the most basic result in anabelian geometry, known as the Neukirch-Uchida theorem.

Let K and L be number fields. Then (the theorem says that) isomorphisms between the absolute Galois groups of K and L, up to inner automorphisms, correspond uniquely to isomorphisms between L and K.

In particular, the absolute Galois group of a number field characterizes it up to an isomorphism.

Iterations of quadratic polynomial functions over finite fields

(28/06/2017)

Amotz Oppenheim (Tel-Aviv University)

I will present a recent result by Heath-Brown on the behavior of iterations of quadratic polynomials over finite fields. The result shows that the size of the image of (iterations of) a quadratic polynomial function, satisfying some natural conditions, has the  size predicted by a probabilistic argument.

This also gives some information on the size of cycles under quadratic maps.

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