Field Arithmetic workshop
Research workshop of the Israel Science Foundation
GTEM workshop
June 13 (Sunday) - June 17
(Thursday), 2010, Tel-Aviv University
Talks and abstracts
Eva Bayer-Fluckiger:
"On isometries of inner product spaces"
Abstract:
Let k be a field of characteristic not 2, and let V
be an inner product space over k.
In 1969, Milnor raised the
following question:
which monic, irreducible polynomials can
be characteristic polynomials of isometries of V?
He gave the
answer to this question when k is a local field.
The aim of
this talk is to treat the case where k is an algebraic number
fields, to discuss relationships of this problem with embedding
questions of tori in semi-simple groups, as well as with some
problems concerning isometries of unimodular lattices, in
particular a conjecture of Gross and McMullen.
Mirela Çiperiani:
"Divisibility of Heegner points and Selmer ranks"
Abstract:
Let E be an elliptic curve over the rationals of even analytic rank
and p a prime of ordinary reduction.
We consider an imaginary quadratic extension of the rationals
such that the corresponding twist of E
has analytic rank 1 and the Heegner hypothesis is satisfied.
We will analyze the way the Heegner module sits
inside the p-Selmer module of E
over the anticyclotomic extension
and see what this implies about
the Selmer group of E over Q
in view of a conjecture of Perrin-Riou.
This is work in progress.
Pierre Dèbes:
"The Regular Inverse Galois Problem and the Hilbert-Grunwald property".
Abstract:
Our main result combines a Grunwald-Wang type conclusion
for arbitrary groups, a new version of Hilbert's irreducibility theorem
and a p-adic form a la Harbater, but with good reduction, of the
Regular Inverse Galois Problem. As a consequence we obtain
a statement that questions the validity of the RIGP over Q.
(joint work with Nour Ghazi).
Ido Efrat:
"Filtrations of absolute Galois groups".
Abstract:
A main theme in Field Arithmetic is to recover
arithmetical information on a field from its various canonical
Galois groups. Standard filtrations of the absolute Galois group
(the p-central filtration or the Zassenhaus filtration)
give rise to such very small canonical Galois groups, which encode
the entire mod-p cohomology, and therefore also the essential
information about valuations and orderings. We describe these
Galois groups, and determine the minimal quotient of the absolute
Galois group which encodes this information.
Arno Fehm:
"Galois theory over rings of arithmetic power series"
Abstract:
Let R be the ring of those 'arithmetic' power series
in Z[[t]] which are
holomorphic on the closed complex disc of radius 0 < r < 1.
Harbater studied
these rings in the 80's and developed patching methods in order to show
that every finite group is a Galois group
over the quotient field K of R.
We strengthen this result by proving that K is both ample and fully
Hilbertian, and hence has a semi-free absolute Galois group. The proof
of ampleness relies on a generalization of Pop's result that the
quotient field of a Henselian ring is ample.
(Joint work with Elad Paran)
Wojciech Gajda:
"On l-adic representations and Abelian varieties over finitely generated fields"
Abstract:
We will discuss some recent computations of images of Galois
representations attached to Abelian varieties over number fields, and
over finitely generated fields of positive characteristic. In the second
part of the talk applications of these computations will be presented -
both to geometry (e.g., Hodge, Tate and Mumford-Tate conjectures) and to
arithmetic (e.g., Geyer-Jarden conjecture on torsion).
Wulf-Dieter Geyer:
"Ramblings in Field Arithmetic"
Abstract:
The talk will give some aspects of joint work with a younger colleague of mine,
centering around the notion of a PAC field.
Main points are the construction of such fields, the arithmetic properties
of varieties over such fields and generalizations of this notion.
Barry Green:
"On the number of automorphisms of curves over algebraically closed fields"
Abstract:
The aim of this talk is
to report on selected results and open questions
on the automorphisms groups of curves over algebraically closed fields
and there reductions in the case of good reduction.
We also draw attention to bounds on the size of these groups,
comparing what is known in characteristic zero
with questions and conjectures in positive characteristic.
Moshe Jarden:
"Ample fields"
Abstract:
A field K is said to be ample if every absolutely
irreducible K-curve with a K-rational simple point has
infinitely many K-rational points.
Equivalently, K is existentially closed in K((t)).
For example each PAC field and each Henselian field is
ample.
The most striking property of an ample field K is the each
finite split embedding problem over K(x) has a regular
solution.
Indeed, no other fields beside ample fields are known to
have that property.
Alex Lubotzky:
"Profinite groups methods in discrete group theory"
Abstract:
Profinite groups originated as Galois groups of infinite Galois
field extensions but in the last three decades became a basic tool in
discrete group theory. They serve as a tool of transporting knowledge from
finite group theory to infnite group theory. We will survey these methods
and illustrate them with some old and new examples and problems.
Ján Mináč:
"Absolute Galois pro-p-groups and their small quotients"
Abstract:
Describing the structure of absolute Galois pro-p-groups of fields
is a difficult open problem.
Recently however M. Rost and V. Voevodsky proved
the Bloch-Kato conjecture
which describes Galois cohomology via generators and relations.
This is very powerful information
about the structure of absolute Galois groups,
but it is encoded information
and it is still challenging to derive
group theoretic and field theoretic consequences.
In recent work with S. Chebolu and I. Efrat,
we consider the quotients of absolute Galois groups
by their third p-central series,
or alternatively by the third Zassenhaus series.
I. Efrat explains in his talk
how these rather small quotients
of absolute Galois groups
encode information about the existence
of non-trivial valuations and the entire Galois cohomology.
In my talk I will explain some technical details of our work,
and outline some applications
on ruling out certain pro-p-groups as absolute Galois groups.
I will also provide a review of some related results obtained
with D. Benson, N. Lemire, A. Schultz and J. Swallow
on the Galois cohomology H*(GE,Fp)
viewed as a Galois module over Gal(E/F),
where E/F is a cyclic extension of a field F
containing a primitive pth root of unity and
H*(GE,Fp) is the Galois cohomology of E
with coefficients in the field Fp with p-elements.
Ambrus Pál:
"Homotopy sections and rational points on algebraic varieties"
Abstract:
It is possible to generalise Grothendieck's anabelian section conjecture by substituting the arithmetic fundamental group with the etale homotopy type. I will discuss under which conditions this homotopy theoretical section map is injective or surjective over various fields of arithmetic interest.
Elad Paran:
"Hilbertianity of fields of power series"
Abstract:
Let R be any domain that has a minimal prime ideal,
and let R[[X]] be the ring of formal power series over R.
We prove that the quotient field of R[[X]] is Hilbertian.
This result gives a positive solution to an open problem of Jarden,
and as a consequence,
enables generalization of Galois-theoretic results
over fields of power series,
obtained by Lefcourt, Harbater-Stevenson, Pop, and the speaker.
Parimala:
"Hasse principle for homogeneous spaces over
function fields of p-adic curves"
Abstract:
Hasse-Minkowski's theorem says that a quadratic form
over a number field is isotropic if it is isotropic over
completions at all places of the number field. One could look for
a Hasse principle in the function field setting with respect to all
discrete valuations of the function field. This is particularly
interesting for Qp(t) :
a Hasse principle for isotropy
of quadratic forms would lead to the fact: every quadratic form over
Qp(t) in at least nine variables has a nontrivial zero. We discuss the
Hasse principle for Qp(t) in the more general context of homogeneous
spaces under connected linear algebraic groups. We also discuss
how these results lead to an understanding of the Rost invariant
associated to a torsor under a simple simply connected group
over these fields. The case of split E8 is particularly interesting.
Sebastian Petersen:
"Kuykian fields"
Abstract:
A field K is called Hilbertian, if for every polynomial
f(T,X)∈ K[T,X]
which is separable and irreducible in K(T)[X] there
are infinitely many t∈ K such that f(t,X) is irreducible
in K[X]. Hilbertian fields play an important role in modern Galois
theory. It is classically known that finitely generated infinite fields
are Hilbertian.
Let K be a Hilbertian field. An important theorem of Kuyk asserts that
every (possibly infinite) abelian Galois extension L/K is Hilbertian again. In
particular every intermediate field of K(μ∞)/K is Hilbertian. Here
K(μ∞) is the field obtained from K by adjoining all torsion
points of the multiplicative group
Gm(K~)=(K~)x.
One might wonder, whether one can replace Gm by other algebraic
groups in this statement. Concerning the case of abelian varieties,
Moshe Jarden proposed the following
Kuykian Conjecture. Let K be a Hilbertian field and
A/K an abelian variety. Then every intermediate field of
K(Ator)/K is Hilbertian.
Fields that satisfy the conjecture are called Kuykian fields.
Classical results of Serre and modern developments like Haran's
diamond theorem allowed Moshe Jarden to prove that every number field is Kuykian.
We can generalize this to the following
Theorem. Every finitely generated Hilbertian field is Kuykian.
Furthermore we give several examples of Kuykian fields
which are not finitely generated.
The Kuykian conjecture in the general case remains open, however.
(Joint work with Arno Fehm and Moshe Jarden)
Peter Roquette:
"News of the Grunwald-Wang story"
Abstract:
Originally, the Grunwald theorem was used by Hasse
in 1930 when he proved the cyclicity of simple algebras over
number fields, and more. The story about this theorem and the
"comedy of errors" connected with it has been told several
times. Recently, another error has been discovered in one of
Hasse's paper on the subject. It could be corrected (by Patrick
Morton). On the other hand, taking into account that for Hasse's
proof only a weak form of the Grunwald theorem is needed, it
has turned out recently that this weak form is valid in an
arbitrary multi-valued field K. Thus given cyclic extensions
locally for finitely many valuations of K,
let n be the lcm of their
degrees. Then there exists a cyclic extension of K
of degree n
whose local degrees coincide with the degrees of the given local
extensions. The proof is almost trivial by Kummer theory and
class field theory is not needed at all for this. The only problem
arises when the prime number 2 divides n. In such cases the
Grunwald theorem in its full generality has been refuted by
Wang but it has turned out that the weak form can be treated by
Kummer theory as well.
Jack Sonn:
"On the minimal ramification problem for semiabelian groups"
Abstract:
The minimal ramification problem is a refinement of the inverse Galois problem.
Given a finite group G,
what is the minimal number ram(G) of ramified primes
in a Galois extension of the rationals Q with group G?
This problem arose in the context of p-groups;
however for any G there is a group theoretic lower bound d(G),
the minimal number of conjugacy classes of G which generate G.
The minimal ramification problem for G is whether
this lower bound can be attained.
For example, if G is the symmetric group Sn, then d(G) =1,
and the problem is still open except for a few small values of n.
For finite nilpotent groups G,
d(G) coincides with the minimal number of generators of G.
It has recently been proved that this problem has an affirmative answer
for a substantial class of finite nilpotent groups
(all finite semiabelian nilpotent groups).
(Joint work with Hershy Kisilevsky and Daniel Neftin).
Jakob Stix:
"On the passage form local to global in the section conjecture"
Abstract:
The passage from local to global has a long tradition in number theory.
The talk will discuss Grothendieck's section conjecture in this respect.
We will present results on Brauer-Manin obstructions for sections
and the relation of the descent obstruction
to sections of the fundamental exact sequence.
The latter is joint work with David Harari.
Pavel Zalesskii:
"Normal subgroups of profinite groups of positive deficiency"
Abstract:
We shall present the results of a joint work with Fritz
Grunewald, Andrei Jaikin-Zapirain and Aline Pinto, where we
initiate the study of profinite groups of non-negative deficiency.
Starting with explaining the term of deficiency for a profinite
groups (the definition was given by Alex Lubotzky in his seminal
paper "Profinite Presentations") and giving several classical
examples, we focus on showing that the existence of a finitely
generated normal subgroup of infinite index in a profinite group
G of non-negative deficiency gives rather strong consequences
for the structure of G. In the case of positive deficiency of
G we compute the cohomological dimension of G and practically
are able to give the structure of Sylow subgroups
of N and G/N.
A pro-p Poincaré duality group G of dimension 3
(PD3-group)
has deficiency 0 in the category of pro-p groups.
A complete description of such groups having finitely generated
normal subgroups will be presented. The result is also generalized
to the profinite case.
We show how these results can be used to deduce structure
information on the profinite completions of several groups of
geometric nature. In particular, the description of ascending
HNN-extensions of free profinite groups (and so also the
profinite completion of ascending HNN-extensions of free groups
known also as mapping tori of free group endomorphisms) will be
presented.