Field Arithmetic workshop

ISF logo    Research workshop of the Israel Science Foundation        Marie Curie actions logo   GTEM workshop

June 13 (Sunday) - June 17 (Thursday), 2010, Tel-Aviv University

Home Program Registration Useful information Participants

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.