
Intercity Number Theory SeminarFebruary 15, Leiden. Room 40712:1513:00  Hendrik Lenstra, Degrees of field automorphisms Abstract. Consider a field extension where the top field is
algebraically closed and of finite nonzero
transcendence degree over the bottom field. The
lecture describes a large discrete quotient of the
relative automorphism group.
 14:0014:45  Jonathan Hanke, The 290Theorem and Representing Numbers by Quadratic Forms Abstract. This talk will describe several finiteness theorems for quadratic
forms, and progress on the question: "Which positive definite
integervalued quadratic forms represent all positive integers?". The
answer to this question depends on settling the related question
"Which integers are represented by a given quadratic form?" for
finitely many forms. The answer to this question can involve both
arithmetic and analytic techniques, though only recently has the
analytic approach become practical.We will describe the theory of quadratic forms as it relates to
answering these questions, its connections with the theory of modular
forms, and give an idea of how one can obtain explicit bounds to
describe which numbers are represented by a given quadratic form.
 15:0015:45  ChiaFu Yu, On geometric mass formulas Abstract. In this talk we will start by discussing the EichlerDeuring mass
formula concerning supersingular elliptic curves. We then discuss a result
on geometric mass for superspecial principally polarized abelian varieties
due to Ekedahl and some others, and its function field analogue for
supersingular Drinfeld modules. We shall also describe a generalization to
the quaternion unitary cases.
 16:0016:45  Lenny Taelman, Cubic rings, cubic forms Abstract. After Gan, Gross, Savin and Deligne:
see Deligne's letter to Gan, Gross and Savin and
Deligne's letter to Edixhoven 

Lfunctions and friendsFebruary 29, VU Amsterdam. Room M129. Today's program is part of Rob de Jeu's Algebra seminar. The seminar finishes early because of the inaugural lecture of Rob van der Vorst at 15:45.11:0012:00  Tejaswi Navilarekallu, Galois actions and Lvalues Abstract. Let K/Q be a finite Galois extension with Galois group G. For every character χ of G, the special value L(χ,0) of the Artin Lfunction carries arithmetic information about the extension. The equivariant Tamagawa number conjecture predicts the precise relation between the Lvalues and certain arithmetic invariants. In this talk, we shall give a formulation of the conjecture and indicate some techniques of verification.  13:0014:00  Don Zagier, Modular Green's functions Abstract. The functions of the title arose many years ago in connection with heights of Heegner points and special values of Lfunctions, but turned out to have further interesting properties, including a conjectural algebraicity statement for their special values. This conjecture has recently been proved in many cases by Anton Mellit. We will discuss this and related work.  14:1515:15  XavierFrançois Roblot, Computing values of padic Lfunctions over a real quadratic field Abstract. Following the works of P. CassouNogučs, D. Barsky, N. Katz and P. Colmez, I will give an explicit construction of a continuous padic function interpolating (some of) the values at negative integers of the Hecke Lfunction of a real quadratic field, the socalled padic Hecke Lfunction. I will show how this construction allows one to compute (approximations of) the values of this function. 

Intercity Number Theory Seminar: Van der Kallen CelebrationMarch 14, Utrecht. room BBL 160. Today's seminar is dedicated to the 61st birthday of Wilberd van der Kallen. After the talks at 4PM there will be a reception in the library.10:3011:30  Jan Draisma, Phylogenetic tree models and classical invariant theory Abstract. Tree models are families of probability distributions that are
used in reconstructing evolutionary trees from genetic data. They are given
by a parameterisation, whereas an implicit description, by means of
polynomial equations, would facilitate testing whether a given empirical
distribution lies in the family. These equations are hard to find, but I
will explain how to use classical invariant theory to reduce the quest for
them to the case of simple trees.
 11:4512:45  Wim Hesselink, Euclidean skeletons of digital images in linear time by the integer medial axis transform Abstract. Roughly speaking, a skeleton of a twodimensional image is a line
drawing that represents the form of the image. The aim of the talk
is to resolve the vagueness of this description.
Digital image and volume data are given as grey values associated to
grid points (pixels or voxels). There can be millions of grid
points. Therefore efficient algorithms are needed. One of the
efficient algorithms available is the feature transform that, for a
given set of background points in a rectangular box, determines in
linear time for every grid point in the box a closest background
point. The integer medial axis IMA is a way to use this for
skeletonization. We discuss theorems and conjectures about IMA,
one of which was both refuted and proved by Wilberd van der Kallen.
 13:4514:45  Johnny Edwards, The chocolate bar conjecture Abstract. In a recent paper, Wilberd van der Kallen proves an inequality satisfied
by quadruples of points in the lattice Z^{2}. This inequality is at the
heart of an algorithm in image processing. See the last lecture. In his
paper, van der Kallen conjectures a similar inequality for points in the
3 dimensional lattice Z^{3}. I will explain the chocolate bar conjecture
and show that it is true in dimension 2 and 3. From this I will deduce
that van der Kallen's conjecture is true, modulo at most a finite set of
exceptional quadruples of lattice points.
 15:0016:00  Vincent Franjou, Finite generation for (higher) invariants Abstract. A classical problem in invariant theory, often referred to as
Hilbert's 14th problem, asks, when a group acts on a finitely generated
algebra by algebra automorphisms, whether the ring of invariants is still
finitely generated. Nagata's work on this problem led to the notion of
geometric reductivity. I shall present a variant of this notion, "power
reductivity", which is better adapted to working over a general
commutative ring. I shall then present progress on the corresponding
conjecture of van der Kallen for higher invariants. This is work in
progress with Wilberd van der Kallen.


Intercity Number Theory SeminarApril 4, Groningen. room 105 of the new Bernoulliborg building, Nijenborgh 9: the big blue building on your right when bus 15 enters the campus.11:4512:45  Cécile Poirier, Stacks, vector bundles and `geometric Langlands'.  13:3014:30  FaiLung Tsang, Skew rings and geometric method in convolutional codes Abstract. There is a natural way to assign a degree structure on the
module F[z]⊗V (V a finite dimensional Fspace), namely the
one induced by the polynomial ring F[z]. One has the question, for any
given sequence of positive numbers d_{1},dots,d_{r}, can one find a
submodule such that (1) there is a basis with degrees d_{i} and (2) the
sum of the degrees of this basis is minimal. The answer is affirmative for
F[z]^{n}.In this talk, we study the case where F[z]⊗V is a skew
(polynomial) ring. We will construct a free vector bundle on
P^{1} and the subbundle associated to the submodule. It turns out
that we can give a positive answer when ∑d_{i} ≤n. The same
technique is applicable to problems concerning polynomial matrices, I will
talk about a question posed by A. Dijksma and B. Ćurgus, and give an
answer to the problem using the geometric method developed. This is joint
work with M. van der Put.
 14:4515:45  Steve Meagher (Freiburg), Genus three curves revisited  16:0017:00  Evgeny Verbitskiy, Marius van der Put, Abelian sandpiles and discrete difference equations 

Intercity Number Theory Seminar: genus 2 dayApril 18, Utrecht. room BBL 160 (BuysBallotLab)11:0011:50  Marco Streng, Igusa class polynomials Abstract. Igusa class polynomials are the genus 2 analogue of the classical Hilbert
class polynomial. We explain both notions and discuss the differences
between the classical (elliptic) case and the genus 2 case, mostly from a
computational perspective.
 12:0012:50  Jeroen Sijsling, Humbert Surfaces and Shimura Curves Abstract. Humbert surfaces are special subvarieties of the modular threefold of genus
2 curves over the complex field, parametrizing the curves whose Jacobian
has an endomorphism algebra containing a real quadratic extension of the
rationals. Certain Shimura curves, moduli curves parametrizing curves for
which the endomorphism algebra of the Jacobian contains a quaternion
algebra over the rationals, can be obtained as a curve in the modular
threefold as the intersection of Humbert surfaces.
We will explain these notions in more detail, then work towards some
concrete results by Hashimoto/Murabayashi and Lange/Wilhelm on the
equations defining Humbert surfaces in terms of Igusa invariants.
 14:3015:20  Andrew Hone, Somos Sequences and genus two addition formulae Abstract. Somos sequences are nonlinear recurrence sequences defined by a
quadratic relation. They arise in number theory (Morgan Ward's
elliptic divisibility sequences), combinatorics (Fomin &
Zelevisnky's cluster algebras) and integrable systems (discrete
Hirota equations, QuispelRobertsThompson maps). In the fourth
order (Somos4) and fifth order case (Somos5) they correspond to
sequences of points on elliptic curve. After reviewing the elliptic
case, we present some results in genus 2, mainly concentrating on
Somos6, for which we give a formula in terms of Kleinian sigma
functions.
 15:3016:20  Dan Bernstein, Hyperellipticcurve cryptography Abstract. The only publickey cryptographic systems
currently recommended by the United States National Security Agency
are ellipticcurve systems. I'll explain how elliptic curves are used in
cryptography and how genus2 hyperelliptic curves can do better; in
particular, I'll discuss recent progress in genus2 scalar
multiplication and in constructing secure genus2 curves. To balance the
picture I'll also discuss recent progress in ellipticcurve
computations. 

Intercity Number Theory SeminarMay 9, Leiden. room 207 Huygens building12:0013:00  Hugo Chapdelaine, An introduction to the 12th Hilbert problem Abstract. The 12th Hilbert problem consists in finding a way
of constructing explicitly the maximal abelian extension of
a given number field K. In the first half of the talk we will illustrate
fragmentary results which are known on this problem, namely in
the case where K is the field of rational numbers or an imaginary quadratic
number field. For the second half of the talk we will mention some recent
padic constructions of conjectural punits in abelian extensions of real
quadratic fields. Hopefully this should bring some new insight towards
the 12th Hilbert problem.  14:0016:00  Cristian Popescu, On the CoatesSinnott Conjectures Abstract. The conjectures in the title were formulated in
the late 1970s as vast generalizations of the classical theorem of
Stickelberger. They make a subtle connection between the
Z[G(L/k)]module structure of the Quillen Kgroups K_{*}(O_{L}) in an abelian extension L/k of
number fields and the values at negative integers of the
associated G(L/k)equivariant Lfunctions Θ_{L/k}(s).
These conjectures are known to hold true if the base field k is
Q, due to work of CoatesSinnott and Kurihara. In this
talk, we will provide evidence in support of these conjectures
over arbitrary totally real number fields k.
 16:1517:15  Remke Kloosterman, Computing the MordellWeil group of elliptic threefolds Abstract. In this talk we discuss a method to compute the rank of
E(Q(s,t)) for a class of elliptic curves E defined over Q(s,t).This method relies on an explicit method to compute H^{4}(Y,C) for a class of
singular threefolds Y. This is joint work with Klaus Hulek. 

Intercity number theory seminarSeptember 12, Leiden. Room 41214:0014:45  Gabriel Chênevert, The quartic fields method Abstract. In this talk I want to explain Serre's quartic fields method, which provides
a way to decide whether two 2dimensional, 2adic absolutely irreducible
Galois representations are equivalent or not. As an example, the method will
be used to determine the modular form corresponding to a part of the
cohomology of a certain smooth cubic fourfold admitting an action by the
symmetric group S_{7}.
 15:0015:45  Sylvain Brochard, Picard functor and algebraic stack Abstract. The Picard functor of a scheme, classifying invertible
sheaves on it, has been studied extensively in the 60's. However, the
work of Giraud, Deligne, Mumford and Artin gave birth in the 70's to
the notion of an algebraic stack, which generalizes that of a scheme.
The following question arises then: does the Picard functor of an
algebraic stack behave like that of a scheme ?
In this talk I will briefly recall what the Picard functor of a scheme
is and what it is designed for. Then I will explain in few words what
an algebraic stack is and try to answer the preceding question.
 16:0016:45  Ronald van Luijk, Density of rational points on diagonal quartic surfaces Abstract. It is a wide open question whether the set of rational points
on a smooth quartic surface in projective threespace can be
nonempty, yet finite. In this talk I will treat the case of diagonal
quartics V, which are given by ax^{4}+by^{4}+cz^{4}+dw^{4}=0 for
some nonzero rational a,b,c,d. I will assume that the product
abcd is a square and that V contains at least one rational
point P. I will prove that if none of the coordinates of P is zero,
and P is not contained in one of the 48 lines on V, then the
set of rational points on V is dense. This is based on joint work
with Adam Logan and David McKinnon.


Standard models of finite fieldsSeptember 26, Nijmegen. Two days after ECC08
this DIAMANT Intercity
will address the theoretical and practical aspects of defining finite fields algorithmically. The lectures will take place in
room HG 00.071 of the Huygensgebouw, which is one of the
lecture halls natural sciences at the Faculty of Science (FNWI),
Heyendaalse weg 135 (green building)  see also this page. To reach the building take bus 10 from central station Nijmegen to Heyendaalse weg. Paid parking is available in a car park under or behind the Huygens building (entrance
Driehuizerweg). 11:3012:15  Frank Lübeck, Conway polynomials Abstract. I will give the definition of the Conway polynomials which define
finite fields, mention some cases where they are used, and explain how they
can be computed. Then I will address the problem that the Conway polynomials
which are not yet known are very difficult to compute. On the other hand
one would like to know them for any field GF(q) for which the factorization
of (q1) is known (these are the fields in which elements can be tested for
primitivity). I will propose a modification of the
definition such that the modified polynomials can be computed in reasonable
time.  13:3014:15  Wieb Bosma, Dealing with finite fields in Magma Abstract. In computations with finite fields it is essential to maintain
subfield relations in a consistent way. In this talk I will describe
the different representations for finite fields in the computer algebra
system Magma, and the mechanism used for ensuring that
subfield diagrams commute.  14:4515:30  Bart de Smit, Consistent isomorphisms between finite fields Abstract. We give a deterministic polynomial time algorithm that on input two finite fields of the same cardinality produces an isomorphism between the two.
Moreover, if for three finite fields of the same cardinality
one applies the algorithm to the three pairs of fields then one obtains
a commutative triangle. The algorithm depends on the definition given in
the next talk.
 16:0016:45  Hendrik Lenstra, Defining F_{q} Abstract. The lecture provides a definition of F_{q} as an actual field
of cardinality q, as opposed to a field just defined up to isomorphism. The
definition is complicated enough that it occupies most of the lecture. No
easier definition is known that has the attractive algorithmic properties
needed in the previous talk. Notes: Standard models of finite fields: the definition [PDF].


Intercity number theory seminarOctober 10, Leiden. The first talk is in room 402 and the others in room 174.12:0012:45  Karen Aardal, Integer programming and some connections to number theory Abstract. We introduce integer programming, some aspects of polyhedral
combinatorics, and links to number theory.
 13:4514:30  Andrea Montanari, Torus based cryptography Abstract. We give a brief description of recent developed techniques for
publickey cryptography over finite fields relying on algebraic torus. In fact
in some cases it is possible to get a compressed representation of elements in
the torus. We discuss an explicit compression/decompression map for elements of
the torus in quadratic finite field extensions.  14:4515:30  Willemien Ekkelkamp, Predicting the sieving effort for the number field sieve Abstract. In order to estimate the most timeconsuming step of the number field
sieve,
namely the sieving step in which the socalled relations are generated, we
present and discuss a new method for predicting the sieving effort.
Our method takes relations from a short sieving test as input and simulates
relations according to this test. After removing socalled singletons, we
decide how many relations we need for the factorization according to
the simulation and this gives a good estimate (within 2%) for the real
sieving.


Intercity number theory seminarNovember 14, Groningen. room 267 of the mathematics building Bernoulliborg11:3012:30  Andrey Timofeev, Indexcalculus in the Brauer groups with arithmetic applications Abstract. The Brauer group is an important invariant of field. In this talk the
definitions and some basic properties will be given and considered how
can we apply Index Calculus algorithm for computing Euler's totient
function as well as for solving the discrete logarithm problem in finite fields with approach of
Brauer groups.  13:3014:30  Felix Fontein, A Concise Interpretation of the Infrastructure of a Global Field Abstract. In this talk, we will present an interpretation of the
infrastructure of a global field. We will describe explicit arithmetic
and relate the infrastructure to the (Arakelov) divisor class group.
This extends work by D. Shanks, H. Lenstra, R. Schoof and others.  14:4515:45  Marius van der Put, A geometric approach to Painlevé differential equations Abstract. The singularities of the solutions of a linear differential equation (in
one variable) coincide with the singularities of the equation. This is not
the case for non linear differential equations. Painlevé introduced the
notion `no moving singularities' to obtain interesting equations. The ones
with order one were classified in a more or less satisfactory way. The
chaos of the order two equation with the above `Painlevé's property'
have been put into the cages PIPVI. They remained there until about
1980 when the subject obtained new impulses from physics. The research on
Painlevé equations is ever growing. In this lecture I will present new
results, obtained in collaboration with MasaHiko Saito, on the `moduli'
and the `monodromy spaces' associated to the Painlevé equations.
 16:0017:00  Jaap Top, Maximal curves over finite fields Abstract. A (smooth, complete, geometrically irreducible) curve C defined
over a finite field F_{q} is called maximal, if the
cardinality of the set of F_{q}rational points on
C equals q+1+gm. Here g denotes the
genus of the curve, and m is the largest integer not exceeding
2√q.
We consider the following question: given a curve over Z[1/N],
for which prime powers q coprime with N, is C⊗F_{q} maximal over
F_{q}?
Although in general this is a difficult problem (already for genus 1!), it
turns out that in certain examples one can give a complete answer. In the
talk this is done for a particular hyperelliptic curve of genus 3. 

Intercity number theory seminarDecember 12, UvA Amsterdam. Morning program in room P0.19 of the Euclides building. Tom Koornwinder gives his farewell address in the afternoon at 15:00.10:3011:15  Arjen Stolk, Fast group operations on Jacobians Abstract. Kamal KhuriMakdisi has described a construction which reduces the problem of
computing with points on the Jacobian of a general curve to linear algebra in
certain RiemannRoch spaces. These algorithms are elegant and fast. In this talk
I will explain the basic ideas behind KuhriMakdisi's approach.
 11:3012:15  Peter Bruin, Finding random points on curves over finite fields Abstract. Consider a smooth, complete, absolutely irreducible curve X of
genus g over a finite field of q elements, given by a
projective embedding as described by K. KhuriMakdisi
(Math. Comp. 73 (2004) and 76 (2007)). We give an algorithm
for picking uniformly random elements of the set of rational points on
X, with expected running time polynomial in g and
log q.
 12:3013:15  Sylvain Brochard, On De Smit's conjecture on flatness on Artin rings Abstract. Let f: A →B be a flat morphism of Artin local rings with the same
embedding dimension (the embedding dimension is the dimension of the
tangent space). Bart de Smit conjectured that any finite Bmodule that
is Aflat is Bflat. We will give some partial results in this
direction, and explain the proof for a particular example. 


