FP6 Research and Training Network
Galois Theory and Explicit Methods

Annual meeting 2007: abstracts

Marco Antei (Lille), Comparison between the fundamental group scheme of a relative scheme and the one of its generic fibre.
Abstract: Let X be a reduced, irreducible faithfully flat scheme over a Dedekind scheme B, and let xX(B) be a point. Gasbarri defines the ``fundamental group scheme π1(X,x)'' that generalize the one defined by Nori, when B is the spectrum of a perfect field. Let η be the generic point of B (that we assume to be affine); we define a morphism π1(Xη,xη) ----> π1(X,xB Bη and we prove that it is surjective. We will explain how the kernel of this morphism measures the obstruction for a torsor over Xη (under the action of a finite Bη-group scheme) to be extended to a torsor over X.

Maite Aranés (Warwick), Modular forms and elliptic curves over imaginary quadratic fields
Abstract: This is a survey talk describing the almost entirely conjectural correspondence between certain modular forms and elliptic curves defined over imaginary quadratic fields. I will also discuss the development of explicit constructions used to approach the problem.

Burcu Baran (Rome), A modular curve of level 9 and the class number one problem
Abstract: For any positive integer n, let Xns+(n) denote the modular curve associated to the normalizer of a non-split Cartan subgroup of level n. The curve Xns+(n) classifies isomorphism classes of elliptic curves with a certain type of "non-split" level n structure. If every prime p that divides n is inert in an imaginary quadratic order O of class number one, then the associated elliptic curve with complex multiplication by O gives rise to an integral point on Xns+(n). Some years ago, Serre pointed out that the solutions by Heegner and Stark of the class number one problem can be viewed in this way as the determination of the integral points of the modular curveXns+(24).

We derive an explicit parametrization for the modular curve Xns+(9) and determine its integral points. This gives yet another solution to the class number one problem.

Lior Bary-Soroker (Tel Aviv), Dirichlet's theorem for polynomial rings
Abstract: In this talk we present a Dirichlet theorem for a polynomial ring in one indeterminate over a PAC field F: For any relatively prime polynomials a,b in F[X] and for every sufficiently large positive integer n there exists a polynomial cin F[X] for which a+bc is a degree n irreducible polynomial, provided that F has a separable extension of degree n. We also discuss how to extend this theorem for more general fields F.
See ArXiV.

Alp Bassa (Lausanne), Lecture 3: towers of algebraic function fields over finite fields
Abstract: In this talk, I will give a short introduction to towers of algebraic function fields and give some examples of explicit wild towers, which are asymptotically good. I will introduce a new tower over cubic finite fields, whose limit attains Zink's lower bound. Many features of this tower are very similar to those of an optimal tower of Garcia-Stichtenoth over quadratic finite fields, whose modularity was shown by Elkies.

Alp Bassa (Lausanne), Lecture 1: rational points on curves over finite fields
Abstract: Although the Riemann Hypothesis for the classical Riemann zeta function is an open question for almost 150 years, its analogue in the case of curves over finite fields has already been proved in 1933 by Hasse (for elliptic curves) and in the 1940's by Weil (for curves of any genus). It is directly related with the number of rational points on these curves, which will be the focus of this talk. Hence we will study the Riemann Hypothesis for curves over finite fields and some of its consequence.

Alp Bassa (Lausanne), Lecture 2: algebraic geometry codes
Abstract: Around 1980, Goppa introduced a construction of error-correcting codes using curves over finite fields, which resulted in a renewed interest in these curves and their rational points. In this talk, I will try to explain this beautiful application of algebraic geometry in coding theory. I will try to introduce basic notions of coding theory, algebraic geometry codes and asymptotic bounds obtained from them.

Erwin Dassen, Introduction to layered lattices
Abstract: In this talk we give an introduction to layered lattices pointing out differences with the classical theory. Motivation will be given in the form of problems from the classical theory that can be better stated in the generalized setting.

Luca Demangos (Lille), The Galois inverse problem
Abstract: I will present the Galois Inverse Problem with his main methods of study: Noether program, Rigidity and Embedding Problems. I will also give some practical example of finite groups realized as Galois groups over the field of rational numbers by the methods previously cited.

Arno Fehm (Tel Aviv), Definability in function fields
Abstract: This talk approaches function fields from a model theoretic point of view. It focuses on the problem of the definability of the field of constants, surveying results from 1960 to 2007, including results of the speaker (as part of a thesis supervised by W.-D. Geyer). Furthermore, the importance of the topic for algorithmic decidability is discussed.

Dieter Geyer (Erlangen), Higher dimensional class field theory
Abstract: Higher dimensional class field theory, i.e. the theory of abelian coverings of higher dimensional arithmetical schemes including varieties over finite fields, was started in case of regular schemes in the 1980's by Bloch, Kato and Saito in several papers using higher dimensional Milnor K-theory. Subsequent papers by Jannsen, Stevenhagen, Spiess, A. Schmidt and others followed. I will speak on a new approach by Goetz Wiesend, using only K0 and K1 groups, and a thesis of Walter Hofmann generalizing Wiesend's results from regular schemes to singular schemes.

Julia Hartmann (Heidelberg), An introduction to differential Galois theory
Abstract: Differential Galois theory is the algebraic theory of linear differential equations. It mimicks and generalizes the usual finite Galois theory. The talk will give a quick introduction to the subject, including the classical characteristic zero theory introduced by Picard and Vessiot as well as the iterative theory in positive characteristic, due to Matzat and van der Put.

Julia Hartmann (Heidelberg), Patching and differential Galois theory
Abstract: Algebraic and formal patching are important tools for attacking inverse problems and embedding problems in usual Galois theory. In a recent project with D. Harbater, we have generalized and adjusted these methods to make them applicable to differential modules. The talk explains this new version of patching over fields together with its applications to differential Galois theory (like embedding problems and inverse problems).

Florian Heiderich (Barcelona), Picard-Vessiot theory for linear partial differential equations
Abstract: In this talk we will present two generalizations of Picard-Vessiot theories to treat linear partial differential equations. The first one is based on the well known differential Galois theory in characteristic zero while the second one extends the iterative differential Galois theory in characteristic p>0. When dealing with linear partial differential equations integrability conditions are necessary in order to ensure the existence of Picard-Vessiot extensions in both cases. We will explain the necessity of these conditions and how the existing theories can be adapted to work in this new setting.

Hendrik Hubrechts (Leuven), Elliptic and hyperelliptic curve point counting through deformation
Abstract: In this talk we will try to explain the basic ideas behind the use of deformation in point counting algorithms for curves over finite fields. By combining Kedlaya's approach that uses Monsky-Washnitzer cohomology with a one dimensional deformation, one can obtain asymptotic improvements of Kedlaya's result for hyperelliptic curves. For elliptic curves an even faster algorithm can be constructed, that performs also very well in practice. The talk will be concluded by presenting some timing results obtained using an implementation in Magma of some of the algorithms.

Mirjam Jöllenbeck (Paris), The action of the absolute Galois group GQ on the fundamental group of the projective plane P1Q\{0,1,∞}
Abstract: Grothendieck suggested in his 'Esquisse d'un Programme' to study the absolute Galois group GQ of Q via its geometric actions, i.e. its actions on fundamental groups of geometric objects (varieties, schemes, stacks, ...). The action of GQ on π1(P1Q\{0,1,∞}) is the first non trivial instance of the Galois action on the fundamental group of a moduli space Mg,n, namely here M0,4. Via this action an element σ∈GQ can be parametrized as a pair (χ(σ), fσ), where χ is the cyclotomic character and fσ is an element of the commutator group of the profinite completion of the free group on two generators F2. The talk explains this action and the parametrization.

Alan Lauder (Oxford), Ranks of elliptic curves over function fields
Abstract: I present experimental evidence to support the belief that one half of all elliptic curves have infinitely many rational points. Previous experimental work suggested instead that two-thirds did.

Hendrik Lenstra (Leiden), Heuristics for class groups
Abstract: The Cohen-Lenstra heuristics provide a conjectural description of the distribution of class groups of algebraic number fields. The lecture will explain the model behind the conjecture and the intuition on which it is based.

Gunter Malle (Kaiserslautern), On class groups of cyclic cubic fields
Abstract: We present calculations of class groups of cyclic cubic number fields and compare them to the Cohen-Lenstra heuristic. While the results forodd primes are close to the predictions, they behave quite differently for the prime 2. This might indicate that the Cohen-Lenstra heuristic has to be modified for the prime 2.

Adam Mohamed (Essen), Classical Lubin-Tate theory
Abstract: It is not known in general how to generate the maximal abelian extension of a global number field. In contrast, Lubin-Tate theory explicitly describes the maximal abelian extension of a local field. My talk is concerned with this explicit approach to local class field theory.

Anna Morra (Bordeaux), Counting cubic extensions of number fields
Abstract: We want to list the cubic extensions of a number field K with bounded relative discriminant. Using a theorem of Taniguchi, we construct a general algorithm, which works in polynomial time in the output size if K is imaginary quadratic. We describe our implementation over K = Q(i).

Andreas Röscheisen (Heidelberg), Inseparable differential Galois extensions
Abstract: In differential Galois theory in positive characteristic, one is faced with the problem of solution fields that are inseparable over the iterative differential field one started with, or at least with the existence of intermediate differential fields over which the solution field is inseparable. These intermediate differential fields cannot be obtained as the fixed field of a group of automorphisms, hence they don't occur in the Galois correspondence. In this talk, we present an approach to handle this problem. The key point is to define the Galois group as a group scheme (and not only as a group) and to use functorial invariants. We will also see that in this setting, the intermediate differential fields over which the solution field is inseparable are obtained as the 'fixed' fields of the non reduced closed subgroup schemes of the Galois group scheme.

Klaas-Tido Rühl (Lausanne), Annihilating ideals of quadratic forms over local and global fields
Abstract: It was already known by Witt that the Witt Ring of a field K is integral, i.e. every equivalence class of quadratic forms over K is a root of a polynomial with integer coefficients. Naturally the same holds for the Witt-Grothendieck Ring of isometry classes of quadratic forms over K. But only in 1987 did Lewis introduce specific annihilating polynomials. He constructed polynomials that annihilate all isometry classes of quadratic forms with a given dimension over an arbitrary field.

In this talk we will improve Lewis' results for local and global fields. Over those fields it is possible to find annihilating polynomials of dramatically lower degree. We will give a complete and minimal set of generators for the annihilating ideal of a given quadratic form, i.e. the ideal consisting of all annihilating polynomials of a quadratic form.

Xavier Taixés i Ventosa (Essen), An algorithm to compute congruences of representations attached to modular forms
Abstract: We describe an algorithm to compute congruences mod ln of Galois representations attached to modular forms. Then we show some experimental results and we determine explicitly the image of the inertia in some specific cases.

Erwin Torreao Dassen, Introduction to layered lattices
Abstract: In this talk we give an introduction to layered lattices pointing out differences with the classical theory. Motivation will be given in the form of problems from the classical theory that can be better stated in the generalized setting.

Jan Tuitman (Leuven), Point counting on nondegenerate curves
Abstract: I will first introduce nondegenerate (toric) curves and give an outline of an algorithm (by Castryk, Denef and Vercauteren) to compute their zeta functions. Then if time permits I will indicate how deformation techniques can be applied to these curves.

Michael Wibmer (Heidelberg), Splitting fields and coinvariant algebras
Abstract: The splitting field of a polynomial is equipped with a natural filtration which is stable under the action of the Galois group G. In this talk we will explain how the G-module structure of this filtration is related to the coinvariant algebra of G.

Gabor Wiese (Essen), Modular forms in inverse Galois theory
Abstract: Modular forms which are eigenfunctions for all Hecke operators give rise to 2-dimensional mod p representations of the absolute Galois group of the rationals. In the talk we will show how these representations, and hence modular forms, can be used to derive results on the occurrence of groups of the type PSL2(Fpr) as Galois groups over the rationals.

Christian Wuthrich (Nottingham), Computations about the Tate-Shafarevich group using Iwasawa theory
Abstract: In analogy to the zeta function for varieties over finite fields, the p-adic L-series of an elliptic curve E over the rational field can provide us with interesting arithmetical information via Iwasawa theory. I will present an algorithm that can compute upper bounds on the order of the p-primary part of the Tate-Shafarevich group E. This is joint work with William Stein.