Intercity Number Theory SeminarMarch 1, Utrecht. A programme around gonalities. Room 611 of the Hans Freudenthal building (the mathematics institute).
- Dion Gijswijt Delft & CWI, Graph gonality a la Baker-Norine: basic concepts, theorems and some open problems
- Anna Cadoret Ecole Polytechnique, Paris, Gonality, isogonality and points of bounded degree on curves over finitely generated fields in positive characteristic
- Maarten Derickx Leiden, Gonality of modular curves
- Janne Kool Utrecht, A combinatorial Li-Yau inequality and rational points on curves
Intercity Number Theory SeminarMarch 15, Leiden. Snellius, room 312. The talks are followed by a reception at the Lorentz center in the Snellius building (adjacent to the math library).
- Pieter Moree MPI Bonn, Sister Beiter and the Novices: Cyclotomic Coefficients and the Young
- Reinier Bröker Brown, Fourier coefficients of Siegel modular forms
- Henri Cohen Bordeaux, Enumerating D_\ell number fields of degree \ell and quartic fields with given resolvent
3rd Belgian-Dutch Algebraic Geometry dayApril 19, Leuven. Room A00.225, Celestijnenlaan 200. See also the map here.
- Alejandro Soto Regensburg, Toric Varieties over a Valuation Ring
One of the main features of toric geometry is that we can study the geometry of algebraic varieties by studying the combinatorics of some objects in convex geometry, namely cones and fans. The crucial result allowing this fact is Sumihiro's theorem. It is well known that this result holds for toric varieties defined over a field and over a discrete valuation ring. The aim of this talk is to show that the crucial correspondence between fans and toric varieties can be extended to the case of toric varieties over a valuation ring of rank one, as introduced by Gubler.
- Sébastien Boucksom Jussieu, A uniform version of Izumi's theorem
A basic result of Izumi states that any two divisorial valuations sharing the same center on a given algebraic variety are comparable, up to multiplicative constants. I will present a joint work with C. Favre and M. Jonsson where we prove a uniform version of Izumi's theorem, showing that the optimal constants vary in a Lipschitz continuous way when the valuation moves inside the dual complex of a given simple normal crossing divisor. As a consequence, we obtain a compactness result for semipositive singular metrics on certain Berkovich spaces, which lies at the heart of a variational approach for solving non-Archimedean Monge-Ampère equations.
- Ariyan Javanpeykar Leiden, Bounds for Arakelov invariants - algorithmic and Diophantine applications
present explicit polynomial bounds for Arakelov invariants of curves in terms of the Belyi degree. Our initial motivation was "algorithmic". In fact, we prove a conjecture of Edixhoven, de Jong, and Schepers on the Faltings height of a cover of curves. This result is a first step towards a polynomial algorithm for computing the ell-adic Galois representation associated to a surface over Q. Finally, we also present a "Diophantine" application. In fact, we prove a special case of Szpiro's small points conjecture (1985). Namely, we prove Szpiro's conjecture for hyperelliptic curves.
Intercity Number Theory SeminarMay 3, Groningen. The lectures are in Room 293 of the Bernoulliborg
- Christina Delfs Oldenburg, Isogenies between Supersingular Elliptic Curves over Fp
The problem of computing an isogeny between two given elliptic curves has been studied by many authors and has several applications. In the case of ordinary elliptic curves the computation is based on the volcano-like structure of the isogeny graph, which provides a connection to ideal class groups. The arising algorithm is sufficiently fast under the assumption of GRH.
In the supersingular case though, the isogeny graph is very irregular and this approach does not work. The currently fastest algorithm for finding isogenies between supersingular curves performs a random walk on the fully-connected supersingular isogeny graph over mathbbFp2 and is considerably slower than the algorithm for ordinary curves.
In this talk we will restrict to isogenies between supersingular elliptic curves over the prime field mathbbFp and examine the corresponding subgraph of the supersingular isogeny graph. We will show some results about this graph that bear resemblance to the ordinary case and hopefully help to adapt the algorithm to this situation. Assuming one small still open conjecture we are able to give an algorithm to construct isogenies between supersingular curves over mathbbFp that works as fast as in the ordinary case. Finally we shortly discuss the possibilities to use this algorithm to obtain an improved low-storage algorithm for the computation of isogenies between arbitrary supersingular elliptic curves.
This is joint work with Steven Galbraith from University of Auckland in New Zealand and still work in progress. We hope to present a complete description of this problem in a paper soon.
- Harm Voskuil, Unitary groups and p-adic uniformisation of curves
Let K be a local field of characteristic zero and let L be the unramified quadratic extension of K. Let SU(3,L) denote the unitary group in three variables.
For a discrete co-compact arithmetic subgroup of the unitary group SU(3,L), we construct a p-adic analytic space of dimension one on which the discrete group acts discretely. The quotient space is a proper algebraic curve.
The construction uses the building of the unitary group. The building of SU(3,L) is a tree. A covering of the building of the unitary group SU(3,L) by buildings of subgroups SU(2,L) invariant under the discrete subgroup will be described.
The components of the reduction of the analytical space are hermitian curves and projective lines. The analytic space itself is closely related to Drinfelds first etale covering of the p-adic upper halfplane.
- Afzal Soomro Groningen, Special curves of genus four and five over finite fields
In this talk we discuss some curves of genus 4 and 5 over a finite field with many points. Maxim Hendriks' thesis (2013) contains models of curves of genus 4 and 5 over some number field admitting a platonic map. We decompose the Jacobian of some of these curves, and reduce modulo primes to obtain curves of genus 4 and 5 over a finite field with many points. Also we discuss some examples of the curves with many points obtained as the fibre product of genus one curves.
- Jaap Top Groningen, First order nonlinear differential equations
This is joint work with L.X. Chau Ngo, K.A. Nguyen, and M. van der Put.
We introduce a notion of equivalence of first order differential equations and study the problem of algorithmically testing this equivalence. This is related to properties of algebraic curves.
RISC / Intercity number theory seminarJune 7, CWI Amsterdam. Room L016.
- Ted Chinburg UPenn, Taking on new capacities
Capacity theory originated in the study of how electrical charges distribute themselves in order to to minimize energy. In this talk I will give an overview of connections between this subject and number theory.
- Juan Garay AT&T Labs, Resource-based Corruption in Secure Computation
The notion of computing in the presence of an adversary which controls or gets access to parts of the system is at the heart of modern cryptography. In such a setting, however, the "corruption" of a party has been viewed as a simple, uniform and atomic operation, where the adversary decides to get control over a party and this party immediately gets corrupted. In this work, motivated by the fact that different entities running a multiparty protocol may require different resources to get corrupted, we put forth the notion of *resource-based corruptions*, where the adversary must invest some resources in order to corrupt them. If the adversary has full information about the system configuration then resource-based corruptions would provide no fundamental difference from the standard corruption model. However, in a resource "anonymous" setting, in the sense that such configuration is hidden from the adversary, much is to be gained in terms of efficiency and security. We showcase the power of such "hidden diversity" in the context of the popular paradigm known as "secure multiparty computation" with resource-based corruptions, and prove that it can effectively be used to circumvent known impossibility results.
Joint work with David Johnson (AT&T Labs), Aggelos Kiayias (U. of Athens) and Moti Yung (Google).
- Iwan Duursma U. Illinois Urbana-Champaign, From abstract curves to efficient secret sharing
The three families of Deligne-Lusztig curves arise in connection with representations of the algebraic groups 2A2 (unitary group), 2B2 (Suzuki group) and 2G2 (Ree group). From their abstract definition it is clear that in principle the curves are suitable for constructing long error-correcting codes or secret sharing schemes with many participants. We describe the following results from the April 2013 thesis of Abdulla Eid: A smooth model for the Ree curve, the determination of the Weierstrass semigroup at a rational point, and its application to curve-based secret sharing.
- Chaoping Xing NTU, On torsion limit of algebraic curves over finite fields
The Ihara limit (or -constant) has been a central problem of study in the asymptotic theory of global function fields (or equivalently, algebraic curves over finite fields). It addresses global function fields with many rational points and so far, most applications of this theory do not require additional properties. Motivated by recent applications, we require global function fields with the additional property that their zero class divisor groups contain at most a small number of torsion points. We capture this by the torsion limit, a new asymptotic quantity for global function fields. It seems that it is even harder to determine values of this new quantity than the Ihara constant. In this talk, we survey some recent progress on this topic. (This is a joint work with Ignacio Cascudo and Ronald Cramer)
4th Belgian-Dutch Algebraic Geometry dayJune 14, Leiden. Part of the workshop Heights and Moduli Spaces at the Lorentz Center.
- Walter Gubler Regensburg, Normal toric varieties over valuation rings of rank 1
Tropicalizations lead to the study of normal toric varieties over valuation rings of rank 1. In this talk, some of their properties are discussed including their classification in terms of fans.
- Gaël Rémond Bordeaux, Isogenies and polarizations
In a joint work with Éric Gaudron, we give several explicit estimates for the geometry of abelian varieties over number fields. In particular, we establish the existence of a small polarization, whose degree is bounded in terms of the Faltings height of the variety, its dimension and the degree of the base field. We further improve and make explcit the isogenies and factorization estimates of Masser and Wüstholz. The proofs are based on some geometry of numbers arguments in euclidean lattices of morphisms between abelian varieties, together with a period theorem.
- Jean-Benoît Bost Orsay, Pluripotential theory, heights, and algebraization on arithmetic threefolds
We present a construction of algebraic vector bundles on projective surfaces over number fields, starting from formal and analytic data along some ample divisor. It involves pluripotential theory and complex analysis on pseudoconcave domains, and the use of hermitian vector bundles of infinite rank.
- José Ignacio Burgos Gil ICMAT Madrid, The singularities of the invariant metric of the sheaf of Jacobi forms on the universal elliptic curve
A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, provided with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key to being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford's result remains valid for line bundles on mixed Shimura varieties. In this talk we will examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We will show that Mumford's result can not be extended to this case and that a new interesting kind of singularities appear. We will discuss some preliminary results. This is joint work with G. Freixas, J. Kramer and U. Kühn.
Intercity Number Theory SeminarSeptember 6, Leiden. Celebration of Rob Tijdeman's 70th birthday. Room 412 of the Snellius (math department). The lectures are followed by a reception. See also the programme and abstracts.
- Frits Beukers Utrecht, Divisor sums and higher powers
- Cameron Stewart Waterloo, On a question of Wintner
- Cor Kraaikamp Delft, Random continued fraction expansions
- Herman te Riele CWI, A cube whose sum of divisors is again a cube
- Ionica Smeets Leiden, Toy problems for the retired
Intercity Number Theory SeminarOctober 11, Groningen. Bernoulliborg, room 105.
- Yonatan Harpaz Nijmegen, From linear equations in primes to the fibration method
Let F : X →P1 be a proper surjective map of algebraic varieties over a global field K whose generic fiber is geometrically integral and whose fibers are all geometrically reduced. Assume in addition that the Brauer-Manin obstruction is the only one for the Hasse principle on all but finitely many fibers. A natural question is then whether or not the Brauer-Manin obstruction is the only one for the Hasse principle on X itself. The fibration method (Colliot-Thélène, Sansuc, Harari, Skorobogatov, Swinnerton-Dyer and others) is a collection of tools and techniques designed to attack such questions under various circumstances. For example, if one assumes Schinzel's hypothesis, one can apply the fibration method to cases where all the bad fibers split under an abelian extension of the base field and in addition almost all the fibers satisfy the Hasse principle. In this talk we will explain how, in the case where the ground field is Q and all the bad fibers are defined over Q, one can replace Schinzel's hypothesis with additive combinatorics results steaming from recent work of Green, Tao and Ziegler. If time permits, we will mention some work in progress applying recent results of Matthiesen and Browning towards removing the abelianness assumption and relaxing the Hasse principle assumption on the fibers. Similar generalizations can be obtained for the case of 0-cycles without relying on any additive combinatorics results. This is joint work with Alexei Skorobogatov and Olivier Wittenberg.
- Andreas Stein Oldenburg, Faster sieving methods for solving discrete logarithms in Jacobians of hyperelliptic curves
We describe novel and improved versions of index-calculus algorithms for solving discrete logarithm problems in Jacobians of medium-genus to high-genus hyperelliptic curves defined over even characteristic fields. One important improvement is based on the effective adaptation of sieving techniques known from the number field sieve, the function field sieve, and the arithmetic of number fields to the curve setting. Hereby, we make use of a highly efficient polynomial sieve that can be easily parallelized. Our new algorithms are applied to concrete problem instances arising from Frey's Weil descent attack methodology for solving the elliptic curve discrete logarithm problem, demonstrating significant improvements in practice. This is joint work with Michael J. Jacobson Jr, Wilke Trei, and Mark D. Velichka.
- Marius van der Put Groningen, Differential equations of order one (characteristic zero)
This talk discusses differential equations f(y',y,z)=0 where f(X,Y,z) is a polynomial in X,Y with coefficients in some finite extension of the rational function field C(z) over the complex numbers C. Moreover, it is assumed that f has positive degree in each of X,Y, and f(X,Y,z) is irreducible as a polynomial in X,Y over an algebraic closure of the rational function field C(z). Questions concerning such differential equations are, for example, the existence of (infinitely many) algebraic solutions. This is joint work with L.X. Chau Ngo, K.A. Nguyen, and J. Top.
- Jaap Top Groningen, Differential equations of order one (positive characteristic)
This is joint work with Marius van der Put. We recall the definition from Grothendieck's [EGA4] of a certain non-commutative algebra D of differential operators in characteristic p>0, serving as an analogue of the classical Weyl algebra generated by z and d/dz. The notion "stratified order one differential equation f(y',y,z)=0" in characteristic p>0 is defined in terms of the algebra D. In the talk, relations of this definition with more classical notions such as super-singularity and the Cartier operator are given. The definition allows one to formulate an analogue of the Grothendieck-Katz conjecture, which in the present instance asserts that an order one equation in characteristic zero admits admits infinitely many algebraic solutions, precisely when for almost all primes p, a reduction modulo p of the equation is stratified. We can prove this conjecture in the special case of autonomous equations.
Intercity Number Theory SeminarOctober 25, Leiden. The talks are in room B03 of the Snellius building. The PhD Defense of Chao Zhang takes place in the Akademiegebouw in the centre of Leiden (Rapenburg).
- Torsten Wedhorn Paderborn, Newton strata for Shimura varieties of Hodge type
I will define the Newton stratification for the models defined by Kisin and Vasiu for Shimura varieties of Hodge type at places of good reduction. I will explain a result of Daniel Wortmann who uses the main theorem of the PhD thesis by Chao Zhang to prove that the ordinary locus of the Newton stratification is open and dense in the special fiber of the model.
- Fabrizio Andreatta Milano, A p-adic criterion for good reduction of p-adic curves
I will present a result, joint work with Minhyong Kim and Adrian Iovita, stating that a proper semistable curve of genus g≥2 over a p-adic field K has good reduction if and only if its pro-p unipotent fundamental group is a crystalline GK-representations.
- Bas Edixhoven Leiden, The Andre-Oort conjecture for Ag, under GRH, after Pila and Tsimerman
In their arxiv preprint of June 2012 Jonathan Pila and Jacob Tsimerman prove that the Andre-Oort conjecture for the moduli spaces of principally polarised abelian varieties is a consequence of the generalised Riemann hypothesis for number fields. I will discuss their proof. Some essential ingredients are the so-called Ax-Lindemann theorem, proved in the same article, the ``counting theorems'' by Pila and Wilkie, and a lower bound for the number of points in Galois orbits of CM-points by Tsimerman. Unfortunately I have not yet studied more recent work by Klingler, Ulmmo and Yafaev on the Ax-Lindemann conjecture for general Shimura varieties (arxiv, July 2013).
- Chao Zhang Leiden, PhD Defense: G-zips and Ekedahl-Oort Strata for Hodge Type Shimura Varieties
Intercity Number Theory SeminarNovember 8, Eindhoven. Location: LG 1.105.
Note that this is NOT in the old Main building, NOR in the new Metaforum building, but FURTHER DOWN THE TU/e CAMPUS. It on the first floor of building number 83 (Laplace Gebouw, LG) on the following map. The entrance is to the East, on the Laplace square. Please keep the additional walking time from the railway station into account when planning your travel.
- Bernd Sturmfels UC Berkeley / MPI Bonn, Tropicalization of Classical Moduli Spaces
Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry. Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry. Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli spaces of marked del Pezzo surfaces. Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G32. This is joint work with Qingchun Ren and Steven Sam.
- Rob Eggermont TU Eindhoven, Noetherianity for infinite-dimensional toric varieties
We consider a large class of monomial maps respecting the action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples where Noetherianity was recently proved or conjectured. In particular, our results imply Hillar-Sullivant's Independent Set Theorem and settle several finiteness conjectures due to Aschenbrenner, Martin del Campo, Hillar, and Sullivant. We introduce a matching monoid and show that it is Noetherian up to symmetry. Our approach is then to factorize an equivariant monomial map into two parts going through this monoid. The kernels of both parts are finitely generated up to symmetry: we find an explicit degree bound for the kernel of the first part, while for the second part, the finiteness follows from the Noetherianity of the matching monoid.
- Annette Werner Frankfurt, Faithful tropicalization of the Grassmannian of planes
We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We also determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. This is joint work with Maria Angelica Cueto and Mathias Häbich.
- June Huh University of Michigan / MPI Bonn, Geometric Chevalley-Warning conjecture
Geometric Chevalley-Warning conjecture predicts that a hypersurface of degree d ≤n in n-dimensional projective space is 1 modulo the class of the affine line in the Grothendieck ring of varieties. I will construct a quartic threefold with mild singularities which is not stably rational over the field of complex numbers, disproving the conjecture over any field of characteristic zero. Considering the same threefold over finite fields leads to interesting questions.
5th Belgian-Dutch Algebraic Geometry dayNovember 22, Nijmegen. Speakers: Nero Budur, René Schoof, and Stefan Schröer. See this page.
DIAMANT SymposiumNovember 28, Lunteren. Two days. See this page.
Wonder afternoonDecember 13, Delft. Aula, lecture room B.
- 13.00-13.45: Coffee/tea
- 13.45-15.30: Lectures on the work of Abel prize winner 2012 Pierre Deligne
- 13.45-14.30: Frans Oort, Pierre Deligne and his proof of the Weil conjectures, part I
- 14.45-15.30: Ben Moonen, Pierre Deligne and his proof of the Weil conjectures, part II
- 15.30-16.00: Coffee/tea
- 16.00-16.30: Award ceremony of the Stieltjes Prize for the best Mathematics PhD thesis in the Netherlands in 2012 to dr Sonja Cox (ETHZ), on behalf of the Foundation Compositio Mathematica:
- Laudation by the chair of the selection committee, Michel Dekking
- Award by rector of the TUD, Prof Ir Karel Luyben
- 16.30-17.00: Lecture by Stieltjes Prize winner
- 17.00-18.00: Reception (location: "Frans van Hasseltzaal")