
Research workshop of the Israel Science Foundation


Technion Center for Mathematical Sciences


Technion Research & Development center

All talks
will be in 232 Amado building. Sunday
Monday
Tuesday
Wednesday
Thursday
Friday


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 wellknown 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 ringsinvariant,
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
BarySoroker
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 HardyLittlewood 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 nontrivial valued fields.
Then we prove that the theory $T$
of nontrivial valued fields in an
appropriate first order language has a model
completion $\tilde T$.
The models of $\tilde T$ are
nontrivial 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 wellknown 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
2power 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.
JeanPierre Tignol
Title: Valuation theory for
algebras with involution
Abstract:
Valuation theory plays a central
role in the solution of various
problems concerning finitedimensional division algebras, such as the
construction of noncrossed products and of
counterexamples to the
KneserTits conjecture. However, relating valuations
to Brauergroup
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 BlochKato 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 "periodindex" 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 noncommutative
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 valuationlike
maps for
finitedimensional 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 F_{q},q
= p^{f}^{ }, are realized as Galois
groups over F_{p}(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 nonsplit 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 wellknown 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 Lfunctions for close to two centuries. We will
discuss generalizations of the class number formula to the context of equivariant Artin Lfunctions
which capture refinements of the BrumerStark and
CoatesSinnott conjectures. These generalizations
relate various algebraicgeometric invariants associated to a global field,
e.g. its Quillen Kgroups and etale
cohomology groups, to various special values of its
Galoisequivariant Lfunctions. 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
nonsolvable 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
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
padic 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 2extension. In the present
work we generalize the main theorem to nonGalois 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 $GCD(a^n1,b^n1)$
and a generalization
Abstract:
There has been interest during
the last decade in properties of the sequence {gcd(a^n1,b^n1)},
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^n1,b^n1)
over F_q[T] was proved by Silverman; we prove a
corresponding generalization (without GRH). (Joint work with Yossi Cohen)
Cesar PolcinoMilies
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 YetterDrinfeld 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), 6192. 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^{n1}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 pZassenhaus 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 pth 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 Fadequate,
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 Fadequate,
and that if L/F is separable,
then L is Fadequate iff the Galois
closure of L is
Fadequate. D.B.LeepT.L.SmithR.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 Fadequate, then L is
Fadequate.
Following LeepSmithSolomon,
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 Fadequate if and only if
K is Fadequate.
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 LeepSmithSolomon.
Here we generalize these results. Combined with the
aforementioned results, and results of D.ChillagJ.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 Fadequate if and only
if the Sylow
subgroups of G are metacyclic, and K is Fadequate.
This is a joint work with Gil
Kaplan and Arieh Lev of the Academic College of
TelAvivYaffo.
JungMiao 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 3cyclic 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 BlochKato prop 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 pth root of unity is in a field F. Then F
is said to be prigid 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 prigid simply by small quotients of the absolute Galois group in a purely
grouptheoretical way or by the cohomological
dimension of G_F(p), where G_F(p) is the maximal prop
Galois group of F. A fundamental role in these results is played by the Galois cohomology of G_F(p) and by
BlochKato prop 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 pextensions. Last, but not
least, it provides a new direct foundation of rigid fields which does not rely
on valuation techniques.
Uriya First
Title: NonClassical 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 wellknown correspondence
between antiautomorphisms 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 antiautomorphisms 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 antiautomorphism (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 Gcrossed 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.