Intercity Number Theory Seminar

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)


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.