Intercity Number Theory Seminar

Spring 2005

Special day on Mathematics of Cryptology
January 21 2005

On January 21 we organize a joint meeting with the RISC seminar. The lectures will take place at Room 201 (Huygens Building) of the Lorentz Center in Leiden.


10:30−11:00 Coffee
11:00−11:45 Carles Padró (Barcelona) Secret Sharing Schemes, Error Correcting Codes and Matroids
Abstract Error correcting codes and matroids have been widely used in the study of secret sharing schemes. This talk deals mainly with the connections between codes, matroids and a special class of secret sharing schemes, namely multiplicative linear secret sharing schemes (MLSSS). Such schemes are known to enable multi-party computation protocols secure against general (non-threshold) adversaries. Two open problems related to the complexity of multiplicative linear secret sharing schemes will be considered. Hirt and Maurer proved that such a scheme can be costructed whenever the set of players is not the union of two unqualified subsets. Cramer, Damgard and Maurer proved that, in this case, a multiplicative linear secret sharing scheme can be efficiently constructed from any linear secret sharing scheme by increasing the complexity by a constant factor of 2. The first open problem we consider is to determine in which situations a multiplicative scheme can be obtained without increasing the complexity. We study this problem in an extremal case. Namely, to determine whether all self-dual vector space access structures admit an ideal MLSSS. By the aforementioned connection, this in fact constitutes an open problem about Matroid Theory, since it can be re-stated in terms of representability of identically self-dual matroids by self-dual codes. We introduce a new concept, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. We prove that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids. The second open problem deals with strongly multiplicative linear secret sharing schemes. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. We prove a property of strongly multiplicative LSSSs that could be useful in solving this problem. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, we show that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults.
11:45−12:00 Coffee break
12:00−12:45 Salil Vadhan (Harvard) Randomness Extractors and their Cryptographic Applications
Abstract: Over the past two decades, a rich body of work has developed around the problem of constructing randomness extractors --- algorithms that extract high-quality randomness (i.e. nearly uniform and independent bits) from low-quality randomness (i.e. sources of biased and correlated bits). Although some of the early results on randomness extraction came from the cryptography literature, most of the subsequent theory has been developed in the setting of computational complexity, where extractors have unified the study of a number of fundamental objects (such as pseudorandom generators, expander graphs, and list-decodable error-correcting codes). In the past few years, the relevance of extractors to cryptography has been re-discovered, with increasing variety of applications being found. In this talk, I will survey the basic theory of randomness extractors, give a sense of the current state-of-the-art, and describe some of their cryptographic applications.
12:45−13:45 Lunch
13:45−14:30 Rafi Ostrovsky (UCLA) Survey on Private Information Retrieval
Abstract: Consider a setting where a user wishes to retrieve an item from a database, without letting the database administrator know which item is being retrieved. Of course, a trivial (but expensive) solution is for the user to request contents of the entire database, thus concealing from the database administrator which item is of interest to the user. Can this be done with less communication? Perhaps somewhat surprisingly, the answer is yes, under various assumptions and settings. In this talk, I'll survey much progress that has been achieved on this problem and point our some interesting connections to other problems in coding theory and several hardness results in cryptography.
14:30−15:00 Tea break
15:00−15:45 Phong Nguyen (ENS Parijs) From Euclid to Lenstra-Lenstra-Lovasz: Revisiting Lattice Basis Reduction
Abstract: Lattices are simple yet fascinating mathematical objects. Roughly speaking, they are linear deformations of the n-dimensional grid Z^n. Lattices have many applications in mathematics and computer science. Of particular importance is lattice basis reduction, which is the problem of finding "nice" representations of lattices. For instance, lattice basis reduction is the most popular tool to attack public-key cryptosystems. In this talk, we will revisit lattice basis reduction, from Euclid's gcd algorithm to the celebrated LLL algorithm. We will also briefly discuss recent results. Curiously, it is possible to obtain a Euclid-like complexity for lattice basis reduction: in some sense, one can compute a reduced basis (without fast integer arithmetic) in essentially the same time as the elementary method to multiply integers.
15:45−16:15 Tea break
16:15−17:00 Steven Galbraith (London) The Eta Pairing
Abstract: We introduce a new pairing on certain supersingular curves which is very closely related to the Tate pairing, but which has some implementation advantages. We interpret the results of Duursma and Lee in terms of this pairing and we describe a fast pairing on genus 2 curves in characteristic 2.
17:00−18:00 Drinks

February 4 Special program in Utrecht on the occasion of Johnny Edwards' PhD defense
The afternoon talks are in room 018 at Kromme Nieuwegracht 66, number 11 on this map
We can have lunch at "Trans 10", which is number 14 on the map
10:30 PhD defense of Johnny Edwards in the Academiegebouw, number 1 on this map
13:00−14:00 Michael Stoll (Bremen), Proving non-existence of rational points on curves
Abstract. Let C be a curve of genus at least two over Q (or, more generally, over a number field K). An important problem is to decide whether C has any rational points. The first step in trying to solve this problem is to check if C has points everywhere locally (which can be done effectively). If this is the case, but C does not appear to have global points, there is the possibility that the absence of global points can be explained by the Brauer-Manin obstruction. It turns out that this is equivalent to the existence of a descent obstruction coming from some abelian covering of C.
We will discuss how we can check for such an obstruction, focussing on genus 2 curves, and we will give experimental results obtained by Victor Flynn, Nils Bruin and myself.
14:15−15:15 Andreas Schweizer (KIAS Seoul), Construction of quadratic function fields whose class groups have m-rank 4.
Abstract. Fix a finite field Fq of odd characteristic and an odd integer m that is not divisible by the characteristic of Fq. We are interested in the m-rank of the divisor class groups of quadratic extensions L of the rational function field Fq(T).
First we show: If there is one L with m-rank at least r, then there are infinitely many L with m-rank at least r. Then for q = 1 mod 4 we explicitly construct an L with m-rank at least 4. Similar constructions give slightly weaker results if q = 3 mod 4 and also for the ideal class group of the integral closure of Fq[T] in L.
15:30−16:30 Andreas Weiermann (Utrecht/Muenster), Analytic combinatorics of the transfinite
Abstract: We consider a natural well-ordered subclass H of Hardy's 1917 orders of infinity. Let H be the least set of functions from the non negative integers into the non negative integers such that: 1) the constant zero function belongs to H, 2) with f and g the function idf+g belongs to H where id denotes the identity function. Let < be the ordering of eventual domination on H. A norm function is a function from H into the non negative integers such that for any non negative integer n there are only finitely many elements in H having norm below n. For a given norm N and a given f in H let cfN(n) be the number of elements g in H such that g<f and N(g)≤ n. We are going to classify various of these count functions. Partition functions appear as special cases of this construction. We use tools from Tauberian theory and generating function methodology. Moreover we indicate where the count functions can be used in logic.
February 18 Leiden, room 312 (first talk), 405 (other talks).
12:00−13:00 Robin de Jong (Leiden), Arakelov geometry and its applications to number theory
Abstract. The purpose of this talk is to make popular the idea that Arakelov geometry can be used to obtain effective results in number theory. We discuss in this respect the existence of an efficient algorithm to compute the Ramanujan tau-function at a given prime number, as well as the effective Shafarevich conjecture trying to make quantitative the (proven) statement that given a number field K, a positive integer g and a set S of primes of K, there exist only finitely many K^alg-isomorphism classes of curves of genus g, defined over K and having good reduction outside S. The former topic is work in progress together with Edixhoven and Couveignes, the latter topic is probably still far out of reach, although recently strong results have been obtained by Heier in the function field case.
In order to bring Arakelov theory into action, it is essential to obtain explicit representations of the various objects that occur in this theory. As an example we discuss an explicit formula for the Arakelov-Green function of a compact and connected Riemann surface of positive genus.
14:00−14:45 Eleonora Pellegrini, Power bases for pure cubic fields
Abstract. In this talk we will consider the problem of the monogenicity of a number field, with a particular attention at cubic fields of the form Q(3m), where m is a cubefree positive integer. After giving a criterion of monogenicity, we will discuss the distribution of those m that give rise to monogenic pure cubic fields.
15:00−15:45 Clemens Fuchs (Leiden), Polynomial generalisations of a problem of Diophantus
Abstract. A Diophantine m-tuple is a set of m positive integers with the property that theproduct of any two distinct elements plus one is a square of an integer. In my talk I will start with a survey on Diophantine m-tuples and possible generalisations with an emphasis on variants of the problem for polynomials withinteger coefficients. E.g. I will discuss a recent joint result with A. Dujella and G. Walsh: we proved that there does not exist a set of more than 12 polynomials with integer coefficients, not all constant, and with the property that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients.
16:00−16:45 Capi Corrales Rodrigañez (Madrid), On the unit group of an order in a non-split quaternion algebra
Abstract. I will speak on some joint work with E. Jespers, G. Leal and A. del Río, in which we give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra H(K) over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of H(Z[(1+√-7)/2]). As a consequence, we get a presentation for the orthogonal group SO3(Z[(1+√-7)/2]). These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.
March 4 Universtiteit Gent, building S22 at Galglaan 2, lokaal 14. On the map of Gent this is between numbers 14 and 18, and on the Campus map it is building 40-22 [further info].
13:30-14:30 Joost van Hamel (Leuven), Abelianised Galois cohomology of reductive groups
14:45-15:45 Lenny Taelman (Groningen), On analytic unformization of abelian varieties and Anderson modules
16:15-17:15 Jan Schepers (Leuven), Introduction to motivic integration
March 18 Groningen, room A901 at Broerstraat 9, next to the Academiegebouw (see the map)
11:15−12:00 Matthias Schuett (Hannover), Extremal elliptic K3 fibrations
Abstract. In this talk we will consider elliptic K3 surfaces. After a brief introduction we will especially emphasize the extremal fibrations. With view to their classification and Frits Beukers' talk, we would like to discuss how many of them arise as pull-back from rational elliptic surfaces by a base change of low degree.
12:15−13:00 Ronald van Luijk (Berkeley), K3 surfaces with Picard number one and infinitely many rational points
Abstract. Not much is known about the arithmetic of K3 surfaces in general. Once the Picard number, which is the rank of the Neron-Severi group, is high enough, more structure is known and more can be said. But still we don't know of a single K3 surface whose set of rational points has been proved to be neither empty, nor Zariski dense.
Also, until recently, not even a single K3 surface was known with Neron-Severi rank 1 and infinitely many rational points. We will give an explicit example of such a surface over Q, where even the Picard number over the algebraic closure is equal to 1. This solves an old problem, that has been attributed to Mumford. The method used has been extended by Remke Kloosterman to find elliptic K3 surfaces of rank 15.
13:45−14:30 Frits Beukers (Utrecht), Computation of extremal elliptic K3 fibrations
Abstract. In a joint effort with H. Montanus we computed all extremal semistable elliptic K3 fibrations over P1. Although the subject matter is geometrical, this talk will be computational. We focus our attention on the determination of 188 plane 'dessins d'enfant' and their associated Belyi maps.
14:45−15:30 Jozef Steenbrink (Nijmegen), The billiard problem on an ellipse
Abstract. I will show how one may visualize the solution to the billiard problem on an ellipse and how this gives rise to a nice deformation of a double hyperbola. It is a demonstration of the use of the CABRI-geometry program on academic level (I hope).
16:15 PhD defense of Remke Kloosterman in the Academiegebouw
April 1 Utrecht, room K11
11:00−12:00 Roelof Bruggeman (Utrecht), Period functions and Maass cusp forms
Abstract. For holomorphic modular cusp forms there is a well established relation with period polynomials. In the case of modular Maass cusp forms, Lewis and Zagier have studied the associated period functions. I'll speak about ongoing work of Lewis, Zagier and me on these period functions and their cohomological interpretation.
12:15−13:15 Yiannis Petridis (Bonn), Modular symbols and spectral theory
Abstract. A conjecture of Goldfeld about periods of elliptic curves implies a weak version of ABC. On the other hand this conjecture is equivalent with the modular symbol conjecture about their relative growth. The modular symbols encode geometric information about the cohomology and the fundamental group of the modular curves. Through trace type-identities we can relate modular symbols and the spectral theory of the Laplace operator on the modular curve. The objects to study are: families of automorphic forms, Eisenstein series twisted by modular symbols and Selberg-zeta functions. The technique is perturbation theory of operators. The result is the distribution of the (normalized) modular symbols: a Gaussian law.
14:15−15:15 Byoung Ki Seo (Utrecht), Asymptotic behaviors of the first return time of translations on a torus
Abstract. We investigate asymptotic behaviors of the first return time of translations on a torus using Diophantine approximations. It is known that the first return time to an element of the equipartition equals to the 1 over the size of the element asymptotically for almost but not all irrational translation on a 1 dimensional unit interval. We expand the results to the case for an irrational translation on a multidimensional torus.
15:30−16:30 Gunther Cornelissen (Utrecht), Complexity of the rational numbers and conjectures about elliptic curves
Abstract. We don't know whether or not there is an algorithm to decide whether a diophantine equation has a rational solution or not ("Hilbert's tenth problem for Q", HTP(Q)), but Julia Robinson has proven in 1949 that one cannot decide the truth of more general statements about the rationals. I will discuss conjectures about elliptic curves that allow us to improve upon Robinson's result and - in a sense that I will make precise - closer to a solution of HTP(Q). The keyword is "Zsigmondy type theorems for elliptic divisibility sequences with extra conditions". I will present some heuristics to support the conjectures.
April 15

Special day on Lambda rings

Nijmegen, room CZ N7 (=N1004 on the first floor of building N1)
On the map look for the red dot with green arrow "oudbouw". Enter there and go up two floors. For parking you might get lucky at the bottom of the map near "INGANG VANAF BRAKKESTEIN" on the Driehuizerweg, which is easier to spot on this map
Speakers: Jim Borger (Max Planck), Frans Clauwens (Nijmegen), Bart de Smit (Leiden).
12:00−13:00 Jim Borger (Max Planck), Lambda rings for beginners part I
14:00−15:00 Jim Borger (Max Planck), Lambda rings for beginners part II
15:15−16:15 Frans Clauwens (Nijmegen), Natural operations on lambda rings
16:30−17:30 Bart de Smit (Leiden), Integral lambda ring structure on finite étale Q-agebras
Abstract. For a lambda ring K which is finite étale as a Q-algebra we determine whether K has a lambda subring R of finite rank over Z so that K=RQ. We also determine the maximal such R if it exists. This is joint work with Jim Borger.
May 13 Leiden, room 412
13:30−14:15 Joost Batenburg (Leiden/CWI), Small, smaller, smallest. Steps toward the atomic resolution microscope.
Abstract. Building an electron microscope that can view samples at atomic resolution is a longstanding goal in the microscopy community. It has recently become possible to acquire images of projected crystalline structures in which separate atom columns can be resolved. By themselves, these images do not provide sufficient information to determine the positions and types of individual atoms inside the crystal. However, when projections from several viewpoints are combined, it is possible to make a full 3D reconstruction of the crystal by performing a tomographic inversion procedure.
Because the number of measured projections is typically very small, methods from continuous tomography, which are used in medical imaging, cannot be used for this application. The young field of discrete tomography is suited particularly well for computing reconstructions from few projections. Discrete tomography is an interdisciplinary field, with links to number theory, combinatorics, operations research and coding theory.
In this talk I will describe how discrete tomography can be used to compute atomic resolution 3D reconstructions of crystals and discuss a variety of problems that still need to be solved.
14:30−15:15 Oliver Lorscheid (Utrecht), Completeness and compactness for varieties over local fields [math.AG/0410346]
Abstract. For a complex variety it is well known that it is complete if and only if the rational points form a compact space in the complex topology. The only essential property of the complex numbers for this statement to hold is their locally compactness. One finds the following generalisation: let K be a local field and X a variety over K, then X is complete if and only if for every finite field extension L of K, X(L) is compact in its strong topology.
15:30−17:00 Pre-Journées program
Three PhD students from Leiden present their proposed talks for the Journées Arithmétiques

Willem Jan Palenstijn, Computing near-primitive root densities
Abstract. Artin's primitive root conjecture gives, for an integer x, an expression for the density of primes q for which x is a primitive root modulo q. In this talk, we consider the following generalization: given a number field K, a non-zero element x of K, and a positive integer d, what is the density of primes q of K for which the subgroup of the multiplicative group of the residue class field of q generated by x has index dividing d?

Bas Jansen, Mersenne primes and Lehmer's observation
Abstract. The Lucas-Lehmer test is an algorithm to check whether a number of the form M=2p-1, with p an odd positive integer, is prime. The algorithm produces a sequence of p-1 numbers modulo M, starting with the number 4 and each time squaring the previous number and subtracting 2. Then the last number is zero modulo M if and only if M is prime. Lehmer observed that if the last number is zero then the penultimate number can be either PLUS or MINUS 2(p+1)/2 modulo M. Gebre-Egziabher showed that if you start your sequence with 2/3 instead of 4, then the test also works, and the sign will be plus if and only if p is 1 modulo 4 (p>5). In this talk we generalize this result.

Reinier Bröker, Class invariants in a non-archimedean setting
Abstract. One of the goals of explicit class field theory is to compute a generating polynomial for the Hilbert class field of a given number field. For imaginary quadratic fields, the theory of complex multiplication provides us with an elegant solution. The classical approach has two improvements. Firstly, one can use `smaller' functions than the classical j-function, leading to smaller polynomials. This leads to the theory of class invariants, which was initiated by Weber. Secondly, one can work in a non-archimedean setting, avoiding the problem of rounding errors in the classical approach. In this talk we will combine both improvements, i.e., we will show how to use class invariants in a non-archimedean setting.