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.
Intercity Number Theory SeminarSeptember 22, Utrecht. room K11
|11:15-12:00||Frits 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.
|13:00-14:00||Vasily 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.
|14:15-15:15||Jan 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
The talk touches in several places on the subject of
`special values of zeta functions'.
|15:30-16:15||Sander 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
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 seminarOctober 13, Leiden. Room 312 of the Mathematical Institute (directions)
This is the first seminar of the
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.
|15:00-15:50||Heinrich 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 SeminarNovember 3, Groningen. Room BB217, Blauwborgje 8, Zernike campus (bus 15 from the train station)
|11:30-12:30||Andy Pollington, Badly approximable numbers and Littlewood's conjecture in Diophantine approximation|
Abstract. Littlewood's conjecture in Diophantine approximation is
lim inf q ||qx|| ||qy|| = 0 for all pairs of real numbers
This result is true if either x or y is not a badly
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
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.
|13:00-13:45||Jeroen Sijsling, Dessins d'enfant(s), Platonic or down-to-earth?|
|13:45-14:30||Lenny Taelman, Permutation groups, linear groups, Galois groups in characteristic p|
Abstract. The three parts of the title refer to: Galois
Tannakian Categories, and something that relates to both.
The lecture will be introductory and will assume no prior
Galois or Tannakian categories.
|14:45-15:45||Robert 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.
|16:00-17:30||Mohamed 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
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 latticesNovember 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).
|11:00-12:00||Hendrik 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.
|13:30-14:30||Phong 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.
|14:45-15:45||Friedrich 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.
|16:00-17:00||Karen 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-TateNovember 24, Leiden. Room 412 (first talk), and 312 (others)
|René Schoof, Equidistribution and L-functions|
|13:30-14:30||Gerard van der Geer, Chebotarev for finitely generated fields|
|14:45-17:00||Bas 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