Research workshop of the Israel Science Foundation

Technion Center for Mathematical Sciences

Technion Research & Development center

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Program

Participants

Useful information

 

 

All talks will be in 232 Amado building.

 

Sunday

 

9:30 - 10:15

David Saltman

Finite u-invariant and bounds on cohomology symbol length

10:30 - 11:15

Louis Rowen

Valuations of polynomials in central simple algebras (slides)

11:45 - 12:30

Murray Schacher

Zero divisors in tensor products of division rings (slides)

14:30 - 15:15

Lance Small

Infinite Dimensional Division Algebras

15:25 - 16:10

Adrian Wadsworth

Value functions and valuation rings for central simple algebras (slides)

16:40 - 17:25

Lior Bary-Soroker

Number theory over a finite field with q elements, in the limit q tends to infinity

17:30 - 18:00

Ehud Meir

On orders of finite dimensional semisimple Hopf algebras

 

Monday

 

9:30 - 10:15

Moshe Jarden

Model completeness of PAC fields

10:30 - 11:15

Pierre Debes

From geometry to arithmetic in inverse Galois theory

11:45 - 12:15

Timo Hanke

Density calculations in the Brauer group of a global field

14:15 - 15:00

Arne Ledet

Quaternion groups as Galois groups

15:30 - 16:30

Jean-Pierre Tignol

Valuation theory for algebras with involution (Colloquium Talk)

16:40 - 17:25

Daniel Krashen

Bounding the symbol length in Galois cohomology

17:30 - 18:00

Mauricio Ferreira

Value functions and Dubrovin valuation rings on simple algebras

19:00-

Dinner

 

 

Tuesday

 

9:30 - 10:15

Heinrich Matzat

Frobenius Modules and Field Restriction

10:30 - 11:15

Andy Magid

Some inverse Galois problems in Differential Galois Theory

11:45 - 12:30

Cristian Popescu

The arithmetic of special values

14:30 - 15:15

Leila Schneps

Grothendieck-Teichm"uller theory and the inverse Galois problem

15:25 - 16:10

Nuria Vila

On the tame inverse Galois problem

16:40 - 17:25

Dan Haran

Uniform patching via Wiener algebras

17:30 - 18:00

Leonid Stern

On the distribution of norm groups of algebraic number fields

 

Wednesday

 

9:00 - 9:45

Jack Sonn

Upper and lower bounds of sequences of the form

$GCD(a^n-1,b^n-1)$ and a generalization

10:00 - 10:45

Juan Cuadra

On the Brauer group of Sweedler Hopf algebra

11:05 - 11:50

Cesar Polcino-Milies

Finite group algebras and coding theory

12:30 - 19:00

Excursion to Akko

 

 

 

Thursday

 

9:30 - 10:15

Alex Lubotzky

The Galois group of random elements of linear groups

10:30 - 11:15

Hershy Kisilevsky

Chebotarev sets

11:45 - 12:15

Eli Matzri

Symbol length over C_n fields

14:30 - 15:15

Ido Efrat

On the Zassenhaus filtration of a profinite group

15:25 - 16:10

Gunter Malle

Structure constants and applications (slides)

16:40 - 17:25

Avinoam Mann

Adequate field extensions and Frattini subgroups

17:30 - 18:00

Jung-Miao Kuo

On cyclic twists of elliptic curves

 

Friday

 

9:30 - 10:00

Claudio Quadrelli

Rigid fields, small powerful Galois groups and Bloch-Kato pro-p groups

10:10 - 10:40

Uriya First

Non-Classical Bilinear Forms: Two Applications

10:50 - 11:20

Danny Neftin

Galois groups of tamely ramified subfields of division algebras

 

 

Abstracts

 

 

Sunday

 

David Saltman

 

Title: Finite $u$ Invariant and Bounds on Cohomology Symbol Lengths


Abstract:
In this work we answer a question of Parimala's, showing that fields
with finite $u$-invariant and characteristic 0 have bounds on the
symbol lengths in their $\mu_2$ cohomology in all degrees.

 

Murray Schacher

 

Title: Zero divisors in tensor products of division rings.

 

Abstract:
We discuss an example of Rowen and Saltman that constructs
nxn matrices inside a tensor product of division rings over an
algbebraically closed field. The construction requires cohomology
and the theory of elliptic curves.

 

 

Lance Small

 

Title: Infinite Dimensional Division Algebras

Abstract: We will discuss various constructions of infinite dimensional division algebras and their relation to some well-known problems. Additionally, there will be some remarks on enveloping algebras of certain infinite dimensional Lie algebras.

 

 

Adrian Wadsworth

 

Title: Value functions and valuation rings for central simple algebras

 

Abstract: We give a general overview of valuation theory on division algebras and central simple algebras, emphasizing how it differs from the theory over fields. Three different kinds of noncommutative rings--invariant, total, and Dubrovin valuation rings, generalize different but equivalent aspects of commutative valuation theory. Passage to the associated graded ring of a valuation has proved very useful in recent years. We describe gauges on central simple algebras and their associated graded rings.

 

 

Lior Bary-Soroker

 

Title: Number theory over a finite field with q elements, in the limit q tends to infinity

 

Abstract:

Number theory is concerned with the arithmetic properties of the integers and in particular of prime numbers, as the building blocks of the integers. Early in the development of number theory, it was observed that there is a deep analogy between the integers and the polynomials over a finite field with q elements.

 

In this talk I will present some new results --e.g. the Hardy-Littlewood tuple conjecture which generalizes the twin prime conjecture and the Goldbach conjecture -- for polynomial rings over a finite field with q elements, when q tends to infinity

 

 

Ehud Meir Ben Efraim

 

Title: On orders of finite dimensional semisimple Hopf algebras.

Abstract:
Finite dimensional semisimple Hopf algebras were studied extensively in the last two decades. The simplest examples for such algebras are the group algebra and the dual group algebra of a finite group. These are known as cocommutative and commutative Hopf algebras respectively.
In general, there are many examples for Hopf algebras which are neither commutative nor cocommutative. Nevertheless, all the known examples for these Hopf algebras (over C) arise from some group theoretical data.

These Hopf algebras are already defined over some number field K, and in many interesting cases they are already defined over the ring of integers of K (the group algebra and the dual group algebra, for example, are already defined over the ring of rational integers Z). It turns out that the existence of an order (i.e. the possibility to define the algebra over the ring of algebraic integers) has some representation theoretic implications, and it gives us a good amount of information on the Hopf algebra.

In this talk we will give some examples of these Hopf algebras, and will discuss the question of existence of orders. We will describe a mechanism to find some orders, and we will show how it can be used to prove that certain Hopf algebras do not admit an order over any ring of algebraic integers. Also, we will explain how it can be used to show that certain Hopf algebras have at most one possible order over any ring of integers (the group algebras, for comparison, can have
many).

This talk will be based on a joint work with Juan Cuadra.

 

 

Monday

 

Moshe Jarden

 

Title: Model Completeness of PAC fields

 

Abstract:

We present a theorem of Koll\'ar on the density property of

PAC fields and a theorem of Abraham Robinson on the model

completeness of the theory of algebraically closed non-trivial valued fields.

Then we prove that the theory $T$ of non-trivial valued fields in an

appropriate first order language has a model completion $\tilde T$.

The models of $\tilde T$ are non-trivial valued fields $(K,v)$ that are

$\omega$-imperfect, $\omega$-free, and PAC.

 

 

Pierre Debes

 

Title: From Geometry to Arithmetic in Inverse Galois Theory

 

Abstract:

I will revisit some connections between geometric topics

from Inverse Galois Theory like the Regular Inverse Galois Problem

or generic extensions and more arithmetic ones like the Inverse

Galois Problem or the Grunwald Problem.

 

Timo Hanke

 

Title: Density calculations in the Brauer group of a global field

 

Abstract:

There is a well-known notion of "natural density" that allows to measure infinite subsets of the set of natural numbers.
A similar concept allows us to measure infinite subsets of the Brauer group of a global field F.
As an application, building on previous results about division algebras over Laurents series fields over F,
we measure the density of crossed products, noncrossed products and rigid division algebras over those fields.
This is joint work with J.Sonn.

 

Arne Ledet

 

Title: Quaternion groups as Galois groups.

Abstract:

We consider quaternion groups of 2-power order as Galois groups,
and provide a generic description of the Galois extensions under suitable
conditions on the base field. The construction is fairly elementary.

 

 

Jean-Pierre Tignol

 

Title: Valuation theory for algebras with involution

 

Abstract:

Valuation theory plays a central role in the solution of various
problems concerning finite-dimensional division algebras, such as the
construction of noncrossed products and of counterexamples to the
Kneser-Tits conjecture. However, relating valuations to Brauer-group
properties is particularly difficult because valuations are defined
only on division algebras and not on central simple algebras with
zero divisors. This talk will present a more flexible tool recently
developed in a joint work with Adrian Wadsworth, which applies to a
broad spectrum of noncommutative situations. In particular, central
simple algebras with anisotropic involution over Henselian fields are
shown to carry a special kind of value function, which is an analogue
of Schilling valuations on division algebras.

 

Danny Krashen

 

Title: Bounding the symbol length in Galois cohomology

Abstract:

As a result of the recently proved Bloch-Kato conjecture, it
is known that one may write classes in certain Galois cohomology
groups as sums of symbols, that is, cup products of classes from the
first cohomology group. In this talk, I will describe some results on
bounding the number of symbols necessary to write a given cohomology
class in terms of the arithmetic of the underlying field and the
relationship between this problem and the "period-index" problem.

 

Mauricio Ferrira

Title: Value functions and Dubrovin valuation rings on simple algebras

 

Abstract: In this work we study the connection between two theories of non-commutative
valuation: Dubrovin valuation rings and gauges. Dubrovin valuation rings
were introduced in 1982 as a generalization of invariant valuation rings
to Artinian simple rings. Gauges are valuation-like maps for
finite-dimensional semisimple algebras over valued fields. Gauges were
introduced much more recently in 2010 by Tignol and Wadsworth. Just as for
valuations on fields, we can define a ring associated to a gauge, which we
call gauge ring. We introduce the concept of minimal gauge on central
simple algebras, which are gauges that the degree zero part of the
associated graded ring has the least number of simple components. We show
that the ring associated to a minimal gauge is an intersection of a finite
set of Dubrovin valuation rings having an extra property introduced by
Gräter in 1992, which is called Intersection Property. We also obtain an
existence theorem of minimal gauges for central simple algebra over a
field with a finite rank valuation.

 

 

 

Tuesday

 

B. Heinrich Matzat

 

Title: Frobenius modules and field restriction


Abstract:

Finite groups of Lie type defined over the field Fq,q = pf , are realized as Galois groups over Fp(t) using field restriction.

 

 

Andy Magid

 

Title: Some inverse Galois problems in Differential Galois Theory

 

Abstract: The inverse problem of differential Galois theory for a differential field F and a (pro)algebraic group G can be solved by constructing a derivation D on F[G] which (a) commutes with the G action and (b) is such that the D constants in F(G) are those of F. Assuming that such a derivation exists, we prove how to extend it to the F coordinate ring of a non-split extension of G by the additive group, thus solving the inverse problem for this extension.

 

 

Cristian D. Popescu

 

Title: The arithmetic of special values

Abstract:
The well-known analytic class number formula, linking the special value at s=0 of the Dedekind zeta function of a number field to its class number and regulator, has been the foundation and prototype for the highly conjectural theory of special values of L-functions for close to two centuries. We will discuss generalizations of the class number formula to the context of equivariant Artin L-functions which capture refinements of the Brumer-Stark and Coates-Sinnott conjectures. These generalizations
relate various algebraic-geometric invariants associated to a global field, e.g. its Quillen K-groups and etale cohomology groups, to various special values of its Galois-equivariant L-functions. This is based on joint work with Greither, Dodge and Banaszak

 

 

Nuria Vila

 

Title: On the tame inverse Galois problem

 

Abstract:

The talk concerns the so called Tame Inverse Galois Problem.

This problem is a strengthening of the Inverse Galois Problem posed by B. Birch.

I will present tamely ramified Galois realizations over Q for some families of non-solvable groups.

 

 

Dan Haran

 

Title: Uniform patching via Wiener algebras


Abstract:
We discuss a unified approach to patching of Galois groups over the field of rational functions over a complete field, archimedean or ultrametric. This approach uses a generalization of Wiener algebras.

A joint work with A. Fehm and E. Paran.

 

 

Leonid Stern

 

Title: On the Distribution of Norm Groups of Algebraic Number Fields

Abstract:

Let X be a subgroup of a group Y . The interval (X; Y ) is the set of subgroups of Y that contain X including X and Y . By local class field theory the interval (N_{K/k}K^*; k^*) contains a finite number of norm groups for any finite extension K of a

p-adic number field k. In the present work we investigate the number of norm groups in the interval (N_{K/k}K^*; k^*) for a given finite extension K/k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree n such that the normal closure of K over k has the Galois group isomorphic to A_n or S_n, then the interval (N_{K/k}K^*; k^*) contains only the obvious two norm groups. Also, the interval (N_{K/k}K^*; k^*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups isomorphic to the dihedral group of order 8 for which the corresponding interval contains a finite number of norm groups. The main theorem in our earlier work states that the interval (N_{K/k}K^*; k^*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension. In the present work we generalize the main theorem to non-Galois extensions. We then use this theorem to prove that the interval (N_{K/k}K^*; k^*) contains infinitely many norm groups for any Galois extension with the Galois group isomorphic to the cyclic group C_8 of order 8 or to the quaternion group Q_8 of order 8.

 

 

Wednesday

 

Jack Sonn

Title: Upper and lower bounds of sequences of the form $GCD(a^n-1,b^n-1)$ and a generalization

 

Abstract:

There has been interest during the last decade in properties of the sequence {gcd(a^n-1,b^n-1)}, n=1,2,3,..., where a,b are fixed (multiplicatively independent) elements in either the rational integers, the polynomials in one variable over the complex numbers, or the polynomials in one variable over a finite field. In the case of the rational integers, Bugeaud, Corvaja and Zannier have obtained an upper bound exp(\epsilon n) for any given \epsilon >0 and all large n, and demonstrate its sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound \exp(\exp(c\frac{log n}{loglog n})) for infinitely many n, where c is an absolute constant. The upper bound generalizes immediately to gcd(\Phi_N(a^n), \Phi_N(b^n)) for any positive integer N, where \Phi_N(x)$ is the Nth cyclotomic polynomial, the preceding being the case N=1. The lower bound has been generalized in Yossi Cohen's Ph.D. thesis to N=2. In this paper we generalize the lower bound for arbitrary N under GRH (the generalized Riemann Hypothesis), using an effective version of the Chebotarev density theorem due to Lagarias and Odlyzko. The analogue of the lower bound result for gcd(a^n-1,b^n-1) over F_q[T] was proved by Silverman; we prove a corresponding generalization (without GRH). (Joint work with Yossi Cohen)

 

 

Cesar Polcino-Milies

 

Title: Finite Group Algebras in Coding Theory

 

Abstract:

Finite group algebras with minimal number of simple components.

Applications to coding theory: Cyclic codes of length 2p^n; cyclic vs abelian codes; Codes over Chain Rings; Metacyclic codes and combinatorial equivalence.

 

 

Juan Cuadra

 

Title: On the Brauer group of Sweedler Hopf algebra

 

Abstract: In 1994 Caenepeel, Van Oystaeyen and Zhang defined the Brauer group of a Hopf algebra with bijective antipode by considering Yetter-Drinfeld module algebras. They extended so a previous construction by Long for commutative and cocommutative Hopf algebras. Since then it was a maingoal to compute the Brauer group of the smallest noncommutative noncocommutative Hopf algebra, namely, Sweedler four dimensional Hopf algebra. In this talk we will report about the current state of knowledge on this problem.

The results to be presented are based on a joint work with Giovanna Carnovale [Israel J. Math. 183 (2011), 61-92. ArXiv:0904.1883].

 

 

Thursday

 

Alex Lubotzky

 

Title: The Galois group of random elements of linear groups

Abstract:

Let G be a finitely generated subgroup of GL(n,F) where F is a finitely generated field of characteristic zero.
We show that for a random element g of G, the Galois group over F of the characteristic polynomial of g has generic
behavior depending on the Zariski closure of G and its connected components. Some interesting counter examples
will be presented when F is not finitely generated. (Joint work with Lior Rosenzweig. )

 

 

Hershy Kisilevsky

 

Title: Chebotarev sets

 

Abstract:
We give conditions for a set of primes of a number field to be a Chebotarev/Frobenius set. We construct a set of density 1/2 which is provably not such a set.

(With Mike Rubinstein)

 

Eli Matzri

 

Title: Symbol length over C_n fields.

 

Abstract:

We will show that a csa, A, over a C_n field, F, of exponent m is similar to the product of at most m^{n-1}-1 symbols of degree m.

As a result we get a bound on the index of A in terms of the exponent.

This result can be extended to fields finitely generated over a C_k field.

 

 

Ido Efrat

 

Title: On the Zassenhaus filtration of a profinite group

 

Abstract: For a prime number p the p-Zassenhaus filtration of a profinite group G is the fastest
descending sequence (G_n) of subgroups such that G_1=G, G_i^p\leq G_{ip} and [G_i,G_j]\leq G_{i+j} for all i,j.
Let G=G_F be the absolute Galois group of a field F containing a p-th root of unity.
In their 1996 paper in Annals Math., Minac and Spira showed that for p=2 the subgroup G_3 is the
intersection of all normal subgroups N of G such that G/N is either cyclic of order 2, cyclic of order 4,
or the dihedral group D_4 of order 8.
We will present a generalization of this remarkable fact for arbitrary primes p and higher subgroups G_n in the filtration.

 

Gunter Malle

Title: Structure constants and applications

Abstract:
We review some results on structure constants for finite nearly
simple groups and explain methods for calculating or estimating them.
We then present various applications, for example to Beauville surfaces
and to Galois realizations, but also to various conjectures about finite
simple groups.

 

 

Avinoam Mann

 

Title: Adequate field extensions and Frattini subgroups

Abstract:

Let L/F be a finite field extension. We always assume that F is a global field.

The field L is F-adequate, if there exists a division algebra with center F and a

maximal subfield isomorphic to L. This definition is due to M.Schacher (1968), who

proved, among many other results, that if F< K< L, then K is also F-adequate,

and that if L/F is separable, then L is F-adequate iff the Galois closure of L is

F-adequate. D.B.Leep-T.L.Smith-R.Solomon (2002) proved a partial converse: if

L/F is a Galois extension with Galois group G, K is the fixed field of the Frattini

subgroup Phi(G) of G, and if G has a certain property, which the authors call Frattini

closed, and K is F-adequate, then L is F-adequate.

Following Leep-Smith-Solomon, we first define, for each finite group G, a certain

characteristic subgroup, which we call the local Frattini subgroup of G, and denote

by Psi(G). This subgroup is contained in the Frattini subgroup, and using it we can

remove the assumption of frattini closure, proving

Theorem 1. Let F be a global field, let L/F be a Galois extension with Galois

group G, and let K be the fixed field of Psi(G). Then L is F-adequate if and only if

K is F-adequate.

We provide several characterization of the local Frattini subgroup, and then

return to the Frattini closed groups. These turn out to be those groups for which

Phi(G) = Psi(G). They were considered already by J.S.Rose (1980), which gave

several classes of such groups, including the ones that were given later independently

by Leep-Smith-Solomon. Here we generalize these results. Combined with the

aforementioned results, and results of D.Chillag-J.Sonn (1981) we obtain, e.g.

Corollary 2. Let L/F be a Galois extension with group G, where either F = Q,

the field of rationals, or F is a global field of characteristic p > 0, and [L : F] is

prime to p, and let K be the fixed field of Phi(G). Then L is F-adequate if and only

if the Sylow subgroups of G are metacyclic, and K is F-adequate.

This is a joint work with Gil Kaplan and Arieh Lev of the Academic College of

Tel-Aviv-Yaffo.

 

 

Jung-Miao Kuo

 

Title: On cyclic twists of elliptic curves

 

Abstract:

Let $E$ be an elliptic curve over a field $k$ and let $C$ be a principal homogeneous space for $E$. Lichtenbaum showed that if the period of $C$ equals its index, then the elements of the relative Brauer group $Br(k(C)/k)$ can be parametrized by the group $E(k)$. However, his mapping is hard to apply for computation. Recently, Ciperiani and Krashen described the parametrization of $Br(k(C)/k)$ by $E(k)$, when $C$ is what they call a cyclic twist of $E$, in terms of a cub product formula. We will determine 3-cyclic twists of elliptic curves and as an application we describe explicitly $Br(k(C)/k)$ for certain cubic curves $C$.

 

Friday

 

Claudio Quadrelli

 

Title: Rigid fields, small powerful Galois groups and Bloch-Kato pro-p groups

Abstract:
I plan to discuss my recent joint work with S. Chebolu and J. Minac. Let p be an odd prime and assume that a primitive p-th root of unity is in a field F. Then F is said to be p-rigid if only those cyclic algebras are split which are split for trivial reasons. I will present new characterizations of such fields and their Galois groups. In particular, it is possible to detect whether a field F is p-rigid simply by small quotients of the absolute Galois group in a purely group-theoretical way or by the cohomological dimension of G_F(p), where G_F(p) is the maximal pro-p Galois group of F. A fundamental role in these results is played by the Galois cohomology of G_F(p) and by Bloch-Kato pro-p groups. Our work extends, illustrates and simplifies some previous results by R. Ware, A. Engler, J. Koenigsmann and I. Efrat, and provide new points of view on maximal p-extensions. Last, but not least, it provides a new direct foundation of rigid fields which does not rely on valuation techniques.

 

 

Uriya First

Title: Non-Classical Bilinear Forms: Two Applications

 

Abstract:

I will discuss a new definition of bilinear forms over rings (without involution!) and two of its applications.

 

To get the grasp of what these new forms are, let F be a field and consider the well-known correspondence between anti-automorphisms s of M_n(F) satisfying (s|_F)^2 = id and sesquilinear forms over the vectors space F^n (considered up to a suitable equivalence relation).

The new bilinear forms would correspond to anti-automorphisms s of M_n(F) that do not satisfy (s|_F)^2 = id.

 

Using a generalization of the correspondence just described, I will present an easy proof for a result of Osborn about semisimple rings with involution, and also a partial solution to a problem that was suggested to me by D. Saltman. Namely, I will show that under some finiteness assumptions, a ring which is Morita equivalent to its opposite rings is Morita equivalent to a ring with an anti-automorphism (Saltman has proved the latter for Azumaya algebras).

 

Danny Neftin

Title: Galois groups of tamely ramified subfields of division algebras

 

Abstract:

For any number field K, it is unknown what are the groups G for which there exists a G-crossed product division algebra over K. We shall discuss an extension of a theorem of Neukirch on embedding problems with prescribed local conditions which leads to the solution of the tame version of the above problem for solvable groups.