Groningen, Room 31 WSN Gebouw (see map)
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.
A MacWilliams identity for convolutional codes
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
distribution alone does not carry enough information about the code.
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
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.
The L-series of a cubic fourfold
Let X be a cubic fourfold. Suppose X contains a surface
is not a complete intersection. In this case, we call X
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 of a
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 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
"interesting" part of the L-function of X equals the
L-function of a
Some cubic curves over function fields
We discuss the equations
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.
Nijmegen, DIAMANT intercity: Special day on radical extensions
|| Hendrik Lenstra, Algorithmic Galois theory
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
Bart de Smit, Entangled radicals
For a field K of characteristic 0, a radical group is an
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
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.
Willem Jan Palenstijn,
Computing field degrees of radical extensions
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.
Some radical algorithms in Magma
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.
Leiden, room 403
E. Friedman (U. of Chile), Survey of Lehmer's Conjecture
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
Let f∈Z[X] be a monic polynomial of degree
n≥2. Denote by
Δ(f) the discriminant of f. We want to determine
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
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.
Utrecht, Room K11 (basement)
E. Friedman (U. of Chile), Lehmer's Conjecture and
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
[L:K]=2 and NormL/K(α)=1.
Thus α is a ``relative unit'' of the extension
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.
F. Beukers (Utrecht), Lower bounds for heights of algebraic
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
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.
Utrecht Room K11 (basement)
Marco Streng, Elliptic divisibility sequences with complex
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
Gunther Cornelissen, Deformations of weakly ramified
coverings of curves
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.
Jakub Byszewski, A universal deformation ring that is not a
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
Oliver Lorscheid, Toroidal integrals
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.
Groups and characters, a farewell symposium for Rob van der
Universiteit van Amsterdam, room P.227 of the Euclides
See the UvA
announcement for directions.
||Hendrik Lenstra (Leiden),
Factoring polynomials over solvable closures
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.
||Gabriele Nebe (Aachen),
Codes and invariant theory
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.
||Afternoon program |
To attend the afternoon program and the closing reception
please register by sending an email to
Evelien Wallet at
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
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