Intercity Number Theory Seminar
September 2 Special program around PhD defense of Gabor Wiese.
Location of the talks: Spectrumzaal of Studentencentrum Plexus, Kaiserstraat 25, Leiden ( directions)
10:00−10:45 Gabor Wiese (Leiden), Modular Forms of Weight One Over Finite Fields.
Abstract. The talk having the same title as my thesis to be defended in the afternoon will start by presenting the main motivation for my research, namely (some of) the number theoretic information really and conjecturally provided by modular forms. Next, there will be a short overview over the thesis. Finally, I will sketch the proof of one of the results.
11:00−11:45 Loïc Merel (Paris), The formula relating modular symbols to L-values.
Abstract. There are well known formulas relating values of L-functions of modular forms to modular symbols. These formulas enable to construct p-adic L-functions etc. Modular symbols contain a finite generating set consisting of the so-called Manin symbols. I will describe how one can express a Manin symbol at level N in terms of L-values obtained by twisting modular forms of level N by characters of level dividing N and of a few local invariants. Here is an elementary corollary of this formula : the regular representation of Gal(QN)/Q), where QN) is the field generated by a primitive N-th root of unity, does not occur in the group of QN)-rational points of an elliptic curve E over Q of conductor N.
14:15−15:00 Thesis defense of Gabor Wiese at the Academiegebouw, Rapenburg 73 in Leiden.

Special day on the ABC-conjecture, September 9 2005

This is the kick-off meeting of an NWO sponsored "Leraar in Onderzoek" project that will help Kennislink to take ABC to the masses.

September 9 Leiden, room 412.
11:15-12:00Frits Beukers (Utrecht), Introduction to the ABC conjecture [PDF]
12:15-12:45Jaap Top (Groningen), Finding good ABC triples, part I; notes in Dutch (PDF)
14:00-14:30Johan Bosman (Leiden), Finding good ABC triples, part II; notes in Dutch (PDF)
14:45-15:30Hendrik Lenstra (Leiden), Granville's theorem; notes in Dutch (PDF)
Abstract. Barry Mazur defined the `power' of a number to be the logarithm of the number to the base its radical. For example, every perfect square has power at least 2. How many integers up to a large bound have power at least a given number? This question is answered by Granville's theorem. It is of importance both in understanding why the ABC-conjecture has a chance of being true, and in analyzing an algorithm for enumerating ABC-triples.
15:45-16:15Willem Jan Palenstijn (Leiden), Enumerating ABC triples; notes in Dutch (PDF)
Abstract. An ABC triple is a triple of coprime positive integers a, b, c with a + b = c and c larger than the radical of abc. In this talk we present an algorithm that enumerates all ABC triples with c smaller than a given upper bound N with a runtime essentially linear in N.
16:30-17:00Carl Koppeschaar (Kennislink), Reken mee met ABC [PPT]

September 23 Utrecht room K11
Joint session with Intercity Arithmetic Geometry
11:00-11:45 Bas Edixhoven, Introduction to Serre's conjecture [PDF]
Abstract. The conjecture will be stated, and put in its historical context and in the wider context of the Langlands program. Serre's level and weight of a 2-dimensional mod p Galois representation will be defined. Khare's result will be stated. An overview will be given of what will be treated in the seminar.
12:00-12:45 Johan Bosman, Galois representations associated to modular forms [PDF]
Abstract. Modular forms, Hecke operators and eigenforms will be defined. The existence of Galois representations associated to eigenforms will be stated. The construction of these representations will be postponed until later, in the more general case of Hilbert modular forms. Something about weight 2 being easier and about weight 1 being special can be said.
13:30-15:15 Gunther Cornelissen, Remarks on a conjecture of Fontaine and Mazur [PS]
Abstract. The following statement will be explained and proved: Let K be a number field, and S a finite set of places of K. Let KS be a maximal algebraic extension of K in which all places in S are completely split. Let X be a smooth irreducible quasi-projective scheme over K, such that for every v in S the set X(Kv) is non-empty. Then X(KS) is Zariski dense in X.
October 7 Leiden, room 312.
Joint session with Intercity Arithmetic Geometry
12:15-13:00 Sander Dahmen, Lower bounds for discriminants,
Abstract. Lower bounds for discriminants of number fields will be roved, in terms of their degree only; see Odlyzko
14:00-14:45 Frits Beukers, Upper bounds for discriminants,
Abstract. Let p be prime. Suppose that r is an irreducible representation of the absolute Galois group of Q on a 2-dimensional vector space over an algebraic closure of Fp, with p in {2,3}. The image of such a representation is finite, hence determines a number field K that is Galois over Q. An upper bound for the discriminant of K will be proved that contradicts the lower bound of the previous lecture in the case that p is in {2,3} and r is unramified outside p. The audience will draw the logical conclusion. For p=2 this result is due to Tate, and for p=3 it is due to Serre.
15:00-15:45 Bas Edixhoven, Overview of Khare's proof
Abstract. An overview will be given of Khare's proof of Serre's conjecture in level one. Those who will not attend the rest of the seminar will have an idea of Khare's proof, and it is hoped that those who will attend the rest of the seminar will now be sufficiently motivated to digest the more technical parts that are to come.
16:00-16:45 Johan de Jong, Kloosterman lecture: Rational points and rational connectivity
Abstract. Recently, Graber, Harris and Starr proved that any family of rationally connected projective varieties over a smooth curve has a section. A complex projective variety (or manifold) M is rationally connected when every two points in M lie on a rational curve in M.
In this lecture we will explain a generalization of this result, joint with Jason Starr, to the case when the base of the family is a surface.
In the other three lectures (October 14, November 11 and November 25) we will explain the idea behind the proof of the theorem, the analogy with Tsen's theorem, the connection with weak approximation and the connection with the period-index problem for Brauer groups.
[October 14: Intercity Arithmetic Geometry Utrecht]
October 21 Delft, Snijderszaal (1e verdieping laagbouw EWI), Mekelweg 4.
11:00-11:45 Klaas Pieter Hart (Delft), Ultrafilters and combinatorics
Abstract. In this talk I describe how ultrafilters may be used to prove combinatorial theorems about subsets of the set of natural numbers. For example, an ultrafilter helps in making various choices in a standard proof of Ramsey's theorem. For more intricate applications one needs to bring in some algebra. One can extend ordinary addition to the whole set of ultrafilters and thus obtain a compact semi-topological semigroup. An idempotent in this semigroup is instrumental in proving Hindman's finite-sum theorem and a special idempotent can be used to prove Van der Waerden's theorem on arithmetic progressions.
12:00-13:00 Valerie Berthe (Montpellier), Some applications of numeration systems
Abstract. The aim of this talk is to survey several applications of numeration systems, first in discrete geometry, and second, in cryptology. We first detail some natural connections between the study of discrete lines (and more generally discrete hyperplanes), and beta-expansions and tiling theory. For the latter application to cryptology, we then focus on some redundant numeration systems in mixed bases.
14:00-15:00 Tom Ward (East Anglia), Heights and highly effective Zsigmondy theorems
Abstract. Elliptic divisibility sequences are known to eventually have primitive divisors (indeed, the primitive part is very large). This talk will describe some families of elliptic divisibility sequences for which effective bounds can be given. For example, every term beyond the fourth term in the Somos-4 sequence 1,1,1,1,2,3,7,23,... is shown to have a primitive divisor.
15:15-16:15 Jan-Hendrik Evertse (Leiden), Linear equations with unknowns from a multiplicative group
Abstract. Let G be a finitely generated multiplicative group contained in the complex numbers, and let a, b be non-zero complex numbers. In 1960, Lang (building further on work of Siegel and Mahler), proved that the equation

(1) ax+by=1  

has at most finitely many solutions x, y in G.
Two equations ax+by=1, a'x+b'y=1 may be called isomorphic if a'=au, b'=bv for certain u, v in G. Clearly, two isomorphic equations (1) have the same number of solutions. It is not difficult to construct equations (1) having two solutions. In 1988, Györy, Stewart, Tijdeman and the speaker proved that for given G, up to isomorphism there are only finitely many equations of type (1) having more than two solutions.
In my lecture I will discuss a generalization to linear equations in more than two variables with unknowns from G. At the end of my lecture I will explain how this fits into a more general geometrical setting, and discuss a recent result by Gael Rémond on semi-abelian varieties, of which the results on linear equations mentioned above are special cases.

16:30-17:15 Robbert Fokkink (Delft), Minimal sets and Euclidean rings
Abstract. Recently J.P. Cerri settled some conjectures on the euclidean minimum of a number field, using results from topological dynamics. This may seem like a remarkable link between two unrelated fields, but it really is not.
November 4 Groningen, Room WSN 31, WSN-building
11:30-12:30 Lenny Taelman, An introduction to Drinfeld's shortest paper
Abstract. The paper referred to in the title is called "Commutative subrings of certain non-commutative rings" (Funkcional. Anal. i Prilozen. 11 (1977), no. 1, 11-14). We discuss its results and applications to number theory and differential equations.
13:15-14:15 Marius van der Put, A local-global problem for differential equations
Abstract. Consider a linear differential operator L:=an(d/dz)n + ... + a1(d/dz) + a0. The question studied in the lecture is: if the inhomogeneous equation L(y)=f has a formal solution everywhere locally, does it follow that a global solution exists?
14:30--15:00 Khuong An Nguyen, d-solvable linear differential equations
Abstract. A linear differential equation over a differential field K is called d-solvable if all its solutions are in a tower KK1Kn of differential fields, in which each Ki+1/Ki is either a finite extension or an extension obtained by adjoining to Ki all solutions and their (higher) derivatives of some linear diff. eq. over Ki of degree at most d. We discuss this notion and give examples.
15:00--15:45 Steve Meagher, Forms of algebraic curves
Abstract. Consider the Fermat curve X4 + Y4 + Z4 = 0; this curve has 96 automorphisms, and thus many `forms' (i.e., curves defined over the same ground field, which are not isomorphic to the given one but which become isomorphic over an extension field). We compute these forms and give a formula for their number of points over a finite field.
16:00--17:00 Heide Gluesing-Luerssen, A survey on convolution codes
Abstract. Convolutional codes are (free) submodules of F[z]n, for F a finite field. They can be interpreted as encoding arbitrarily long sequences of message blocks into sequences of codeword blocks. As is the case for classical block codes, a certain weight function is needed for describing and investigating the error-correcting properties of the code. In this talk I will explain the basic notions for convolutional codes. Thereafter, I will sketch some current research projects.
[November 11: Intercity Arithmetic Geometry, Leiden]
[November 16-18: DIAMANT/EIDMA symposium]
December 2: no seminar
[December 9: Intercity Arithmetic Geometry, Utrecht]
December 16 Leiden, Room 412
11:30-12:30 Sven Verdoolaege (LIACS, Leiden), Enumerating parametric convex integer sets and their projections
Abstract. Given a convex set of integer tuples defined by linear inequalities over a fixed number of variables, where some of the variables are considered as parameters, we consider two different ways of representing the number of elements in the set in terms of the parameters. The first is an explicit function which generalizes Ehrhart quasi-polynomials. The second is its corresponding generating function and generalizes the classical Ehrhart series. Both can be computed in polynomial time based on Barvinok's unimodular decomposition of cones. Furthermore, we can convert between the two representations in polynomial time. Finally, we present some ideas on how to handle projections of convex integer sets.
13:30-14:15 Willemien Ekkelkamp (CWI/Leiden), A variation of the MPQS factoring algorithm: analysis and experiments
Abstract. One of the methods used for factoring is the multiple polynomial quadratic sieve factoring algorithm. In this talk I will present a variation of this algorithm and some experimental results. Further, we will take a look at the (theoretical) ratio of the different types of relations coming out of the sieve. This ratio will be used in a simulation program for estimating the time needed for factoring a number.
14:30-15:15 Jeanine Daems (Leiden), A historical introduction to mathematical crystallography
Abstract. What is a crystallographic group? How many crystallographic groups are there? The first part of Hilbert's 18th problem deals with crystallographic groups. In 1910 Bieberbach solved this problem by proving that the number of crystallographic groups in each dimension is only finite. In this talk I will discuss Bieberbach's proof and a modern proof of the same theorem. In dimensions 2, 3 and 4 the exact number of crystallographic groups is known, I will talk about the methods used to find them.
15:30-16:30 Peter Stevenhagen (Leiden), Algorithmic class field theory
Abstract. Class field theory `describes' the abelian extensions of a number field, but does not readily provide generators for these extensions. Finding a `natural' set of generators is a Hilbert problem that is still largely unsolved. We will discuss to which extent the theory furnishes algorithms to generate class fields, and how this can be applied in algorithmic practice.