RISC Seminar/Intercity Number Theory Seminar: Cryptologic AlgorithmsFebruary 6, CWI Amsterdam. Euler Room (Z009, ground floor). For abstracts, see the program.
- Benjamin Wesolowski EPFL, Random Self-Reducibility of the Discrete Logarithm Problem in Genus 2
- Rob Granger EPFL, A tale of two QPAs
- Alina Dudeanu EPFL, Computing Denominators of Igusa Class Polynomials
- Anja Becker EPFL, A Sieve Algorithm Based on Overlattices
- Joppe Bos NXP Semiconductors, Sieving for Shortest Vectors in Ideal Lattices: a Practical Perspective
- Arjen Lenstra EPFL, Factorization Factories
Intercity Number Theory SeminarFebruary 20, Leiden. Snellius building, Room B03.
- Peter Bruin Universiteit Leiden, Computing modular Galois representations
In the quest to understand the absolute Galois group of the rational numbers, the two-dimensional representations of this group (and in particular the modular ones) have been the subject of intensive research. I will try to motivate the study of such representations, explain some aspects of the theoretical work of Edixhoven, Couveignes and collaborators (including myself) on computing such representations, and report on the practical computations that have been done so far by various people, with an eye towards the future.
- Elisa Universiteit Leiden, A Gross-Zagier formula for dimension 3
Gross and Zagier (1985) prove that the product of the differences of the j-invariant of some elliptic curves, with complex multiplication by certain orders, is an integer, and compute its factorization. We will see the generalization of this result for dimension 2 (genus 2 curves/abelian surfaces) and we will discuss recent progress in the dimension 3 case.
- Marta Pieropan Leibniz Universität Hannover, On the distribution of rational points on Fano varieties over number fields
After introducing the natural questions around rational points on varieties over number fields, we focus on Manin's conjecture about the distribution of rational points on Fano varieties. This conjecture predicts an asymptotic formula for the number of points of bounded anticanonical height that is intimately related to the geometry of the variety. We see how Cox rings and universal torsors can be employed to produce parameterizations that are used to verify the conjecture via lattice point counting in some specific examples.
- Peter Stevenhagen Universiteit Leiden, Adelic points of elliptic curves
Angelakis proves in his thesis (2015) that there are many different imaginary quadratic fields that all have the "same" absolute abelian Galois group, i.e., their absolute abelian Galois group is isomorphic to some universal profinite group. In this talk, we show how his techniques can be used to prove similar results for the adelic point groups associated to elliptic curves.
Intercity Number Theory SeminarMarch 13, Groningen. First two lectures in room 165, Bernoulliborg, last two lectures in room 105.
- Ute Spreckels Oldenburg, On the primality of the order of CM abelian varieties over finite prime fields
Let A be a CM abelian variety of dimension g defined over a number field F. For any rational prime l which is unramified in the CM-field of A let A[l] be the group of l-torsion points over an algebraic closure of F. The Galois group of F(A[l])/F can be embedded into a maximal torus of the general symplectic group GSp2g(Fl). In 2014, Weng described how the splitting behavior of l in the CM-field determines this maximal torus. We refer to these techniques and provide an explicit description of Gal(F(A[l])/F) as a matrix group in case that the image of Gal(F(A[l])/F) equals the maximal torus. By counting matrices with eigenvalue 1 we are able to compute the density of primes p such that the order of A mod p is divisible by l. This leads to a heuristic formula for the number of primes such that the reduction of the Jacobian of a fixed algebraic curve of genus g has prime order. Our result generalizes conjectures of Koblitz for elliptic curves and of Weng for abelian surfaces.
- Paul Helminck Groningen, Variations on some constructions by J-F. Mestre
Between 1979 and 1992 Mestre presented various constructions of curves C with the property that Mor(C,E) modulo constant morphisms (for E some elliptic curve) is a fairly large group. This has applications on p-ranks of class groups of quadratic fields, Mordell-Weil ranks of twists of a given elliptic curve, and more. We recall some of his ideas and applications, and discuss some variations. This is part of my master's thesis project completed in summer 2014.
- Remke Kloosterman HU Berlin, An upper bound for the Mordell-Weil rank after Galois base change and a Chevalley-Weil formula for certain surfaces
Recently, Pal obtained an upper bound for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. We will give a geometric proof for this bound. More specifically, let X be a complex hypersurface in a Pn-bundle over a curve C. Let C'→C be a Galois cover with group G and let X' be the fiberproduct X ⨉C C'. Then we describe the C[G]-structure of Hp,q(X') provided that X' is either smooth or n=3 and X' has at most ADE singularities. We then show that this description is sufficient to obtain Pal's upper bound.
- Jaap Top Groningen, Lucas-Lehmer revisited
This talk is inspired by the recent bachelor's project of Joep Hamersma (Groningen, January 2015). In 2005 Dick Gross published a paper in the Journal of Number Theory, describing a variant of the classical Lucas-Lehmer test for determining primality of Mersenne numbers -1+2n, n odd. Gross uses the group of rational points on a specific elliptic curve, whereas the original test uses the group of units in Z[√3]. Following Gross, various people (Gurevich & Kunyavskii, Kirschmer & Mertens, Silverberg, and now Hamersma in his bachelor's project) discuss variations of his work. We present some of this here.
Seminar Aachen-Köln-Lille-Siegen (AKLS) and Intercity Number Theory SeminarApril 24, Utrecht. BBG (Buys Ballot Gebouw), room 065 (ground floor)
- Roelof Bruggeman Utrecht, Holomorphic automorphic forms and cohomology
The relation between holomorphic cusp forms and period polynomials has been studied for many years. Most of the interest has been directed to the case of entire automorphic forms, especially to cusp forms. Eichler, 1957, explicitly included cocycles attached to other automorphic forms in his definition. Recently YoungJu Choie, Nikos Diamantis and I studied cocycles attached to holomorphic automorphic forms on the upper half-plane with arbitrary complex weight, without a restriction by growth conditions.
I'll discuss a number of the ideas from this work. In the talk I'll stick to the modular group and real weights.
- Carel Faber Utrecht, On Teichmüller modular forms
Vector valued Siegel modular forms may be viewed as sections on a toroidal compactification of Ag of the bundles obtained by applying a Schur functor for GL(g) to the Hodge bundle. Similarly, Teichmüller modular forms are sections on Mg or its Deligne-Mumford compactification of the pullbacks of those bundles via the Torelli morphism. I will first recall several results of Ichikawa on scalar valued Teichmüller modular forms, of genus three especially. Then I will report how joint work with Bergström and Van der Geer indicates the existence of many vector valued Teichmüller modular forms of genus three.
- Miranda Cheng Amsterdam, UvA, Optimal mock Jacobi forms
In this talk I will consider the space of optimal mock Jacobi forms of weight one. I define special elements — symmetric optimal mock Jacobi forms— and prove there are 39 of them, in one to one correspondence to the 39 genus zero non-Fricke Atkin-Lehner groups. They generate the space of optimal mock Jacobi forms in a natural way; 23 of them are singled out by a positivity condition and correspond to the 23 cases of umbral moonshine. Moreover, all Ramanujan’s mock theta functions arise from symmetric optimal mock Jacobi forms. Based on work with J. Duncan.
Intercity Number Theory SeminarMay 8, UvA and VU Amsterdam. This seminar will be held at the Vrije Universiteit. Before lunch in room WN-P647 (W&N-building), after lunch in HG-08A05 (main building).
- Lenny Taelman UvA, Complex multiplication and K3 surfaces over finite fields
In this talk I will attempt to describe the set of zeta functions of K3 surfaces over a given finite field. For Abelian varieties this question is completely answered by the Honda-Tate theorem. A significant part of the talk will be devoted to the theory of complex multiplication of complex projective K3 surfaces, which plays an important role in the construction of K3 surfaces over finite fields.
- Abhijit Laskar VU University Amsterdam, On local monodromy filtrations attached to motives
In the first part we formulate and show some special cases of an analogue of the weight monodromy conjecture for motives defined over number fields. In the second part, we give applications to L-functions of motives and the Langlands program.
- Ada Boralevi TU Eindhoven, Linear spaces of matrices of constant rank
Given a complex vector space V of dimension n, one can look at d-dimensional linear subspaces A in Λ2 V, whose (nonzero) elements have constant rank r. The natural interpretation of A as a vector bundle map yields restrictions on the values that r, n, and d can attain. In this talk we will deal with the case r=n-2. I will mention a classification result for the 3-dimensional case, and then try to convince you that 5-dimensional examples cannot exist. Then we will concentrate on the 4-dimensional case (the most interesting one), for which I will give a method to construct new examples. The technique involves instanton bundles of charge 2 and 4 and the derived category of P3, and provides an explanation for what once used to be the only known example, by Westwick. This talk summarizes joint works (some in progress) with J. Buczynski, D. Faenzi, G. Kapustka, P.Lella, and E.Mezzetti.
- Frits Beukers Utrecht, Finite hypergeometric functions
By the end of the 1980's J.Greene introduced complex valued functions on finite fields which are the analogues of the classical analytic hypergeometric functions defined by Euler and Gauss. Beside having an uncanny similarity with their analytic analogues, they can also be used to count points on certain algebraic varieties over finite fields.
Intercity Number Theory SeminarMay 27, Eindhoven. The first two talks are in Auditorium 12. The PhD defense of Rob Eggermont is in Auditorium 4.
- Giorgio Ottaviani Firenze, The Waring decomposition of a polynomial and its uniqueness
A Waring decomposition of a homogeneous polynomial (over complex numbers) is a minimal expression as a sum of powers of linear forms. The Alexander-Hirschowitz Theorem settles the number of summands needed for a general polynomial, while for any polynomial the description is still open. In the talk we give a overview on this topic and then we address the uniqueness of the Waring decomposition.
- Jochen Kuttler Alberta, Modules of differentials for Lie algebras
In this talk, I will attempt to introduce and discuss modules of differentials for Lie algebras modelled after the corresponding notion for rings. There even might be some motivation for doing so. This is joint work with Arturo Pianzola.
- Rob Eggermont Eindhoven, PhD defense
DIAMANT SymposiumMay 29, TBA. This is a two-day event: May 28 and 29.
North German Algebraic Geometry Seminar (NoGAGS) and 8th Belgian-Dutch Algebraic Geometry Day(s)June 12, Nijmegen. The Program. This is a two-day event: June 11 and 12.
Intercity Number Theory SeminarSeptember 25, Leiden. Snellius building. Morning: B2, afternoon: 407/409.
- Christopher Frei TU Graz, The Hasse norm principle for abelian extensions
Let L/K be a normal extension of number fields. The Hasse norm principle is a local-global principle for norms. It is satisfied if any element x of K is a norm from L whenever it is a norm locally at every place. For any fixed abelian Galois group G, we investigate the density of G-extensions violating the Hasse norm principle, when G-extensions are counted in order of their discriminant. This is joint work with Dan Loughran and Rachel Newton.
- Efthymios Sofos Universiteit Leiden, On the fibration method in analytic number theory
The fibration method, very broadly, refers to the situation where one establishes a required property for a variety by fibering it into simpler varieties known to satisfy the said property. Its use in arithmetic geometry has been made chiefly in the context of the Brauer-Manin obstruction, for example in the works of Swinnerton-Dyer and Colliot-Thélène and most recently in that of Harpaz and Wittenberg.
In this talk we will show that this method can be adopted in analytic number theory to tackle a range of important problems. Our first result regards obtaining correct lower bounds for Manin's conjecture for smooth del Pezzo surfaces of degree >1 which have a conic bundle structure. One should notice that the conjecture has not been established for these varieties and there were previously only weak upper bounds for degree 3 and no noteworthy results for degree 2. The second problem we shall refer to is known as the Sarnak saturation number: Given a variety V defined over the rationals, it is defined as the least integer r(V) such that rational points with at most r prime factors are Zariski dense. We shall show that varieties fibered into unirational varieties have finite saturation. This covers many new low dimensional cases unassailable by other analytic methods hitherto used in this problem.
- Alex Bartel Warwick University, Heuristics for Arakelov class groups
The Cohen-Lenstra heuristics, formulated in the early 80s, are a probabilistic model for the behaviour of ideal class groups in families. For example, they postulate that, for an odd prime p, the "probability" that the p-class group of an imaginary quadratic field is isomorphic to a given abelian p-group A is inverse proportional to #Aut(A). This is a very natural kind of behaviour for "random" algebraic objects. Cohen-Lenstra-Martinet also proposed a model for class groups of more general number fields, but the probability weights in this generality, while agreeing well with experiments, look rather mysterious. I will explain how to view the general Cohen-Lenstra-Martinet heuristic as an instance of the rule that random algebraic objects should be distributed inverse proportionally to the number of automorphisms, by passing to Arakelov class groups. The main difficulty that one needs to overcome is that Arakelov class groups typically have infinitely many automorphisms. This difficulty is resolved through a theory of commensurability of certain algebraic objects. This is an ongoing project with Hendrik W. Lenstra Jr.
- Maarten Derickx Universiteit Leiden, Galois closures over integrally closed domains
From a finite separable field extension L/K one can construct a Galois closure N, which is the smallest Galois extension of K containing L. The automorphism group of N/K is then called the Galois group of L/K. In his PhD thesis Owen Biesel defined the notion of G-closure for G ⊂Sn a subgroup, R a commutative ring with unity and A a commutative R-algebra that is locally free of rank n as R-module. He showed that if R = L and A = K, then a G-closure exists if and only if G contains a conjugate of the Galois group of L/K. If one furthermore takes G equal to the Galois group, then the G-closure of L/K is actually isomorphic to the Galois closure of L/K. If one leaves the world of field extensions, one can ask if it is still the case whether the minimal groups for which a G-closure exists are all conjugate. Owen Biesel already proved that this is true if A/R is étale . The main result discussed in this talk is that this is also the case if R is an integrally closed domain with field of fractions K and A is such that for all maximal ideals m ⊂A ⊗R K one has K ⊂(A ⊗R K)/m is a separable field extension, in particular if R is integrally closed and of characteristic 0 there are no conditions on A. Examples showing that the integrally closed and seperability conditions cannot be removed will be given.
Intercity Number Theory SeminarOctober 16, Utrecht. Rode zaal in Ruppertgebouw, Leuvenlaan 21.
- Martijn Kool Utrecht, Donaldson-Thomas invariants of local elliptic surfaces
The Gromov-Witten invariants of a Calabi-Yau 3-fold X are a virtual count of curves on X. We consider the case X is the total space of the canonical line bundle over an algebraic surface S, where S is elliptically fibred. Instead of GW invariants, we consider the closely related Donaldson-Thomas invariants, which are easier to calculate. Using a combination of motivic methods and torus localization, we express the generating function of all invariants in terms of a combinatorial object called the topological vertex. This leads to a closed formula. Joint work with J. Bryan and B. Young.
- Gert Heckman Nijmegen, Moduli of real genus 3 curves
We discuss the PhD work of my former student Sander Rieken, about reality questions for the Kondo period map from the moduli space of genus 3 curves to a certain ball quotient of dimension 6 over Gauss integers. The analysis of the maximal real component (quartic curves with 4 ovals) is particularly nice, and leads to an odd presentation of the Weyl group of type E7, analogous to a presentation of the bimonster group, due to Ivanov, Norton, Conway and Simons.
- Cor Kraaikamp Delft, Natural extensions and Nakada’s alpha-expansions
In 1981, Hitoshi Nakda introduced a new class of continued fraction expansions, the so-called alpha-expansions, where the parameters alpha runs between 0 and 1. At the time these expansions attracted a lot of attention, and for alpha's between 1/2 and 1 their properties are extremely well understood. Recently, work by Laura Luzzi and Stefano Marmi showed that these alpha-expansions have a remarkable behavior when alpha is between 0 and 1/2. Since their paper, Nakada's continued fractions again are widely investigated. Using simple means we will try to explain why these expansions show such a remarkable behavior.
- Frits Beukers Utrecht, A geodesic continued fraction using LLL
Continued fractions in higher dimensions, such as Jacobi-Perron, suffer from the problem that they give approximations which are often far from the best possible ones. In the 1990's J.Lagarias proposed an algorithm which does not have this problem and which is based on Minkowski reduction of quadratic forms. However, this does not scale well to higher dimension than 5 or 6. We discuss the theory and practice of an algorithm based on LLL which does scale to higher dimension.
ANA2015 (Automatic Sequences, Number Theory, Aperiodic Order)October 30, Delft. Room Pi in the EWI building. This is part of a 3-day event: please register.
Intercity Number Theory SeminarNovember 6, UvA and VU Amsterdam. This seminar will be held at the Vrije Universiteit: the morning talk in HG-08A33 (main building) and the afternoon talks in WN-M607 (mathematics and physics building).
- Arno Kret UvA, Galois representations for the general symplectic group
We explain how to construct Galois representations attached to automorphic representations of the general symplectic group over a totally real number field, under local simplifying hypotheses. This is joint work with Sug Woo Shin.
- Rob de Jeu VU, Numerical experiments for p-adic L-functions
p-adic L-functions are obtained by interpolation of classical L-functions at non-positive integers, but their behaviour at other points is much less understood. We discuss some background, and computations of (many) examples concerning zeroes at positive integers, as well as the multiplicities of zeroes of the associated Iwasawa power series. This is joint work with Xavier-François Roblot.
- Lance Gurney UvA, Frobenius lifts and minimal models of CM elliptic curves
Let K be an imaginary quadratic field. We describe a certain structure possessed by the moduli stack M of elliptic curves with CM by OK. The crucial observation being that, for each prime p of K, M admits an automorphism lifting the Np-power Frobenius. Exploiting this structure (or guided by it) we are lead to several new results in the theory of CM elliptic curves. One of which is that any CM elliptic of 'Shimura type' defined over the Hilbert class field of K admits a global minimal model (this was originally proven by Gross under the additional assumption that disc(K/Q) be prime). If time permits we will also explain how this structure leads to a global theory of 'canonical lifts' for elliptic curves with CM by OK.
- Jan Draisma TU/e & VU, Orthogonally decomposable tensors as semisimple algebras
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, /higher-order/ tensors typically do not admit such an orthogonal decomposition. We prove that those /orthogonally decomposable tensors/ form a real-algebraic variety defined by low-degree polynomials. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. To prove our main theorem, we associate an algebra to a tensor; show that if the tensor is orthogonally decomposable, then the algebra satisfies certain polynomial identities; and finally use classifications of simple associative algebras and compact Lie algebras to prove that these identities characterise orthogonally decomposable tensors. [This talk is based on joint work in progress with Ada Boralevi (TU/e), Emil Horobet (TU/e), and Elina Robeva (UC Berkeley).]
DIAMANT SymposiumNovember 27, Lunteren. This is a two-day event: November 26-27
9th Belgian-Dutch Algebraic Geometry DayDecember 11, Leiden. Snellius Building, room 407-409, of the Maths Dept. in Leiden. Organisers: Ben Moonen, Johannes Nicaise, Lenny Taelman, Bas Edixhoven.
- David Holmes Leiden, Models of degenerating jacobians
The problem of how to construct a good model for a degenerating family of curves is well understood - for example, after an alteration of the base, any such family admits a stable model. A degenerating family of curves yields a degenerating family of jacobians, but the problem of constructing a good model for the family of jacobians seems much harder. Over a 1-dimensional base the problem was essentially solved by André Néron in the 1960s, when he constructed the `Néron model' of the family, a smooth group scheme satisfying a good universal property.
Over bases of dimension greater than 1 there are a number of different approaches to constructing models of jacobians with various properties, due to many people including Caporaso, Chiodo and the speaker. The main aim of the talk will be to describe what some of these models look like in a few simple examples. If time allows, we will briefly discuss how models of jacobians arise in attempting to construct Gromov-Witten invariants for Artin stacks.
- Clément Dupont MPIM Bonn, Motives of bi-arrangements
Bi-arrangements of hyperplanes (or more generally of hypersurfaces) are geometric objects which give rise to interesting relative cohomology groups. In particular, they produce interesting examples of mixed Hodge structures of Tate type. In this talk we will explain how to compute these cohomology groups by using geometric and combinatorial tools. This generalizes the classical theory of arrangements of hyperplanes, due to Arnol’d, Brieskorn and Orlik-Solomon.
- Misha Verbitsky ULB (Brussels), HSE (Moscow), Transcendental Hodge algebra
Let M be a projective manifold. The transcendental Hodge lattice of weight p is the smallest rational Hodge substructure in Hp(M) containing Hp,0(M). The transcendental Hodge lattice is a birational invariant of M. I will prove that the direct sum of all transcendental Hodge lattices for M is an algebra, and compute it (using Yu. Zarhin's theorem) explicitly for all hyperkahler manifolds. This result has many geometric consequences. In particular, I will show that dimension d of a family of hyperkahler manifolds containing a symplectic torus of dimension m satisfies m ≤ 2d/2-1.
Geometry in Winter, A memorial symposium for Prof. dr. A.H.J.M. van de VenDecember 18, Leiden. Snellius building, room 312. For more information, including abstracts, see website.
- Ekaterina Amerik Univ. de Paris Sud, Orsay, Some applications of the cone conjecture for hyperkaehler manifolds
- Arnaud Beauville Univ. de Nice, Recent developments on stable rationality
- Fabrizio Catanese Univ. Bayreuth, Interesting Variations of Hodge Structures and related algebraic surfaces
- Klaus Hulek Univ. Hannover, Intermediate Jacobians of cubic threefolds and their degenerations
- Chris Peters Univ. Grenoble I and T.U. Eindhoven, Vector bundles and surfaces, a lifelong fascination