Intercity Number Theory Seminar
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2006

February 10 Groningen, Room 31 WSN Gebouw (see map)
11:30-12:30 Irene Polo, Cubic surfaces
Abstract. Cubic surfaces are very classical objects in algebraic geometry, studied from the middle of the 19th century onward, starting with work of Cayley and Salmon. Every smooth cubic surface over the complex numbers is isomorphic to the blow-up of the projective plane in six points. We describe an algorithm for finding such an isomorphism, starting from a given cubic with a line on it. We also explain for which cubics over the real numbers, such an isomorphism exists over the real numbers. Many examples will be given.
13:15-14:15 Gert Schneider, A MacWilliams identity for convolutional codes
Abstract. The MacWilliams identity that relates the weight enumerator of a block code to that of a dual code has given rise to numerous applications and an elaborate theory in classical coding theory. For convolutional codes it is known that there is no such relation of the weight distributions of a code and its dual, suggesting that the weight distribution alone does not carry enough information about the code. The weight adjacency matrix associated to a convolutional code is a more refined object, from which the weight distribution can be derived. In this talk I will outline a conjecture of a MacWilliams identity between the weight adjacency matrices of a convolutional code and its dual and sketch the proof of this conjecture for a certain non-trivial class of convolutional codes, that includes block codes and reduces to the ordinary MacWilliams identity in this case.
14:30-15:30 Remke Kloosterman, The L-series of a cubic fourfold
Abstract. Let X be a cubic fourfold. Suppose X contains a surface T that is not a complete intersection. In this case, we call X "special". Over the field of complex numbers, and under certain conditions on the degree of T, Hassett proved that the variety F(X) of lines on X is isomorphic to the desingularized second symmetric product S[2] of a K3-surface S. In this talk we prove a stronger statement: Suppose X and S are as above and S is defined over K, a subfield of the complex numbers. Suppose that S has a K-rational point. Then X has a model over K and F(X) is isomorphic to S[2] over K. In the case S is defined over Q, it has a rational point and the Picard number of S equals 20, we can use the above result to prove that the "interesting" part of the L-function of X equals the L-function of a Hecke eigenform.
15:45-16:45 Jaap Top, Some cubic curves over function fields
Abstract. We discuss the equations y2=x3+tn+1 and y2=x3+xtn+x over the rational function field C(t) over the complex numbers C. Both define elliptic curves, and we compute the rank of the group of C(t)-rational points as a function of n, using a method of T. Shioda. We also discuss how one may find generators of these groups.
February 24 Nijmegen, DIAMANT intercity: Special day on radical extensions
12:10-13:00 Hendrik Lenstra, Algorithmic Galois theory [PDF]
Abstract. Algorithmic Galois theory occupies itself with the design and analysis of efficient algorithms concerning algebraic field extensions. The main purpose of the present lecture is to explain the rules of the game. The best results that have been obtained are related to solvability by radicals, and they have been achieved by means of group theory. A number of open but apparently feasible problems will be formulated.
13:00-14:00 Lunch break
14:00-14:50 Bart de Smit, Entangled radicals [PDF]
Abstract. For a field K of characteristic 0, a radical group is an abelian group B containing K* so that each b in B has a power in K*. If all finite subgroups of B are cyclic, then we can embed B in the multiplicative group of an extension field of K. To analyze the radical field extension K(B) of K one needs to understand relations between radicals, such as √5 + √-5 = 4√-100. We will show that these are controlled by the entanglement group. As an application, we formulate Artin's primitive root conjecture over number fields.
15:00-15:50 Willem Jan Palenstijn, Computing field degrees of radical extensions [PDF]
Abstract. In this talk we will present an algorithm that efficiently computes the field degree of finite radical extensions over the rationals, up to a suitably defined cyclotomic part. The main ingredient is the theory developed in the previous lecture.
15:50-16:10 Tea break
16:10-17:00 Wieb Bosma, Some radical algorithms in Magma
Abstract. We will discuss several problems and examples of representing certain elements in solvable number fields as nested radicals in the computer algebra system Magma. The problems have to do with ambiguities arising from multivalued root extraction, with testing for equality, with simplification, and with finding such a representation.
March 10 Leiden, room 403
11:30-12:30, 13:30-14:30 E. Friedman (U. of Chile), Survey of Lehmer's Conjecture
Abstract. Kronecker showed that if the roots of a polynomial P(x), assumed monic, irreducible and with integer coefficients, all lie on the unit circle in the complex plane, then the roots of P(x) are actually just roots of unity. But how close to the unit circle can the roots of P(x) be without them all actually being on the unit circle? This simple-looking question turns out to be difficult and we have only partial answers. Lehmer suggested in 1933 a surprisingly precise answer which has stood the test of time.
I will describe what is known about Lehmer's conjecture and how it relates to some other aspects of number theory. I will give Smyth's simple 1971 proof solving Lehmer's problem for non palindromic polynomials (a polynomial is palindromic if the sequence of coefficients reads the same backwards or forwards). This result gives the problem additional structure, allowing the introduction of tools from algebraic and analytic number theory.
The exposition is meant to be accessible to Master's program students, with essentially no prerequisites (more truthfully, the first hour will require nothing beyond the Taylor series of an analytic function, the second hour will require an acquaintance with basic elements of algebraic number theory such as discriminants, regulators and absolute values).
14:45-15:30 J. Brakenhoff (Leiden), Squarefree discriminants
Abstract. Let fZ[X] be a monic polynomial of degree n≥2. Denote by Δ(f) the discriminant of f. We want to determine the probability that Δ(f) is squarefree. Looking locally at each prime, we can find a heuristic value for this probability, which depends on n.
15:45-16:45 S. Vostokov (St. Petersburg), Reciprocity laws:yesterday and today
Abstract. In the talk the origins of reciprocity laws from Euler to our time will be discussed. We start with the classical version, then go to Hilbert's interpretation of this law and discuss the current state of the problem and basic results. We also discuss applications of the reciprocity law to other problems in arithmetic.
March 24 Utrecht, Room K11 (basement)
11:30-12:30, 13:30-14:30 E. Friedman (U. of Chile), Lehmer's Conjecture and Relative Regulators
Abstract. Smyth showed in 1971 that if a polynomial (assumed monic, irreducible, with integer coefficients and of degree at least 2) is not palindromic, then its roots cannot be too close to the unit circle, in a certain precise sense. (A polynomial is palindromic if the sequence of coefficients reads the same backwards or forwards.) Lehmer's conjecture states that Smyth's conclusion should hold even for palindromic polynomials, as long as they are not cyclotomic. The simplest open case of Lehmer's conjecture is that of Salem numbers, i.e. of a palindromic polynomial having only one root outside the unit circle. Such a root α is called a Salem number. It is an algebraic unit which generates a number field L=Q(α), endowed with a subfield K=Q(α+α-1) with [L:K]=2 and NormL/K(α)=1. Thus α is a ``relative unit'' of the extension L/K.
I will describe regulators of relative units and a conjectural lower bound for them which would imply the Salem number case of Lehmer's conjecture. I will sketch a proof of this conjecture in the high-rank case. Unfortunately the Salem number case is the rank-one case. I will end by presenting a formula which might pave the way for a proof of the low-rank case. This is joint work with N.-P. Skoruppa, published some 6 years ago, but still unfinished in the low-rank case.
14:45-15:30, 15:45-16:30 F. Beukers (Utrecht), Lower bounds for heights of algebraic points
Abstract. We consider the algebraic points on an algebraic variety defined over the algebraic closure of Q and their absolute normalised logarithmic heights. In many cases it turns out that one can give uniform non-trivial lower bounds for these heights. We start with a remarkable elementary result by Zhang-Zagier and then proceed to discuss some wide generalisations, with applications to diophantine equations.
April 7 Utrecht Room K11 (basement)
11:30-12:20 Marco Streng, Elliptic divisibility sequences with complex multiplication
Abstract. A classical elliptic divisibility sequence (indexed by the integers) arises as denominators of the multiples of a fixed rational point of infinite order on an elliptic curve over the rationals. Such a sequence is knownto have primitive divisors from some point on (as follows from height estimates and Siegel's theorem). If the curve has complex multiplication, we show how the cm-ring can be used to index a similar sequence of ideals and prove that it has primitive divisors. The classical proof breaks down and needs to be replaced by an inclusion/exclusion proof with finer diophantine estimates involving ellipticlogarithms
13:30-14:20 Gunther Cornelissen, Deformations of weakly ramified coverings of curves
Abstract. A cover of curves over a field of characteristicp is said to be weakly ramified if all second ramification groups vanish. I will discuss work with Ariane Mézard that computes the (mixed-characteristic) universal deformation ring of such a cover. Using previous work with Bertin and Kato, one only needs to determine the characteristic of that ring, which turns out to be either p or zero.
14:30-15:20 Jakub Byszewski, A universal deformation ring that is not a complete intersection
Abstract. Bleher and Chinburg recently gave an example of a linear representation of a finite group whose universal deformation ring is not a complete intersection, thus answering a question of Flach. Their proof uses some modular representation theory. We will show that the same result can be proven using only standard cohomological obstruction calculus.
15:40-16:30 Oliver Lorscheid, Toroidal integrals
Abstract. In the 1970's, Don Zagier developed the formalism of toroidal automorphic forms, where the usual parabolic condition is substituted by the vanishing of integrals over various tori. We will outline the relation between these spaces and zeros of zeta functions (via Eisenstein series and Tate's thesis), paying particular attention to the case of a split torus (left somewhat aside in the original work) and to explicit calculation of such integral vanishing conditions in the case of global function fields.
April 21 Groups and characters, a farewell symposium for Rob van der Waall,
Universiteit van Amsterdam, room P.227 of the Euclides building.
See the UvA announcement for directions.
10:30-11:00 Coffee
11:00-11:50 Hendrik Lenstra (Leiden), Factoring polynomials over solvable closures
Abstract. One of Rob van der Waall's most celebrated results concerns solvable extensions of number fields, and it is proved by means of group theory. The same applies to the result that the present lecture is devoted to: there is an efficient algorithm for factoring polynomials over the solvable closure of a number field. Some care is required in formulating precisely what this assertion means, because the solvable closure of any number field is of infinite degree over the field of rational numbers. The lecture will provide the context, the definitions, the algorithm, as well as the result from group theory that is crucial in proving the correctness of the algorithm.
12:00-12:50 Gabriele Nebe (Aachen), Codes and invariant theory
Abstract. In 1970 on the ICM in Nice, A.M. Gleason presented his famous theorem that the weight enumerator of a doubly-even self-dual binary code is an element of the polynomial ring generated by the weight enumerators of the Hamming code of length 8 and the Golay code of length 24. The proof uses the fact that this polynomial ring is the invariant ring of the complex reflection group G9 of order 192. In the meantime, many variations of this theorem have been proven. Together with E. Rains and N. Sloane, we develop a theory that allows us to prove that, in a quite general situation, the weight enumerators of codes of a given Type over a not necessary commutative finite ring span the invariant ring of the associated Clifford-Weil group. These are finite complex matrix groups given by explicit generators.
14:00-18:20 Afternoon program
To attend the afternoon program and the closing reception please register by sending an email to Evelien Wallet at ewallet@science.uva.nl.
This part of the day will include a lecture of Bertram Huppert (Mainz) with the title How to shuffle cards, and lectures by Arjeh Cohen and Rob van der Waall.

Further activities in the summer of 2006

Special day on explicit complex multiplication theory

September 8, Leiden. first lecture: room 403, other three: room 412

It has been known since the 19th century that the values of certain analytic functions, such as the exponential function, generate families of algebraic extensions over the rationals. Despite the exponential nature of the phenomenon, computations in "low genus" are possible, and the genus 1 case is more or less classical. We will review the classical theory from various angles before moving on to computations in genus 2, which are still in their infancy.

12:00-12:45Peter Stevenhagen, Algebraic extensions from analytic functions
13:45-14:30Peter Stevenhagen, Explicit computations using Shimura's reciprocity law
14:45-15:30Everett Howe, Complex multiplication of abelian surfaces
15:45-16:30Everett Howe, Explicit computation of Igusa invariants

Intercity Number Theory Seminar

September 22, Utrecht. room K11

11:15-12:00Frits Beukers, Irrationality of p-adic L-values
Abstract. In this lecture we show how to prove irrationality of certain values of p-adic L-series using classical continued fractions a la Stieltjes.
13:00-14:00Vasily Golyshev (Moscow), Spectra and Their Arithmetic
Abstract. I will present a survey of results on the arithmetic of quantum spectra of certain algebraic varieties. The emphasis will be made on explaining a recurring pattern that is still unaccounted for: a classification problem in geometry of Fano varieties can be translated into a statement of purely arithmetic nature whose solution may be translated back into a solution of the original problem.
14:15-15:15Jan Stienstra, Apery-like numbers, differential equations of type DN and dimer models
Abstract. Apery-like numbers can be generated in (at least) two ways. One way is by a recurrence relation, or equivalently a differential equation. Vasily Golyshev observed that the corresponding differential operators (which he called `of type D2') can be written as the determinant of a matrix whose entries are linear DO's. The matrix entries have an interpretation in terms of quantum cohomology and Gromov-Witten invariants of Del Pezzo surfaces. This type of differential operators can be generalized to higher orders and are then conjectured to contain valuable information about Fano varieties.

Another way to generate Apery-like numbers is as constant terms in powers of a two-variable Laurent polynomial. In some interesting cases this Laurent polynomial happens to be the determinant of the Kasteleyn matrix of a dimer model (a certain type of graph, which in math is also known as a `dessin d'enfant of genus 1'). Also this approach has very interesting generalizations. The talk touches in several places on the subject of `special values of zeta functions'.

15:30-16:15Sander Dahmen, Lower bounds for numbers of ABC-hits
Abstract. An ABC-hit is a triple (a,b,c) of relatively prime positive integers such that a+b=c and rad(abc) < c. It is easy to see that there exist infinitely many ABC-hits. I will discuss lower bounds for the number of ABC-hits (a,b,c) with c < x (denoted N(x)) when x goes to infinity. In particular I will prove that for every e > 0 and x large enough

N(x) > exp((logx)1/2-e).


Article: Lower bounds for numbers of ABC-hits [pdf]

GTEM Kick-off seminar

October 13, Leiden. Room 312 of the Mathematical Institute (directions)

This is the first seminar of the GTEM Research Training Network.

This is the first GTEM seminar of the hosted by the Dutch Intercity Number Theory Seminar.

John Cremona, Lattice reduction over function fields, with applications to finding points on curves over function fields
Abstract. Methods for finding rational points on algebraic curves and higher-dimensional varieties based on lattice-reduction first came to attention through Elkies ANTS IV article (2000), which was based on real approximations. This was followed by a p-adic method, often referred to as "p-adic Elkies", which seems to have been thought up independently by several people, including Heath-Brown and Elkies. This method is easy to describe and implement and has been used very successfully, for example, in finding rational points on quadric intersections in P3 (which is useful for 2- and 4-descent on elliptic curves). I will report on joint work with Nottingham student David Roberts showing that a similar method may also be applied to curves defined over Fq(T), replacing LLL-reduction of Z-lattices by the "Weak Popov Form" of an Fq[T]-lattice.
René Schoof, Semi-stable abelian varieties and modular curves
Abstract. We show that for every odd squarefree integer n < 30, every semi-stable abelian variety over Q is isogenous to a power of the Jacobian of the modular curve X0(n).
Michel Matignon, p-Groups and automorphism groups of curves in characteristic p>0
Abstract. I will explain my motivations to look at p-groups of automorphisms of curves, then I will report on old and new results concerning p-cyclic covers of the affine line in char. p>0. I will deduce the notion of big p-group action on a non zero genus curve and use classfield theory in order to produce such actions; then I will begin a classification. If I have enough time I will show how to get examples of p-cyclic covers of the projective line over a p-adic field with a big wild monodromy group.
15:00-15:50Heinrich Matzat, Differential equations and finite groups
Abstract. It is an old question to characterize those differential equations or differential modules, respectively, whose solution spaces consist of functions which are algebraic over the base field. The most famous conjecture in this context is due to A. Grothendieck and relates the algebraicity property with the p-curvature which appears as the first integrability obstruction in characteristic p. Here we prove a variant of Grothendieck's conjecture for differential modules with vanishing higher integrability obstructions modulo p - these are iterative differential modules - and give some applications.

Intercity Number Theory Seminar

November 3, Groningen. Room BB217, Blauwborgje 8, Zernike campus (bus 15 from the train station)

11:30-12:30Andy Pollington, Badly approximable numbers and Littlewood's conjecture in Diophantine approximation
Abstract. Littlewood's conjecture in Diophantine approximation is that lim inf q ||qx|| ||qy|| = 0 for all pairs of real numbers (x,y). This result is true if either x or y is not a badly approximable number. We show that for all badly approximable x and a set of y which are badly approximable and have full Hausdorff dimension this is still true if instead we consider lim inf f(q) ||qx|| ||qy|| where f(q) is any increasing function for which f(q) =o(q logq). This is joint work with Sanju Velani.
13:00-13:45Jeroen Sijsling, Dessins d'enfant(s), Platonic or down-to-earth?
13:45-14:30Lenny Taelman, Permutation groups, linear groups, Galois groups in characteristic p
Abstract. The three parts of the title refer to: Galois Categories, Tannakian Categories, and something that relates to both. The lecture will be introductory and will assume no prior knowledge of Galois or Tannakian categories.
14:45-15:45Robert Carls, A higher dimensional 3-adic CM construction
Abstract. My talk is about joint work with D. Kohel and D. Lubicz. I will sketch a new 3-adic method for the construction of CM curves over number fields. A CM curve is a curve whose Jacobian has complex multiplication. Our method is based on Hensel lifting by means of equations defining a higher dimensional analogue of X0(3). Curves with prescribed complex multiplication are used in primality testing algorithms and as key parameters in pairing based cryptosystems. An essential step in our algorithm is the computation of the theta null point of a canonical lift of an ordinary abelian variety over a finite field of characteristic 3. The lifting algorithm has quasiquadratic time complexity in the degree of the finite field. Explicit examples will be computed.
16:00-17:30Mohamed Barakat, Homological algebra and applications to linear control theory
Abstract. In this talk I will try to explain why a linear control system is equivalent to a module over an appropriate ring. Questions arising in linear control theory have their direct analoga in module theory and vice versa. Homological constructions thus lead to insights in the control system that are independent of its realization. I will introduce the basics of homological algebra and illustrate using our symbolic algebra package "homalg" the above mentioned interconnection by several examples over computable rings. The first complete implemetation of the Quillen-Suslin theorem developed at our work group can now be accessed through "homalg" and enables one to explicitly construct a flat output of a flat control system.

DIAMANT intercity seminar on lattices

November 10, Leiden. This day is organized together with Karen Aardal
The first lecture takes place in room C3, the others in room C1 of the Gorlaeus lab (directions).

11:00-12:00Hendrik Lenstra, A new type of lattices
Abstract. The lecture will start by recalling how one can use a lattice basis reduction algorithm for solving systems of linear equations over the ring of integers. An analysis of this application suggests that one can more appropriately handle it by means of a new notion of lattice, for which the length function takes values in an ordered vector space of dimension greater than one. The full theory of these generalized lattices, as well as the corresponding basis reduction algorithms, remain to be developed. No previous knowledge of lattices is necessary for following the lecture.
13:30-14:30Phong Nguyen, Hermite's constant and lattice reduction algorithms
Abstract. Lattice reduction is a computationally hard problem of interest to both public-key cryptography and public-key cryptanalysis. Despite its importance, extremely few algorithms are known. In this talk, we will survey all lattice reduction algorithms known, and we will try to speculate on future developments. In doing so, we will emphasize a connection between those algorithms and the historical mathematical problem of bounding Hermite's constant.
14:45-15:45Friedrich Eisenbrand, Integer programming: results in fixed dimension
Abstract. In this lecture we will survey results on integer programming in fixed dimension which are obtained by using lattices and lattice basis reduction techniques. After we review the basic principles which lead to polynomial algorithms for integer programming, we also survey structural results concerning the integer hull and outline recent algorithms which show that integer programming in fixed dimension with a fixed number of constraints can be solved with a linear number of arithmetic operations.
16:00-17:00Karen Aardal, Lattices and integer programming formulations
Abstract. We consider the problem of determining whether the system of equations Ax = d has an integer solution x satisfying 0 ≤x u. We reformulate the problem using a reduced basis for a certain lattice. We then take a closer look at the special case where the matrix A has one row. We observe that a certain input structure makes the original formulation computationally hard even in low dimension. The reduced basis used in the reformulation detects this structure, and enables us to search for a feasible solution effectively. We explain this theoretically, both for the reformulation as well as for the original formulation.

Special day on Chebotarev and Sato-Tate

November 24, Leiden. Room 412 (first talk), and 312 (others)

Program:

René Schoof, Equidistribution and L-functions
13:30-14:30Gerard van der Geer, Chebotarev for finitely generated fields
14:45-17:00Bas Edixhoven, Recent results on the Sato-Tate conjecture
Abstract. Elliptic curves over number fields or function fields over finite fields lead to l-adic Galois representations, and Frobenius conjugay classes in SU2. The Sato-Tate conjecture states that these conjugacy classes are equidistributed, if the elliptic curve has no potential complex multiplication. What equidistributed in SUn for general n means for the eigenvalues is precisely Weyl's integration formula. As explained in Schoof's lecture, the equidistribution follows from suitable properties of L-functions. These properties then follow from modularity statements for the symmetric powers of the l-adic Tate modules of the elliptic curve that have been recently proved by Taylor cum suis.