2011

Intercity number theory seminar

11:30-12:20	Tom Ward, Counting fixed points of group automorphism Abstract. Compact group automorphisms form a simple family of dynamical systems, and a natural dynamical invariant is their periodic point data. Understanding this amounts to knowing about how many fixed points an automorphism has, and this is a surprisingly subtle question. This talk will give an overview of some recent approaches to this problem, and some of the number-theoretic issues that arise.
13:30-14:20	Soroosh Yazdani, Local Szpiro's conjecture Abstract. Given an elliptic curve over the rationals, its conductor and minimal discriminant are two integers that keep track of its primes of bad reduction. A conjecture of Szpiro states (in a fairly explicit manner) that the minimal discriminant can't be much larger than the conductor of a given elliptic curve. It is known that this conjecture is equivalent to the ABC conjecture, and has many applications to Diophantine equations. In this talk, we present a similar conjecture which we call Local Szpiro's Conjecture. Specifically, we conjecture that given any semi stable elliptic curve with minimal discriminant Δ, one can always find a prime of bad reduction such that vp(Δ) ≤6. In this talk we present some evidence and application for this conjecture. This is joint work with Mike Bennett.
14:30-15:20	Sander Dahmen, Klein forms and the generalized superelliptic equation Abstract. Let F be a binary form over the integers and consider the exponential Diophantine equation F(x,y)=zn with x and y coprime. For general F it seems very difficult to solve this equation, but as we will explain in this talk, for so-called Klein forms F, the modular method can provide a good starting point. Together with solving certain infinite families of Thue equations, we can show in particular, that there exist infinitely many (essentially different) cubic forms F for which the equation above has no solutions for large enough exponent n. This is joint work with Mike Bennett.
15:40-16:30	Jonathan Reynolds, Modular methods for perfect powers in elliptic divisibility sequences Abstract. For Mordell curves, I will explain why there are finitely many perfect powers in an elliptic divisibility sequence whose first term is greater than 1. The proof uses recent modular methods by Bennett, Billerey, Dahmen, Vatsal and Yazdani.

Intercity number theory seminar

11:15-12:15	James Lewis, An Archimedean height pairing on the equivalence relation defining Bloch's higher Chow groups Abstract. The existence of a height pairing on the equivalence relation defining Bloch's higher Chow groups is a surprising consequence of some recent joint work by myself and Xi Chen on a non-trivial K1-class on a self-product of a general K3 surface. I will explain how this pairing comes about.
13:15-15:15	Jean-Louis Colliot-Thelene, Lectures on the Hasse principle, I Abstract. Hasse principle and weak approximation : elementary fibration method, Brauer-Manin obstruction for rational and integral points, examples. To which extent can we compute the Brauer group and the Brauer-Manin set?
15:30-16:30	Ronald van Luijk, Computability of Picard numbers Abstract. The Néron-Severi group of a variety is the group of its divisor classes modulo algebraic equivalence. The rank of this group is called the Picard number of the variety. After giving a short review of ad hoc methods that compute the Picard number in certain cases, I will sketch an idea of Bjorn Poonen to prove that the Picard number is computable in general. This is joint work in progress with Damiano Testa.

Intercity number theory seminar

11:15-12:15	Damiano Testa, The surface of cuboids and Siegel modular threefolds Abstract. A perfect cuboid is a parallelepiped with rectangular faces all of whose edges, face diagonals and long diagonal have integer length. A question going back to Euler asks for the existence of a perfect cuboid. No perfect cuboid has been found, nor it is known that they do not exist. In this talk I will show that the space of cuboids is a divisor in a Siegel modular threefold, thus allowing to translate the existence of a perfect cuboid to the existence of special torsion structures in abelian surfaces defined over number fields.
13:15-15:15	Jean-Louis Colliot-Thelene, Lectures on the Hasse principle, II Abstract. Hasse principle and weak approximation : classes of varieties for which they hold (in particular, some intersections of two quadrics). Some classes for which they do no hold. Interpretation by means of the Brauer-Manin obstruction.
15:30-16:30	Martin Bright, Some computational aspects of the Brauer-Manin obstruction Abstract. Given a surface X over a number field, how far can we go towards computing the Brauer-Manin obstruction on X? I have been implementing algorithms in this direction with the Magma group in Sydney, and several interesting problems in computational arithmetic geometry have come up along the way. I will discuss a selection of these problems.

Intercity number theory seminar

At 17:00 there will be drinks offered at the Foobar to celebrate that Martin Lübke has been at the department for 25 years.

11:45-12:45	Christine Berkesch, The rank of a hypergeometric system Abstract. Gelfand, Kapranov, and Zelevinsky's introduction of a torus action to the study of hypergeometric systems brought a wealth of combinatorial and geometric tools to the theory. They showed that the dimension, or rank, of the solution space of such a system is constant for generic parameters. I will discuss homological tools developed by Matusevich, Miller, and Walther, which can be used to obtain a combinatorial formula for the rank of a hypergeometric system at any parameter.
13:45-15:45	Jean-Louis Colliot-Thelene, Lectures on the Hasse principle, III Abstract. Continuation of Lecture III. How to compute the Brauer group. Harari's formal lemma, applications. Torsors under algebraic groups and descent. Beyond the Brauer-Manin obstruction.
16:00-17:00	Andrei Teleman, Holomorphic bundles and holomorphic curves on class VII surfaces. The classification problem for class VII surfaces. Abstract. The classification of complex surfaces is not finished yet. The most important gap in the Kodaira-Enriques classification table concerns the Kodaira class VII, e.g. the class of surfaces X having kod(X)=-∞, b1(X)=1. These surfaces are interesting from a differential topological point of view, because they are non-simply connected 4-manifolds with definite intersection form. The main conjecture which (if true) would complete the classification of class VII surfaces, states that any minimal class VII surface with b2>0 contains b2 holomorphic curves. We explain a new approach, based on ideas from Donaldson theory, which gives existence of holomorphic curves on class VII surfaces with small b2. In particular, for b2=1 we obtain a proof of the conjecture, and for b2=2 we prove the existence of a cycle of curves.

Intercity number theory seminar

11:45-12:45	Karl Rökaeus (Amsterdam), Global function fields with many rational places Abstract. The Riemann hypothesis for curves (proved by Weil) give an upper bound on how many rational places there can be in a global function field of genus g with constant field of cardinality q. This bound is rarely met; when g/q is big it can be improved substantially and in general it is unknown how far it is from being achieved. For small q (mostly powers of 2 and 3) and g<50 much work has been done on constructing fields with many places, and this combined with work improving the upper bounds has determined the maximum possible number of places in such a fields, or a small interval in which it lies. These intervals are recorded on the webpage manYPoints.org. In this talk we discuss some (classical) methods from class fields theory that can be used for constructing fields with many places. We then apply them in a systematic computer search. This yields many new lower bounds in cases that had not been much investigated earlier; it also gives some new fields in characteristic 2 and 3.
13:30-15:45	Jean-Louis Colliot-Thelene, Lectures on the Hasse principle, IV Abstract. The Brauer group may be used to define a Brauer-Manin obstruction to the existence of a zero-cycle of degree 1. For arbitrary smooth projective varieties over a global field, is this the only obstruction?
16:00-17:00	Remke Kloosterman, Calculating the Mordell-Weil group for a class of elliptic threefolds Abstract. Let K be a subfield of the complex numbers. In the first part of the talk I will give a heuristic argument why the calculation of the Mordell-Weil rank of a "general" elliptic curve over K(s,t) is easier than the calculation of the Mordell-Weil rank of a "general" elliptic curve over K(t). I will illustrate this by discussing a particular class of elliptic curves over K(s,t) with constant j-invariant 0. For this class of elliptic curves we will give an upper bound for the Mordell-Weil rank in terms of the degree of the discriminant of the elliptic curve that is roughly half a naive upper bound for the rank. We get this bound by considering the syzygies of the ideal of the singular locus of the discriminant curve. These consideration lead to a lot of interesting by-products, i.e., we obtain an example of a "Zariski triple", and we obtain examples of triples (g,k,n) such that the locus {[C] ∈Mg | C admits a g26k and the image of C has at least n cusps } ⊂Mg has much bigger dimension than expected.

Intercity Number Theory Seminar

Today's theme: applications of measure theory in number theory

11:30-12:20	Janne Kool (Utrecht), Measure theoretic rigidity for Mumford curves Abstract. Measure theoretic rigidity (in the style of Mostow) states that two compact hyperbolic Riemann surfaces of the same genus are isomorphic if and only if the "boundary map" associated to their uniformizations is absolutely continuous. In this talk, we will formulate the analog of this result for "p-adic Riemann surfaces", i.e., for Mumford curves. In this case, the mere absolute continuity of the boundary map implies only isomorphism of the special fibers of the Mumford curves, and needs to be enhanced by a finite list of conditions on the harmonic measures (in the sense of Schneider and Teitelbaum) on the boundary to guarantee an isomorphism of the Mumford curves.
13:30-14:20	Tom Ward, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture Abstract. This will be a short introduction to some of the ideas used in the work of Einsiedler, Katok and Lindenstrauss on the Littlewood problem. This will include the connection between the modular surface and continued fractions and an introduction to measure rigidity.
14:40-15:30	Tanja Eisner, Arithmetic progressions via ergodic theory Abstract. We sketch the development from van der Waerden's theorem on arithmetic progressions to the recent Green-Tao theorem and show how methods from ergodic theory have been decisive in this field.
15:40-16:30	Karl Petersen, Some combinatorial and number-theoretic questions related to certain adic dynamical systems Abstract. Trying to figure out dynamical properties of particular systems often leads to interesting (and sometimes very difficult) questions in combinatorics or number theory. This has certainly been the case with the Pascal, Euler, faux multidimensional Euler, and Delannoy adic systems, which have led to results, questions, and conjectures about binomial coefficients, Eulerian numbers, Stirling numbers, and Delannoy numbers. We review how these matters arise and some of what we know and would like to know about them.

Intercity Number Theory Seminar

Intercity Number Theory Seminar

13:30-14:30	Alberto Facchini, Krull-Schmidt Theorem: the case two Abstract. I will mainly present the content of a joint paper with Pavel Prihoda (The Krull-Schmidt Theorem in the case two, Algebr. Represent. Theory 14(3) (2011), 545-570), but also other results obtained jointly with A. Amini, B. Amini, S. Ecevit, M. T. Kosan and N. Perone. Essentially, the Krull-Schmidt-Azumaya Theorem says that if M1,...,Mm,N1,...,Nn are R-modules with local endomorphism rings and M1⊕...⊕Mm≅N1⊕... ⊕Nn, then n=m and there exists a permutation σ of {1,...,n} such that Mi≅Nσ(i) for every i=1,...,n. I will present what happens if the endomorphism rings of the modules Mi and Nj have two maximal ideals instead of only one. Several examples of these modules will be given.
14:45-15:45	Antonella Perucca, Radical characterizations of elliptic curves Abstract. Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime p in S to the size #E(k_p) of the group of points over the residue field. We prove that it suffices to look at the radical of the size. We also replace E(k_p) by the image of the Mordell-Weil group via the reduction modulo p, and solve this analogous problem for a large class of abelian varieties. This is a joint work with Chris Hall.
16:00-17:00	René Schoof, Integral points on a modular curve of level 11 Abstract. Using lower bounds for linear forms in elliptic logarithms we determine the integral points of the modular curve associated to the normalizer of a non-split Cartan group of level~11. As an application we obtain a new solution of the class number one problem for complex quadratic fields. This is joint work with Nikos Tzanakis.

Intercity Number Theory Seminar

13:00-14:00	Valérie Berthé, Adic constructions for fundamental domains for Kronecker sequences Abstract. The aim of this lecture is to provide explicit constructions for fundamental domains for Kronecker sequences in order to get arithmetic codings of the corresponding dynamical system (the underlying toral translation) that preserve their arithmetic properties. It is well-known how to associate such fundamental domains with algebraic (and more precisely Pisot) parameters, when the Kronecker sequence is generated by a substitution (a substitution is a symbolic action that replaces letters by words). These fundamental domains are called Rauzy fractals or central tiles. It is more delicate to consider nonalgebraic parameters. To this end, substitutive symbolic dynamical systems can be extended to so-called S-adic systems governed by multidimensional continued fraction algorithms. These latter systems are obtained by iterating not only one substitution, but a finite number of them. The hierarchies thus produced have not necessarily the same structure at each level, but there are only finitely many possible structures. Our aim here is under suitable convergence properties that play the role of the Pisot assumption to define generalized Rauzy fractals with tiling properties in this framework. This a joint work with W. Steiner and J. Thuswaldner.
14:10-14:55	Milan Lopuhaä (Leiden), Field topologies on countable fields Abstract. In 1972, it was proven that any field K admits exactly 2^(2^|K|) field topologies on it, i.e. topologies such that all field operations are continuous. In my Bachelor Thesis, I modified the proof of the countable case to show that in the case of an algebraic closure of a finite field, these topologies can be made in such a way that all field automorphisms are continuous, and the proof of this will be shown in this seminar.
14:55-15:40	Henk Don, Polygons in billiard orbits Abstract. We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.

Intercity Number Theory Seminar

13:30-14:30	Wouter Castryck, Frobenius statistics for varying curves over fixed finite fields Abstract. Fix a large prime power q and an integer g>0. Let C be a "random" genus g curve over the finite field GF(q). Let Jac(C) be the group of rational points on its Jacobian. For various reasons, one can wonder about the probability that #Jac(C) is divisible by N (for a given integer N) the probability that #Jac(C) is prime, the probability that Jac(C) is cyclic, ... the asymptotic behavior of these probabilities for g going to infinity. We will show how all of this can be estimated (sometimes heuristically, sometimes provably) using a random matrix framework. This is joint work with Amanda Folsom, Hendrik Hubrechts, and Andrew Sutherland.
14:45-15:45	Gabriele Dalla Torre, L-function-preserving isomorphisms of groups of quadratic characters Abstract. Let Mer(C) be the set of meromorphic functions on C. Given a number field K, we denote by LK: K*/K*2 →Mer(C) the function that maps (d mod K*2) to the quadratic L-function associated to the quadratic extension K(√d) of K. The main result of this lecture is the following theorem. Let K and K' be number fields. Then the natural map from the set of field isomorphisms K →K' to the set of group isomorphisms β: K*/K*2 →K'*/K'*2 with the property that LK' o β= LK, is bijective.
16:15-17:15	Peter Jossen, The unipotent part of the Mumford-Tate conjecture Abstract. The Mumford-Tate conjecture is a statement which compares singular cohomology with l-adic cohomology of algebraic varieties, classically abelian varieties, defined over finitely generated fields of characteristic zero. I will show that a small but already useful part of this conjecture is true, and can be formulated in positive characteristic as well.

RISC/Intercity Number Theory Seminar

11:45-12:30	Ariel Gabizon (Israel Institute of Technology - Technion / CWI), Extractors : Background, Applications and Recent Constructions Abstract. Randomness extractors are functions whose output is guaranteed to be uniformly distributed, given some assumption on the distribution of the input. The first instance of a randomness extraction problem comes from von-Neumann who gave an elegant solution to the following problem: How can a biased coin with unknown bias be used to generate `fair' coin tosses? In this case the input distribution consists of independent identically distributed bits. Since then many families of more complex input distributions have been studied. Also, the concept of randomness extraction has proven to be useful for various applications. The talk will give some background on extractors and review applications and techniques used in recent constructions of extractors.
14:00-14:45	Gil Cohen (Weizmann Institute of Science), Non-Malleable Extractors with Short Seeds and Applications to Privacy Amplification Abstract. Motivated by the classical problem of privacy amplification, Dodis and Wichs (STOC '09) introduced the notion of a non-malleable extractor, significantly strengthening the notion of a strong extractor. A non-malleable extractor is a function nmExt that takes two inputs: a weak source W and a uniform (independent) seed S, and outputs a string nmExt(W,S) that is nearly uniform given the seed S *as well* as the value nmExt(W,S') for any seed S' \neq S that may be determined as an arbitrary function of S. In this work we present the first unconditional construction of a non-malleable extractor with short seeds. By instantiating the framework of Dodis and Wichs with our non-malleable extractor, we obtain the first 2-round privacy amplification protocol for min-entropy rate 1/2 + delta with asymptotically optimal entropy loss and poly-logarithmic communication complexity. This improves the previously known 2-round privacy amplification protocols: the protocol of Dodis and Wichs whose entropy loss is not asymptotically optimal, and the protocol of Dodis, Li, Wooley and Zuckerman whose communication complexity is linear and relies on a number-theoretic assumption. Joint work with Ran Raz and Gil Segev.
15:00-15:45	Stefan Dziembowski (Rome), Leakage-Resilient Cryptography From the Inner-Product Extractor
16:00-16:45	Christian Schaffner (UVA/CWI), Randomness extraction and expansion in the quantum world Abstract. Randomness extraction is a fundamental task in cryptography, where it is intimately connected with the problem of privacy amplification. In this talk we will survey the specific challenges posed by this task in the setting where an adversary may hold *quantum* information about the source and give an overview over the known results in the area. In the last part, we touch on recent joint work with Fehr and Gelles. We demonstrate that quantum mechanics allows to expand some initial randomness in a secure way even if the used devices are manufactured by the adversary.

Intercity Number Theory Seminar

13:30-14:30	Pascal Autissier, On the non-density of integral points Abstract. We give nondensity results for integral points on affine varieties, in the spirit of the Lang-Vojta conjecture. In particular, le tX be a projective variety of dimension d>1 over a number field K (resp., over C). Let H be the sum of 2d properly intersecting ample divisors on X. We show that any set of quasi-integral points (resp., any integral curve) on X-H is not Zariski dense.
14:45-15:45	Fabrizio Andreatta, On p-adic families of elliptic overconvergent modular forms Abstract. I will report on a joint project with A. Iovita and G. Stevens. I will show how to construct families of overconvergent elliptic modular forms as global sections of suitable "modular sheaves" even for non-integral weights. These are defined by correcting the Hodge-Tate map using the theory of the canonical subgroup. I will show how we can explain/recover the theory of Coleman in a way which can be generalized to other groups beside GL2,Q.
16:00-17:00	Jan-Hendrik Evertse, Multiply monogenic orders Abstract. An order in an algebraic number field K is a subring of the ring of integers of K which has quotient field K. An order generated over Z by one element, i.e. of the shape Z[w], is called monogenic. Given an order O, the set of w with Z[w]=O falls naturally into equivalence classes, where two elements v, w of O are called equivalent if v-w or v+w is a rational integer. In 1976, Györy proved that for any number field K, and any order O in K, there are only finitely many equivalence classes of w in O such that O=Z[w]. An order O for which there are at least k equivalence classes of w with Z[w]=O is called k times monogenic; if there are precisely/at most k such equivalence classes, it is called precisely/at most k times monogenic. For instance any order in a quadratic number field is precisely one time monogenic. In this talk I fix a number field of degree at least 3 and consider varying orders in this field. The first main result is, that every number field K of degree at least 3 has at most finitely many orders which are three times monogenic. This bound 3 is best possible. The second main result is, that under some additional constraints imposed on K, there are only finitely many two times monogenic orders in K which are not 'of a special type.' This is joint work with Attila Bérczes and Kálmán Györy.

Intercity Number Theory Seminar

11:00-12:00	Matthias Schuett, Arithmetic of quintic surfaces Abstract. The Picard number is a non-trivial invariant of an algebraic surface which captures much of its inner structure. We will discuss the fundamental problem which Picard numbers occur on surfaces on general type for the prototype example of quintics in IP^3. We review what seems to be known and explain a new technique based on arithmetic deformations.
13:00-14:00	Tetsuji Shioda, Cubic surfaces via Mordell-Weil lattices, re-revisited
14:30-15:30	Bas Heijne, PhD defense: Elliptic Delsarte Surfaces.