## 2010

### Intercity number theory seminar

5 February, Leiden. Room 405
12:15–13:00
Peter Bruin, Computing in Picard groups of projective curves over finite fields, part 1
Following the work of K. Khuri-Makdisi (Math. Comp. 73 (2004) and 76 (2006)), I will describe a way of representing a smooth projective curve over a field, and of divisors on it, that allows fast computation of group operations in the Picard group. This is especially interesting in the case of modular curves, where such a representation can be computed from spaces of modular forms. If the base field is finite, there are additional operations such as choosing uniformly random elements of the Picard group and computing Frobenius maps, Frey–Rück pairings and (for modular curves) Hecke operators. I will explain how these operations can be done efficiently in the setting of Khuri-Makdisi's representation of the curve.

Some of this material was explained in the Intercity seminar of 12 December 2008 by Arjen Stolk and myself, but knowledge of this will not be assumed.

14:00–14:45
Peter Bruin, Part 2
15:00–15:45
Lenny Taelman EPFL Lausanne, Towards a characteristic p analogue of the class number formula, part 1
The class number formula gives an expression for the residue at s=1 of the Dedekind zeta function of a number field K in terms of the class number and other arithmetic invariants of K. In this talk I will discuss an analogue of the Dedekind zeta function, which is associated to a characteristic p function field and which takes values in characteristic p. I will formulate a conjecture that expresses the value of this function at "s=1" in terms of arithmetic invariants of the function field and present evidence towards it. Crucial to the formulation of the conjecture is the Carlitz module, which is a function field analogue of the multiplicative group, and most of the talk will in fact serve as an introduction to the Carlitz module and its arithmetic properties. (More info: http://arxiv.org/abs/0910.3142)
16:00–16:45
Lenny Taelman EPFL Lausanne, Part 2

### Intercity number theory seminar

5 March, Utrecht. Buys Ballot Lab, room 205. There will be tea at 15:20.
11:30–12:20
Oliver Lorscheid Bonn, Geometry over the field with one element: an introduction
While the elusive object F1, the so-called field with one element, lurks around in mathematical concepts for more than half a century, in recent years F1-geometry became a buzzword for a whole collection of approaches that try to generalize algebraic geometry as invented by Grothendieck.

In this talk, we will review the philosophy of F1, in particular Tits idea on Chevalley groups over F1 and how to prove the Riemann hypothesis with F1-geometry. We will give an overview of the various attempts towards F1, and, to present some mathematics, we will sketch how to use Connes-Consani's F1-schemes resp. torified varieties to realize Tits' idea.

13:30–14:20
Nikolas Akerblom NIKHEF, Solitons and generalized elliptic functions
Solitons are studied in physics as mathematical models of various natural phenomena, such as "solitary waves" and vortices in superconductors. One particular field theory in physics is the Jackiw-Pi model. The solitonic solutions of the Jackiw-Pi model are determined by the Liouville equation, dating to the year 1853. In this talk, I explain recent progress on obtaining all such Jackiw-Pi solitons in the case where "space" is a torus---that is, in the case where the solitons assemble themselves in a periodic pattern in the plane, also referred to as "vortex condensation". In fact, I present an explicit, formulaic classification of all these soliton solutions in terms of a class of generalized elliptic functions and discuss some of their physical properties.
14:30–15:20
Guyan Robertson Newcastle, Building centralizers in A2~ groups
I will give a brief overview of buildings, followed by some calculations, and I will end with a question to the audience on zeta functions.
15:40–16:30
Dajano Tossici Scuola Normale Superiore di Pisa, On the essential dimension of group schemes in positive characteristic
After an introduction to the essential dimension of functors we will focus on the essential dimension of group schemes. Then I will talk about some recent results I obtained in a joint work with A. Vistoli. We give a general lower bound and a general upper bound for group schemes in positive characteristic. We will also show how to compute, with these two bounds, the essential dimension of some classes of group schemes.

### Intercity number theory seminar

19 March, Groningen. Room 267 Bernoulliborg
11:45–12:45
Bas Heijne Groningen, Elliptic Delsarte surfaces
In 1986 T. Shioda published a method for computing the rank of the group of sections of a class elliptic surfaces which admit an abelian cover by a Fermat surface. In this talk we describe this class explicitly in terms of Newton polygons of plane curves, and we compute ranks for these surfaces.
13:30–14:30
Jan Draisma, A tropical proof of the Brill-Noether theorem
We give an explicit sequence of Brill-Noether general graphs in every genus, confirming a conjecture of Baker and giving a new characteristic independent proof of the Brill-Noether Theorem, due to Griffiths and Harris, on nonexistence of special divisors on general curves. Joint work with Filip Cools, Elina Robeva, and Sam Payne.
14:45–15:45
Jaap Top, Legendre elliptic curves over finite fields
Two aspects of the well-known Legendre family of elliptic curves will be discussed. The first is their relation to the family of plane quartic curves admitting S4 as automorphism group. The second aspect concerns the sets of pairwise isogenous Legendre elliptic curves over a given finite field.
16:00–17:00
Mirjam Dür, On the cones of completely positive and doubly nonnegative matrices and their use in optimization
The convex cone of completely positive matrices and its dual cone of copositive matrices arise in several areas of applied mathematics. In particular, these cones have recently attracted interest in mathematical optimization, since it has been shown that many combinatorial and quadratic binary problems can be reformulated as linear problems over these cones.

Both cones are related to the cones of positive semidefinite and entry-wise nonnegative matrices: every completely positive matrix is doubly nonnegative , i.e., positive semidefinite and component-wise nonnegative, and it is known that up to dimension 4 the reverse statement also holds true. Therefore, the case of 5x5 matrices is of special interest.

The talk will give an overview on the role of all mentioned matrix cones in mathematical programming, and provide some new results about the 5x5 completely positive and doubly nonnegative matrices.

### Intercity number theory seminar

16 April, Leiden. Room 409
11:15–12:00
Ronald van Luijk, Computing Picard groups
In general it is hard to find the Picard group of a given surface defined over a number field. I will start by describing a method from 2004 that computes at least the rank in many cases. We will also describe various improvements that have been made to this method and conclude with the latest related trick, by Shioda and Schuett, which allows us to find not only the rank, but also generators for the Picard group.
12:15–13:00
Bart de Smit, Deformation rings of group representations
For a representation V of a finite group G over the prime field Fp one may wonder if V lifts to a representation over Zp, or over other complete local rings A with residue field Fp. It turns out that such a lift to A exists if and only if A contains a quotient of the so-called deformation ring RV of V. In this talk we address the question which local Zp-algebras can occur as the deformation ring of some pair (G,V). In particular we will see that not all deformation rings are complete intersections, which answers a question of Matthias Flach. This is joint work with Ted Chinburg and Frauke Bleher.
14:00–14:45
Michiel Kosters Leiden, Tameness and rings of integers
Let K be a number field. An important problem in algebraic number theory is to find the integral closure O of Z in K. In general one is given an order A in K and one can ask for a prime p in Z if p divides the index (O:A). We will give an easy criterion for this if A is tame at p'.
15:00–15:45
Sep Thijssen Nijmegen, Recognizing radical extensions of prime degree
Let K/F be a proper extension of infinite fields, with multiplicative groups K* and F*. Brandis proved in 1965 that the quotient K*/F* is not finitely generated. The situation is much clearer for the torsion subgroup of K*/F*. Due to van Tieghem we know that the torsion subgroup of K*/F* is finite when K and F are number fields. Furthermore there is a polynomial time algorithm that constructs all elements of this group.

Elements of K* that are torsion over F* are radicals. So it is not surprising that the algorithm uses techniques to recognize radical extensions. A well known theorem in Galois theory states that a cyclic extension is a radical extension when the ground field contains sufficient roots of unity. Many text books contain a proof that is based on Hilbert 90. The theorem itself originates from ideas of Lagrange and his resolving equations. In the talk I shall present a way to construct radicals without the condition of roots of unity in case of an extension of prime degree.

16:15–17:00
Peter Stevenhagen, Primitive roots and arithmetic progressions

### Intercity number theory seminar

7 May, KNAW Amsterdam. Part of a 2 day workshop on cryptography and lattices.

### Intercity number theory seminar

21 May, Utrecht. Buys Ballot Lab, room BBL 065
11:30–12:20
Noriko Yui Queens, The modularity of certain K3-fibered Calabi-Yau threefolds over Q
We consider certain K3-fibered Calabi-Yau threefolds defined over Q. We will discuss their modularity (automorphicity). Some of the Calabi-Yau threefolds considered here are shown to be of CM type. We establish the modularity of such Calabi-Yau threefolds, and their mirror partners (if exist) in the sense of Arthur and Clozel. Several explicit examples are discussed. This reports on a joint work in progress with Y. Goto and R. Livne.
13:30–14:20
Esther Bod Utrecht, Algebraicity of the Appell-Lauricella and Horn functions
The Appell-Lauricella and Horn functions are generalizations of the Gauss hypergeometric function. In 1873, Schwarz found a list of all rational parameters such that the Gauss function is algebraic over C(z). In 2006, Beukers proved a combinatorial criterion for algebraicity of general GKZ-hypergeometric functions. In this talk, I will explain some basic concepts of GKZ-hypergeometric functions and show how we can use Beukers' criterion to extend Schwarz's list to all Appell-Lauricella and Horn functions.
14:30–15:20
Bart de Smit, The covering spectrum of Riemannian manifolds
The Galois configurations of non-isomorphic number fields with the same zeta functions can also be used to make non-isomorphic Riemannian manifolds with the same Laplace spectrum. In this talk we give a group theoretic criterion for two Riemannian manifolds to have the same covering spectrum, and we show by group theoretic means that the covering spectrum is not a spectral invariant. This is joint work with Ruth Gornet and Craig Sutton.
15:40–16:30
Gunther Cornelissen, Some Dirichlet series in Riemannian geometry
It is known that the spectrum of the Laplace-Beltrami operator doesn't necessarily determine a RIemannian manifold up to isometry. Knowing the spectrum is the same as knowing the zeta function of the manifold. I will define more general Dirichlet series on closed Riemannian manifolds that do capture the manifold up to isometry. This allows one to define a "length" of a homeomorphism of Riemannian manifolds.

### Intercity number theory seminar

11 June, Leiden. Part of the workshop on numeration at the Lorentz Center
09:30–10:30
Mark Pollicott, Dynamical Zeta Functions Revisited
Dynamical zeta functions are complex functions which are used, amongst other things, to asymptotically count dynamical quantities for either discrete maps or flows (e.g., closed orbits). The basic philosophy is that the more you know about the domain of zeta function the more you know about the properties of the dynamical system.

i) Extending the Domain of zeta functions: In these lectures we will describe different approaches to extending zeta functions, culminating in recent progress on extending Ruelle zeta functions for Anosov diffeomorphisms and flows using operator theoretic methods, and their applications.

ii) Special Values of zeta functions: Once a zeta function (or the closely related L-function) has an extension to a larger domain it is a natural question to ask what is the significance of the locations of the zeros, the poles and their residues, and the value of the function at specific values. Frequently, these provide interesting global information about the system. We will describe a number of results of this type.

iii) Dynamical zeta functions and counting: One of the historical motivations for studying (dynamical) zeta functions was to obtain asymptotic estimates, particularly on the number of closed orbits. We will describe both classical results and recent results.

Throughout we will motivate the work by drawing analogies with classical results in number theory.

11:00–12:00
Mike Keane Wesleyan, Numeration dynamincs
Using an idea I developed some time ago, I explain how to write the usual continued fraction transformation on the unit interval as a transformation acting on coefficients of quadratic polynomials, instead of as a mapping on [0,1]. This leads to a geometric picture of this transformation in terms of a so-called baker's transformation, from which the ergodic nature of the transformation can be easily deduced. As a by-product I obtain a derivation of the well-known invariant distribution discovered by Gauss and documented in his notebooks and in a letter to Laplace. Then, according to an idea of Allouche, I use this representation to sketch a simple proof of Lagrange's theorem on the periodicity of continued fractions for quadratic irrationals. Finally, I shall discuss an up to now unsuccessful attempt to use these ideas to prove that the classical Cantor middle-third set does not contain any quadratic irrationals.
13:00–14:00
Vilmos Komornik, Expansions in noninteger bases
Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. Seemingly innocent questions lead to surprisingly deep problems many of which are still open. Starting with a discussion of Renyi's greedy or beta-expansions, we give an overview of various results concerning the number of expansions of some given real number in a given base. Most of the early results in this field are due to P. Erd\H os and his collaborators. Next we concentrate ourselves to the case of unique expansions and we clarify the rich combinatorial, topological and fractal nature of the set of such bases and real numbers. Finally, we study the existence of so-called universal expansions and their relationship to problems of Diophantine approximation.
14:30–15:30
Boris Adamczweski, Automata in Number Theory
Among infinite sequences or infinite sets of integers, some have a rigid structure, such as periodic sequences or arithmetic progressions, whereas others, such as random sequences or random sets, are totally unordered and could not be described in a simple way. Finite automata are one of the most basic model of computation and thus take place at the bottom of the hierarchy associated with Turing machines. Such machines can naturally be used at once to generate sequences with values over a finite set, and as a device to recognize some sets of integers. One of the main interest of these automatic sequences/sets comes from the fact that they are highly ordered without necessarily being trivial. One can thus consider that they lie somewhere between order and chaos, even if, in most respects, they appear to have a rigid structure. In these lectures, I will survey some of the connections between automatic sequences/sets and number theory. Several substantial advances have recently been made in this area and I will try to give an overview of some of these new results. This will possibly includes discussions about prime numbers, the decimal expansion of algebraic numbers, the search for an analogue of the Skolem-Mahler-Lech theorem in positive characteristic and the study of some Diophantine equations that generalize the famous S-unit equations over fields of positive characteristic.
15:30–16:30
Shigeki Akiyama, What is the Pisot conjecture?
In this talk, I wish to give a rathor personal account on Pisot conjecture of substitutive dynamical systems. This conjecture contains many interesting aspects: ergodic, analytic, combinatorial and number theoretical nature. Moreover it can also be understood as a problem of numeration on Pisot number base, which is called Dumont-Thomas number system. In the end, I wish to talk on a possible generalization in higher dimension

### Intercity number theory seminar

3 September, Leiden. Room 174
11:30–12:30
Lenny Taelman, Mass formulas for finite p-groups
Fix a positive integer n. We will study the function which associates to a prime number p the mass of the category of groups of order pn. This mass is defined to be the rational number ∑G 1/#Aut(G), where G runs over the set of isomorphism classes of groups of order pn.

We will make use of the correspondence between nilpotent groups and Lie algebras due to Lazard, and a result coming from the theory of motivic integration due to Denef and Loeser. No prior knowledge of these techniques will be assumed.

13:30–15:30
Andrea Lucchini Padova, The probabilistic zetafunction of finite and profinite groups
See PDF
15:45–16:45
Hendrik Lenstra, Radical Galois groups
Given a field K with separable closure Ksep, let L be the field obtained by adjoining to K all elements x of Ksep for which there is a positive integer n such that xn belongs to K. The lecture is concerned with describing the Galois group Gal(L/K) as well as the kernel of the restriction map of Gal(Ksep/K) to Gal(L/K).

### Intercity number theory seminar

17 September, Eindhoven. DIAMANT seminar room (HG 9.41).
Tea and cookies will be served at 15:15
11:45–12:30
Wouter Zomervrucht Leiden, The complexity of Buchberger's algorithm
Thomas Dubé (1990) proved the existence of a Gröbner basis of a multivariate polynomial ideal with an explicit upper bound for the degree of its polynomials; this bound is doubly exponential in the number of variables. Similar lower bounds are known. These results, however, do not apply to the complexity of Buchberger's algorithm - the standard algorithm for the construction of Gröbner bases. For this algorithm, no general complexity bound appears to be proved in the literature. We will prove such a bound, given in terms of the Ackermann function.
13:30–14:15
Jan Draisma, Gröbner bases in infinitely many variables, or equations for the 2-factor model
In 1928, educational psychologist Truman Lee Kelley published "Crossroads in the Mind of Man", which contained a new "pentad test" for Gaussian factor analysis with two factors. Pentads are degree-5 polynomials that vanishes on all symmetric nxn-matrices that can be expressed as a rank-2 matrix plus a diagonal matrix. In their recent algebraic approach to factor analysis, Drton, Sturmfels, and Sullivant raise the problem of determining all such polynomials. I will report on a finite computation that determines a generating set of polynomials for all n. While the result is very satisfactory, the method---Gröbner bases in infinitely many variables---is likely to have even more striking applications in the future. This is joint work with Andries Brouwer.
14:30–15:15
Christiane Peters, Wild McEliece
We propose using the McEliece public-key cryptosystem with wild Goppa codes''. These are subfield codes over small Fq that have an increase in error-correcting capability by a factor of about q/(q-1). McEliece's construction using binary Goppa codes is the special case q=2 of our construction. The advantage of considering the case q>2 is that larger finite fields allow to use smaller keys at the same security level. We explain how to use wild Goppa codes in the McEliece cryptosystem and how to correct ⌊ qt/2 ⌋ errors where previous proposals corrected only ⌊ (q-1)t/2⌋ errors. We also present a list-decoding algorithm that allows even more errors. This is joint work with Daniel J. Bernstein and Tanja Lange. http://eprint.iacr.org/2010/410
15:45–16:30
Yael Fleischmann Eindhoven, Questions of rationality in Ruan's conjecture on crepant resolutions
After he gave an new definition of a cohomology of orbifolds, Y. Ruan conjectured in 2002 that the cohomology ring of a complex orbifold [Y] and the cohomology ring of a crepant resolution of the underlying space Y are isomorphic, if such a crepant resolution exists. In 2007, J. Bryan and T. Graber gave a conjecture describing this isomorphism. In my talk I will give an overview of my diploma thesis where I proved Ruans conjecture restricted to the 2-dimensional complex case and ADE singularities using Bryans and Grabers results.

### Intercity number theory seminar

1 October, Nijmegen. The first talk will be in room HG00.086 and the last two talks in room HG00.071. This is also the day of the Wiskundetoernooi.
14:30–15:15
Cecilia Salgado, Zariski density of rational points on del Pezzo surfaces of low degree
Let k be a non-algebraically closed field and X be a surface defined over k. An interesting problem is to know whether the set of k-rational points X(k) is Zariski dense in X. A lot of research is done in this field but, surprisingly, this problem is not completely solved for the simplest class of surfaces, the rational, where one expects a positive answer. In this lecture I will define del Pezzo surfaces, a important subclass of rational surfaces. I will talk about the cases already treated (mainly by Manin), as well as the two cases left open, the del Pezzo surfaces of degrees one and two, presenting recent results (in progress) in the field.
15:30–16:15
Rajender Adibhatla Essen, Higher congruence companion forms
This talk will discuss the local splitting behaviour of ordinary, modular Galois representations and relate them to companion forms and complex multiplication. Two modular forms (specifically p-ordinary, normalised eigenforms) are said to be "companions" if the Galois representations attached to them satisfy a certain congruence property. Companion forms modulo p play a role in the weight optimisation part of (the recently established) Serre's Modularity Conjecture. Companion formsmodulo pn can be used to reformulate a question of Greenberg about when a normalised eigenform has CM.
16:30–17:15
David Gruenewald, Explicit Complex Multiplication in Genus 2
In this talk we make explicit the Galois action on the CM moduli for genus 2 Jacobians. By using recently computed (3,3)-isogeny relations, we demonstrate how this can be used to improve the CRT algorithm for computing Igusa class polynomials, providing some examples. This is joint work with Reinier Bröker and Kristin Lauter.

### Stieltjes afternoon

8 October, Leiden.

### Intercity number theory seminar

15 October, CWI Amsterdam. Room L017
14:00–14:45
Alexander Kruppa CWI, The factorization of RSA768
The factorization of the RSA challenge number RSA-768 of 232 digits (768 bits) with the General Number Field Sieve by an international team on December 12th, 2009, set a new record for the factorization of hard, general integers. This talk gives a brief introduction to the Number Field Sieve, describing the different steps of the algorithm, how they were carried out in the case of RSA-768 using computing resources distributed over various countries, some of the difficulties that arose and how they were solved.
15:00–15:45
Florian Luca, Multiperfect repdigits
For a positive integer n, let σ(n) denote the sum of divisors of n. Positive integers n such that σ(n)/n is an integer are called multiperfect. These include the perfect numbers n (for which the above ratio is 2) as well as others such as n=120 for which σ(120)/120=3. Given an integer g>1, a {\it base g repdigit} is a positive integer N whose base g representation consists of a string of identical digits d. Such a base g repdigit is therefore of the form N=d(gm-1)/(g-1), where d∈\{1,ldots,g-1\} is the repeated digit and m is the number of base g digits of N. In this talk, I will show that in any fixed base g>1, there are only finitely many repdigits N which are multiperfect. When g=10, the only such is N=6.
16:00–16:30
Herman te Riele, Rules for construction of amicable pairs
Amicable pairs are pairs of positive integers (m,n), m<n, satisfying σ(m)-m=n and σ(n)-n=m, where σ is the sum-of-divisors function. Smallest example: (220,284). Many `rules'' for finding amicable pairs are known and these rules have played a major role in the construction of the current list of almost twelve million known amicable pairs. Still, it has not yet been proved that the number of amicable pairs is infinite. In this talk some rules will be discussed, including a rule of Erdős, to find amicable pairs, and implications for a possible proof of the existence of infinitely many amicable pairs.
16:45–17:30
Gabriele Dalla Torre, Digits and powers of two
Consider the sequence of powers of two written in base 10. Does the sum of the digits in 2n tend to infinity? Is it eventually increasing? Are there infinitely many powers of 2 with more digits 4 than digits 8? These and other similar questions will be discussed in the talk.

### Intercity number theory seminar

29 October, Leiden. Final day of the workshop Arithmetic of Surfaces at the Lorentz Center.

### RISC/Intercity Number Theory Seminar

18 November, CWI Amsterdam. Special two day seminar on towers of function fields including lectures of Garcia and Stichtenoth.

### DIAMANT Symposium

26 November, Lunteren. Two day workshop with a special session on the ABC conjecture on Friday afternoon. Speakers: Masser, Oesterlé, Palenstijn.

### Intercity number theory seminar

10 December, Groningen. Room 293 Bernoulliborg,
Speakers: Monique van Beek, Peter Dickinson, Florian Hess, Claus Diem
11:45–12:45
Monique van Beek Groningen, The rank of elliptic curves admitting a 3-isogeny
In the sixties John Tate presented an elementary account on how one might succeed in calculating the rank of an elliptic curve over the rational numbers, in case the curve admits a rational isogeny of degree 2. In my master's thesis I do the same for elliptic curves admitting a rational isogeny of degree 3. The talk discusses some aspects of this.
13:30–14:30
Claus Diem, On the discrete logarithm problem in elliptic curves.
It is well known that the classical discrete logarithm problem (the problem to compute indices modulo prime numbers) can be solved in subexponential expected time. In contrast, it is not known whether the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields (the elliptic curve discrete logarithm problem) can be solved in subexponential expected time. Indeed, it was the lack of an obvious algorithm for this computational problem which was faster than "generic" algorithms which lead Miller and Koblitz to suggest the use of the problem for cryptographic applications. In 2004 Gaudry gave a randomized algorithm with which one can -- under some heuristic assumptions -- solve the elliptic curve discrete logarithm problem over all finite fields with a fixed extension degree at least 3 faster than with generic algorithms. Based on this work, in an article which is going to be published in Compositio Mathematica, I have shown that there exists a sequence of finite fields (of strictly increasing cardinality) over which the elliptic curve discrete logarithm problem can be solved in subexponential time. In this talk I want to explain how my previous result can be extended such that over more families of finite fields the elliptic curve discrete logarithm problem can be solved in expected subexponential time.
14:45–15:45
Peter Dickinson, Linear-time checking of sparse matrices for complete positivity
The cone of completely positive matrices and its dual, the cone of copositive matrices, are useful in optimisation, especially in providing convex formulations of NP-complete problems. It has been proven that telling if a matrix is copositive is a co-NP-complete problem and it is widely expected that telling if a matrix is completely positive is an NP-complete problem. In this talk we study how to check sparse matrices for complete positivity. We present linear-time methods for preprocessing a sparse matrix in order to reduce the problem. For some types of matrices these methods do not just reduce the problem but in fact solve it in linear-time.
16:00–17:00
Florian Hess, The Tate-Lichtenbaum pairing and applications
Pairings on elliptic curves over finite fields have been in the focus of one of the most active and important research areas in cryptography during the last ten years. The main objectives are their use in cryptographic primitives and number theoretic, algorithmic issues. We want to discuss some of these aspects with a view towards class field theory.