2007

Intercity Number Theory Seminar

11:30-12:30	Jaap Top, Elliptic surfaces with a high Picard number Abstract. In characteristic zero an elliptic K3 surface has Picard number at most 20. We will present many examples with high Picard number and discuss reduction modulo primes.
13:15-13:45	Nguyen Khuong An (Groningen), The algebraic subgroups of GL2(C). Abstract. The algebraic subgroups of PGL2(C) and SL2(C) are well known. In contrast to this, complete and dependable literature on the algebraic subgroups of GL2(C) seems to be missing.
14:00-15:00	Marius van der Put, Solving linear differential equations Abstract. The theme of solving a linear differential equation in terms of equations of lower order goes back to L. Fuchs around 1880 and G. Fano 1900. We will explain how the representation theory of semi-simple Lie algebras can be used for this problem.
15:15-16:15	Steve Meagher (Freiburg), When is the twist of a Jacobian a Jacobian? Abstract. Over a non algebraically closed field a Jacobian variety may have twists that are not Jacobians. How can one characterize these?
16:30-17:00	Lenny Taelman, An exercise in algebraic topology Abstract. I shall formulate and solve a little exercise. Familiarity with only the most basic algebraic topology will be assumed.

DIAMANT Intercity Number Theory Seminar

11:00-11:45	Peter Montgomery, Parallel block Lanczos Abstract. Some factorization and discrete logarithm algorithms have a linear algebra phase, where a huge sparse system must be solved over a finite field. One avoids a memory explosion by using iterative methods, but run time can remain high. We describe how to parallelize the linear algebra and relate our experiences.
12:00-12:45	Ramarathnam Venkatesan (Microsoft Research), Cryptographic applications involving Spectral analysis of Rapid mixing Abstract. We survey some applications that involve spectral analysis in various domains. First one is the analysis of classic Pollard Rho. Second one stems from the question if all elliptic curves of the same order over a finite field have the same difficulty of discrete log. The third one involves the design of a stream cipher called MV3. This is joint work with Steve Miller (Rutgers), David Jao (Waterloo).
13:45-14:30	Corentin Pontreau, Bogomolov's problem, small points on varieties Abstract. Height functions describe in a certain sense the arithmetic complexity of an algebraic number or more generally of an algebraic variety. We will present which kind of lower bounds for the height of algebraic varieties (Bogomolov's problem) and points of such varieties one can expect. Even if most of the results can be stated for semi-abelian varieties, we will mainly deal with the torus case, roughly speaking Gm×... ×Gm over the algebraic closure of Q.
14:45-15:30	Sierk Rosema (Leiden), Sturmian substitutions, cutting paths and their projections Abstract. From a string of zeros and ones of finite length we construct a stepped line that we call a cutting path. By projecting the integer points on this path onto the y-axis, we form a new string of zeros and ones. If σ is a Sturmian substitution, we apply this process to un=σn(0) to define a sequence of words vn. We will show that if σ has an incidence matrix with determinant 1, then there exists a Sturmian substitution τ such that vn=τ(vn-1) for every n>1.

Intercity Friday of the Mariusfest

Intercity Number Theory Seminar

12:00-13:00	David Kohel, Complex multiplication and canonical lifts Abstract. The j-invariant of an elliptic curve with complex multiplication by K is well-known to generate the Hilbert class field of K. Such j-invariants, or rather their minimal polynomials in Z[x], can be determined by means of complex analytic methods from a given CM lattice in C. A construction of CM moduli by p-adic lifting techniques was introduced by Couveignes and Henocq. Efficient versions of one-dimensional p-adic lifting were developed by Bröker. These methods provide an alternative application of p-adic canonical lifts, as introduced by Satoh for determining the zeta function of an elliptic curves E/Fpn. Construction of such defining polynomials for CM curves is an area of active interest for use in cryptographic constructions. Together with Gaudry, Houtmann, Ritzenthaler, and Weng, we generalised the elliptic curve CM construction to genus 2 curves using 2-adic canonical lifts. The output of this algorithm is data specifying a defining ideal for the CM Igusa invariants (j1,j2,j3) in Z[x1,x2,x3]. In contrast to Mestre's AGM algorithm for determining zeta functions of genus 2 curves C/F2n, this construction pursues the alternative application of canonical lifts to CM constructions. With Carls and Lubicz, I developed an analogous 3-adic CM construction using theta functions. In this talk I will report on recent progress and challenges in extending and improving these algorithms.
14:00-15:45	Johan Bosman, A polynomial with Galois group SL2(F16) Abstract. An interesting computation challenge is to calculate polynomials P in Q[x] that have a prescribed Galois group. By this we mean the Galois group of the splitting field together with its action on the roots of P. Jürgen Klüners and Gunther Malle developed methods that work for many groups, including all transitive permutation groups of degree up to 15. In this talk we will present a polynomial whose Galois group is isomorphic to SL2(F16), a group that Klüners and Malle could not handle with their approach. The computation makes use of Galois representations of modular forms.
16:00-17:00	Reinier Bröker, Lifting supersingular curves Abstract. In this talk we present a p-adic algorithm to compute the Hilbert class polynomial corresponding to an imaginary quadratic order O. This polynomial has integer coefficients and it's roots, say in the complex numbers, are j-invariants of elliptic curves with endomorphism ring O. The prime p that we use in this p-adic algorithm is inert in O, and is therefore quite small. The main step in the algorithm is computing the `canonical lift' of a supersingular elliptic curve over the finite field Fp. Many examples will be given.

Intercity Number Theory Seminar

11:00-12:00	Harm Voskuil, p-Adic uniformisation: introduction and examples Abstract. I briefly define and explain affinoid domains and rigid analytic spaces. Then the subject of p-adic uniformisation of (analytic) varieties is discussed. The main examples treated are those of abelian varieties and algebraic curves. Moreover, I will compare the p-adic and the real uniformisations of these examples.
13:00-14:00	Fumiharu Kato, Topological rings in rigid geometry Abstract. This is a joint-work with Kazuhiro Fujiwara (Nagoya). While classical algebraic geometry only deals with finite type rings over a field, scheme theory involves arbitrary rings; of course, fields and finite type rings over them are still important in scheme theory, because fields are `point objects', and finite type rings over a field are 'fiber objects' for locally of finite type maps between schemes. Rigid geometry a la Tate-Raynaud, on the other hand, has been developed over a-adically complete valuation rings of height 1. It has become recognized by experts that, in order to detect all 'points' of rigid spaces, one has to consider a-adically complete valuation rings of arbitrary height. This leads one to the quest for a reasonable class of topological rings that allows a generalization of `classical' rigid geometry, something compared with the scheme theory as a generalization of classical algebraic geometry, in such a way that a-adically complete valuation rings are `point objects', and that complete rings topologically of finite type over an a-adically complete valuation rings are 'fiber objects' for locally of finite type morphisms. In this talk, we would like to propose a candidate of such a class of topological rings, the so-called, topologically universally adhesive (t.u.a.) rings. This class is closed under topologically finite type extensions, contains any Noetherian ring complete with respect to an ideal, and has several nice topological, ring-theoretical, and homological properties, such as, Artin-Rees property, coherency, etc. Moreover, a deep theorem, which we attribute to Gabber, says that any a-adically complete valuation ring is contained in this class. We would like to indicate that, by means of this notion, one can develop a reasonable theory of formal schemes that admits one to generalize theorems in EGA III, and, based on these foundations, one has a generalization of the notion of rigid spaces.
14:10-15:10	Francis Brown, Multiple zeta values and periods of moduli spaces of genus zero curves Abstract. Let n≥4, and let M0,n denote the moduli space of curves of genus 0 with n marked points. In a recent paper, Goncharov and Manin showed how a pair of boundary divisors on the compactification M0,n defines a mixed Tate motive unramified over the integers. They conjectured that the periods one obtains in this way are multiple zeta values. In this talk I will outline a proof of this conjecture. I will give an explicit construction of the compactification M0,n, and recall some of its geometric and combinatorial properties. I will then explain how to compute the periods by iterated application of Stokes' formula in a suitable algebra of polylogarithm functions on M0,n.
15:30-16:30	Gautam Chinta, The theory of Weyl group multiple Dirichlet series Abstract. A Weyl group multiple Dirichlet series associated to a finte root system Φ of rank r is a Dirichlet series in r complex variables having an analytic continuation to r copies of C and satisfying a group of functional equations isomorphic to the Weyl group of Φ. These series have been the topic of intense study in recent years. I will discuss the history of the subject, describe how to construct these series and indicate some applications.

GTEM day

11:00-11:45	Dieter Geyer, Higher dimensional class field theory Abstract. Higher dimensional class field theory, i.e. the theory of abelian coverings of higher dimensional arithmetical schemes including varieties over finite fields, was started in case of regular schemes in the 1980's by Bloch, Kato and Saito in several papers using higher dimensional Milnor K-theory. Subsequent papers by Jannsen, Stevenhagen, Spiess, A. Schmidt and others followed. I will speak on a new approach by Goetz Wiesend, using only K0 and K1 groups, and a thesis of Walter Hofmann generalizing Wiesend's results from regular schemes to singular schemes.
12:00-12:45	Andrea Surroca Ortiz, On the Mordell-Weil and the Tate-Shafarevich groups of abelian varieties Abstract. This talk is about some conjectures relating the height, the conductor, the regulator and the Tate-Shafarevich group of abelian varieties over number fields. We will see that a result of Goldfeld-Szpiro relating the order of the Tate-Shafarevich group to the conductor of an elliptic curve over Q can be extended to arbitrary abelian varieties over number fields. On the other hand, we will also mention an application to the abc-conjecture, which is a work in collaboration with V. Bosser.
14:00-14:45	Christian Wuthrich, Computations about the Tate-Shafarevich group using Iwasawa theory Abstract. In analogy to the zeta function for varieties over finite fields, the p-adic L-series of an elliptic curve E over the rational field can provide us with interesting arithmetical information via Iwasawa theory. I will present an algorithm that can compute upper bounds on the order of the p-primary part of the Tate-Shafarevich group E. This is joint work with William Stein.
15:00-15:45	Gabor Wiese, Modular Forms in Inverse Galois Theory Abstract. Modular forms which are eigenfunctions for all Hecke operators give rise to 2-dimensional mod p representations of the absolute Galois group of the rationals. In the talk we will show how these representations, and hence modular forms, can be used to derive results on the occurrence of groups of the type PSL2(Fpr) as Galois groups over the rationals.

Intercity Number Theory Seminar

There is a group lunch for the seminar participants, and there will be tea and coffee at 14:30

11:30-12:30	Graham Everest, Elliptic Curves and Hilbert's tenth problem Abstract. Hilbert's Tenth Problem asks if an algorithm can be constructed which will determine if a finite system of Diophantine equations, with rational integer coefficients, has an integral solution. This was answered negatively in 1970 by Yuri Matiyasevic, building on work of Davis, Putnam and Robinson. The same question, except now to determine of there is a rational solution, has not been resolved. Recent work of Poonen has shown the same negative answer for some large subrings of the rationals using the arithmetic of elliptic divisibility sequences. In my talk I will report on Poonen's work as well as give some new results about which subrings of Q are covered by Poonen's methods.
13:30-14:30	Karma Dajani, Ergodic properties of signed binary expansions Abstract. Signed separated binary expansions of integers are expansions of the form n=∑i=0k-1 ai2i, where ai∈{ -1,0,1} and ai ai+1=0. Identifying an integer n with its corresponding sequence of SSB-digits a0,a1,...,ak-1, we consider an SSB-compactification K of Z, namely the set K={ (x0,x1,...) ∈{ -1,0,1}N : xi xi+1=0 for all i∈N}. On K there are two natural transformations, the shift σ and the odometer τ (the latter is analogous to adding 1 mod 2 with carry). In this talk, we discuss the ergodic properties of these transformations.
14:45-15:45	Fritz Schweiger, Multidimensional continued fractions - new results and old problems Abstract. Regular continued fractions exhibit a number of remarkable properties. If one puts pn/qn := 1/(a1+1/(a2+1/...(an-1+1/an)...)), then one obtains "good" approximations to x. The related map Tx=1/x-a1(x) is ergodic and admits an absolutely continuous invariant measure. The algorithm becomes eventually periodic, i.e., Tn+mx=Tmx for some ngeq0, mgeq1, if and only if x is a quadratic irrational number ("Theorem of Lagrange"). Since the days of C.G.J. Jacobi (1804-1851) who invented an algorithm for pairs of cubic irrational numbers, numerous multidimensional continued fraction algorithms have been proposed. In this talk the following related topics will be addressed. Convergence results and Diophantine properties of multidimensional continued fractions Invariant measures for multidimensional continued fractions Algebraic properties of multidimensional continued fractions.

RISC / INTS: Computational Number Theory

12:00-13:00	Ronald van Luijk, Explicit twisting of Jacobians of dimension 2 Abstract. In order to bound the rank of the group of rational points A(k) on an abelian variety A over a number field k, one often does a 2-descent to bound the order of the finite group A(k)/2A(k). This group injects into H1(k,A[2]), where A[2] denotes the group of 2-torsion points. The elements of this galois cohomological group correspond to certain twists of A, which are varieties over k that become isomorphic to A over the algebraic closure of k. Such a twist is in the image of A(k)/2A(k) if and only if it contains a rational point. In this talk I will first explain how twists correspond to cocyles. Then I will show how to find explicit equations of these twists as the intersection of 72 quadrics in P15 in the case that A is the Jacobian of a curve of genus 2.
14:00-15:00	Alexander May (Bochum), Using LLL-Reduction for Solving RSA and Factorization Problems: A survey Abstract. The talk addresses the problem of inverting the RSA function and the problem of factorizing integers. We relax these problems in several ways and show that the relaxations lead to polynomial time solvable problems. In this approach, we model the relaxed problems as polynomial equations which have roots of small size. The roots are then found by a method originally introduced by D. Coppersmith in 1996, which in turn is based on the famous LLL lattice reduction algorithm. We also present a novel application for RSA with so-called small CRT exponents. Namely, we show that the factorization of an RSA modulus N=pq can be found in polynomial time provided that RSA is used with a secret exponent d such that both d (mod p-1) and d (mod q-1) are smaller than N0.073. The existence of such a polynomial time attack answers a long-standing open problem by Wiener.
15:15-16:15	David Freeman, Constructing abelian varieties for pairing-based cryptography Abstract. In recent years, the Weil and Tate pairings on abelian varieties over finite fields have been used to construct a vast number of new and useful cryptosystems. The abelian varieties used in these systems must have small embedding degree with respect to a large prime-order subgroup. Such "pairing-friendly" abelian varieties are rare and thus require specific constructions. In this talk we describe our two recent contributions to the catalogue of pairing-friendly abelian varieties: ordinary elliptic curves of prime order with embedding degree 10, and ordinary abelian surfaces over Fq having arbitrary embedding degree with respect to a prime subgroup of order roughly q1/4. Both results require finding curves with complex multiplication by a specified CM field; making this step feasible while maintaining the pairing-friendly property is the difficult part of such constructions.

Intercity Number Theory Seminar

12:00-12:45	Rob de Jeu, Part I: introduction to K0, K1, K2
13:30-14:15	Rob de Jeu, Part II: K2 of curves over number fields Abstract. We start by giving an introduction to K0, K1 and K2 of rings, with an emphasis on arithmetic. The second talk will concentrate on a special case of a conjecture by Beilinson, relating K2 of a curve over a number field with the value of its L-function at 2.
14:30-16:15	Hendrik Lenstra, Algorithms for ordered fields Abstract. The lectures address the following problem from an algorithmic perspective. Given a finite extension of an ordered field, how can one, in a reasonably explicit manner, write down all orderings of the extension field that extend the given ordering of the base field?

Special day on discrete tomography

11:45-12:30	Rob Tijdeman, Words with any periods Abstract. Let be given positive integers n, p1, ..., pk. In the lecture an algorithm is presented which computes a word of length n with periods p1, ...,pk and among such words one with the maximum number of distinct letters. The algorithm has linear runtime. Some further properties of such words are mentioned. This concerns joint work with Luca Zamboni.
13:30-14:15	Sierk Rosema (Leiden), Beta-substitutions, cutting paths and their projections Abstract. A β-substitution σ is a particular type of substitution over an alphabet of k+1 letters. By projecting the integer points on the cutting path that corresponds to σn(0), we form a k-dimensional rotation word v(n). We define a function φ for which we prove that v(n+1)=φ(v(n)) for every n∈N.
14:30-15:30	Birgit van Dalen, Dependencies between line sums Abstract. In discrete tomography an integer matrix is reconstructed from the line sums in several directions. Between those line sums exist linear relations. We consider the problem of finding these relations. We can distinguish between so-called global and local dependencies. In this talk the local dependencies will be constructed for any set of directions.
15:45-16:45	Arjen Stolk, An algebraic approach to line sum dependencies Abstract. We rephrase the problem of dependencies between line sums in an algebraic way. Using this point of view, we have a good handle on the global dependencies (those which do not depend on the shape of the table). We compute the number of independent global dependencies and indicate how one could construct a basis.

Intercity Number Theory Seminar

11:00-12:00	Victor Abrashkin (Durham), An elementary approach to Breuil's classification of finite flat p-group schemes Abstract. In 2000 Breuil obtained a classification of finite flat commutative p-group schemes over valuation rings of complete discrete valuation fields of characteristic 0 with perfect residue field of characteristic p. This classification is based on a very extensive use of crystalline technique, in particular, it uses the Breen-Berthelot-Messing crystalline Dieudonne theory. In the talk it will be explained a direct way how to establish the main ingredient of this classification - the case of group schemes killed by p.
13:00-14:00	Cameron Stewart, On a refinement of the ABC conjecture Abstract. We shall present a refinement of the original ABC conjecture of Masser and Oesterle together with a heuristic justification for the refinement.
14:10-15:00	Frits Beukers, Nearby perfect powers Abstract. Let m, n be coprime integers >1. According to the ABC-conjecture distinct m-th and n-th powers of integers have a distance of order at least X(1-1/m-1/n-ε), where X is the size of the two powers. We consider the question for which exponents c we can find an infinite number of n-th and m-th powers with distance O(Xc).
15:30-16:30	Attila Bérczes, On pairs of polynomials and binary forms with given resultant Abstract. Let F and G be two polynomials or binary forms and denote by R(F,G) their resultant. Let S be a set of primes and denote by ZS the ring of S-integers. In the talk we consider resultant equations of the type R(F,G) = c to be solved in polynomials F,G with coefficients from ZS, where c is a positive integer.

Special day on cryptology

11:30-12:30	Ivan Damgaard, The Past, Present and Future of Secure Multi-Party Computation
13:30-14:30	Yuval Ishai, Secure Multi-Party Computation in the Head
16:15-17:15	Ronald Cramer, Inaugural lecture (in Dutch)