Intercity Number Theory Seminar


Intercity Number Theory Seminar

14 February, Leiden. Room 412 of the Snellius (math institute). Followed by the oratie of Joost Batenburg.
Wouter Zomervrucht Leiden, Descent of genus 1 curves
A scheme X over a base S can be described on an open cover {UiS} by a compatible collection of schemes Xi over Ui. More generally one can formulate descent data for a scheme over S relative to an étale cover; but such a descent datum does not necessarily descend to a scheme X over S. A classical result states that if each Xi/Ui is a curve of genus g, for some fixed integer g unequal to 1, then the descent datum is effective. Michel Raynaud (1968) proved that this is no longer true in genus 1. In this talk, we provide a simpler example of non-effective descent of curves of genus 1.
David Holmes Leiden, Ranks of twists of elliptic curves and rational points on Kummer varieties
Ranks of Mordell-Weil groups of elliptic curves (and more generally of abelian varieties) over number fields are very mysterious - for example, it is not known whether there exists an elliptic curve over Q of rank at least 29. Given a positive integer r, we present a sufficient condition for an abelian variety A over a number field k to have a quadratic twist of rank at least r, in terms of the p-adic density of rational points on the Kummer variety of the r-fold product of A with itself, for various primes p. Unfortunately, at present it seems very hard to construct interesting examples where our condition holds. However, we can show that if the Brauer-Manin obstruction is the only obstruction to weak approximation on the Kummer variety of A^r, then our condition applies and we can show that A has a quadratic twist of rank at least r (even infinitely many such twists). Note that the sufficiency of the Brauer-Manin obstruction for explaining weak approximation on Kummer varieties is not currently known! This is joint work with René Pannekoek.
Michiel Kosters Leiden, Images of maps between curves over a large field
A field k is called large if every irreducible k-curve with a k-rational smooth point has infinitely many smooth k-points. Let k be a large perfect field. Let C, D be normal projective curves over k and let f: C D be a finite morphism. Suppose that the induced map fk: C(k) →D(k) is not surjective. In this talk we will show |D(k) \fk(C(k))|=|k|.

Intercity Number Theory Seminar

28 February, Groningen. Room 267 in Bernoulliborg.
Jan Steffen Müller Oldenburg, A p-adic BSD conjecture for modular abelian varieties
In 1986 Mazur, Tate and Teitelbaum came up with a p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes p of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces, and the algorithms that we used to gather this evidence.
Masato Kuwata Chuo University, Elliptic K3 surfaces with Mordell-Weil rank 18
The rank of the Mordell-Weil group of an elliptic K3 surface over a field of characteristic 0 can be as high as 18. I will construct many elliptic K3 surfaces defined over Q whose Mordell-Weil group over Q has rank 18. Then, I show the field of definition of the Mordell-Weil group and give an explicit description of its generators. This is joint work with Abhinav Kumar.
Moritz Minzlaff, Three and a half results on p-adic point counting and beyond
From lifting of curves to characteristic 0 and lattices in p-adic cohomology to fast computations of Frobenius morphisms, this talk covers the main results of my PhD thesis. It will explain how they fit into the bigger puzzle of computing zeta functions, to what extent they generalise the previously known state-of-the-art, and how these results might be one further step towards an efficient point counting algorithm for general curves. The talk will include software demonstrations and highlight related unsolved questions such as the existence of unliftable space curves.
Sietse Ringers Groningen, Deformation quantization of Poisson manifolds à la Kontsevich
In Maxim Kontsevich' celebrated paper from 1997 on deformation quantization of Poisson manifolds it is proved that on the ring of smooth functions of the manifold, there is a unique associative (but generally non-commutative) deformation of the pointwise product that is compatible with the Poisson structure. This talk will be an introduction to that paper: we introduce all relevant concepts, state the problem, and illustrate it using as an example the dual of a Lie algebra.

Intercity Number Theory Seminar

28 March, Eindhoven. At the occasion of the afscheidsrede of Arjeh Cohen. The first four lectures are in Auditorium 7.
Willem de Graaf Trento, Unit groups of integral abelian group rings
We describe an algorithm for obtaining generators of the unit group of the integral group ring of a finite abelian group. Subsequently we report on the results obtained using its implementation in Magma. In particular, we obtained the index of the group of Hoechsmann units in the full unit group, for all abelian groups of orders up to 110. (Joint work with Paolo Faccin, and Wilhelm Plesken).
Daan Krammer Warwick, The braid group of Zn
I will talk about my 2007 paper of the same title. Intuitively the braid group of Zn, B(Zn), looks like the fundamental group of the space of subgroups of R2 isomorphic to Zn, which is uncountable. But B(Zn) is given by a countable presentation. The braid group of Zn is to GL(2,Z) what the braid group Bn is to the symmetric group Sn. The main result is that B(Zn) is pseudo-Garside.
Jean-Pierre Tignol Louvain, Triality and compositions
Among Dynkin diagrams, D_4 is the only one that has a symmetry of order 3. As a result, split adjoint or simply connected groups of type D_4 have outer automorphisms of order 3, which are called trialitarian automorphisms. Such automorphisms were constructed by using octonion algebras by Elie Cartan in 1925. In this talk, I will report on a joint work with Chernousov, Elduque and Knus, where we establish a one-to-one correspondence between trialitarian automorphisms and a certain type of operation on quadratic spaces, called symmetric composition. This correspondence yields a classification of trialitarian automorphisms up to conjugation over an arbitrary field.
Stefan Maubach Bremen, The profinite polynomial endomorphisms and the profinite polynomial automorphism group
If Fq is a finite field, then the profinite polynomial endomorphisms PMAn(Fq) contains the set Fq[x1,...,xn]n. PMAn(Fq) is obtained by an inverse limit of certain finite groups induced by Fq[x1,...,xn]n. PMAn(Fq) is a group with addition, and a monoid with composition (just as Fq[x1,...,xn]n is). Inside PMAn(Fq) we find the profinite automorphism group, PGAn(Fq), which contains the group of polynomial automorphisms GAn(Fq)={ FFq[x1,...,xn]n ; F has an inverse in Fq[x1,...,xn]n w. respect to composition }. The motivation for making the groups PGAn(Fq) is that we essentially don't understand GAn(Fq). Well, and if you don't understand something, it sometimes makes sense to make it larger (in some not-completely-uncoordinated way): in this case, PGAn(Fq) can be seen as a completion of GAn(Fq) with respect to a topology. In this talk, I will mainly focus on trying to understand the structure (or ``local size'') of PGAn(Fq). (Joint work with Abdul Rauf.)
Arjeh Cohen Eindhoven, Wiskunde in het Web
Afscheidsrede, in de Blauwe Zaal.

Intercity Number Theory Seminar

25 April, VU Amsterdam. Room M655 of the W&N building.
Nuno Freitas Bayreuth, An asymptotic Fermat Last Theorem for totally real fields
Jarvis and Meekin showed that the strategy leading to the proof of Fermat's last theorem generalizes to proving that the classical Fermat equation xp + yp = zp has no non-trivial solutions over Q(√{2}). Two of the major obstacles to extending this result to other number fields are the modularity of the Frey curves and the existence of newforms in the spaces obtained after level lowering. In this talk, we will discuss how recent modularity theorems, along with applying level lowering twice, allow us to circumvent the aforementioned obstacles. In particular, will describe conditions on a totally real field K that, when satisfied, the asymptotic FLT holds. That is, there is a constant BK (depending only on K) such that, for primes p > BK, the equation xp + yp = zp only admits solutions (a,b,c) satisfying abc=0. Moreover, we will also see that the conditions are satisfied for most real quadratic fields.
Hang Liu VU Amsterdam, On the K_2 of certain families of curves
We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of the regulator. Working over a number field, we show that in some of those families the elements are integral. We determine when those curves are hyperelliptic, finding, in particular, that over any number field we have non-hyperelliptic curves of all composite genera g with g independent integral elements in the kernel of the tame symbol. We also give families of elliptic curves over real quadratic fields with two independent integral elements.
Lenny Taelman Leiden, Characteristic classes of genus one curves
One can attach to a curve of genus one over a field k various invariants ("characteristic classes") in the Galois cohomology groups Hi(k,Z/nZ). These classes vanish if the curve has a rational point. I will discuss the construction of some of these classes, and give some partial results towards the classification of such invariants.

6th Belgian-Dutch Algebraic Geometry Day

16 May, Leuven. Celestijnenlaan 200, Room A00.225. See also this page.
Olivier Benoist Strasbourg, Complete families of smooth space curves
In this talk, we will study complete families of smooth space curves, that is complete subvarieties of the Hilbert scheme of smooth curves in P3. On the one hand, we will construct non-trivial examples of such families. On the other hand, we obtain necessary conditions for a complete family of smooth polarized curves to induce a complete family of non-degenerate smooth space curves. Both results rely on the study of the strong semistability of certain vector bundles.
Frank-Olaf Schreyer Saarbrücken, Matrix factorizations and families of curves of genus 15
In this talk, I explain how certain matrix factorizations on cubic threefolds lead to families of curves of genus 15 and degree 16 in P4. The main result is that the moduli space M = { (C, L) | C a curve of genus 15, L a line bundle on C of degree 16 with 5 sections } is birational to a certain space of matrix factorizations of cubics, and that M is uniruled. My attempt to prove the unirationality of this space failed with this method. Instead one can interpret the findings as evidence for the conjecture that the basis of the maximal rational connected fibration of M has a three-dimensional base.
Carel Faber Utrecht, On the cohomology of the moduli spaces of pointed curves of genus three
Some time ago, we established that Galois representations not associated to Siegel modular forms occur in the cohomology of M3,n. This happens for the first time for n = 17. Recently, we have identified two 6-dimensional Galois representations in that case, using the recent work of Chenevier and his collaborators. Larger representations occur for larger n. How these representations are connected to Teichmüller modular forms is not clear yet. (Joint work with Bergström and van der Geer.)

RISC/Intercity Number Theory Seminar

23 May, CWI Amsterdam. Lectures are in room L120.
Serge Fehr CWI, Reconstructing a Shared Secret in the Presence of Faulty Shares - A Survey
Secret sharing is a fundamental primitive in cryptography. In this talk, I consider the problem of reconstructing a shared secret in the presence of faulty shares, with unconditional security. We require that any t shares give no information on the shared secret, and reconstruction is possible even if up to t out of the n shares are incorrect. The interesting setting is n/3 <= t < n/2, where reconstruction of a shared secret in the presence of faulty shares is possible, but only with an increase in the share size, and only if one admits a small failure probability. In this talk, I give an overview over the different known solutions. The first one, which is due to Rabin and Ben-Or (1989), suffers from relatively large shares of size Omega(k*n), where k is the security parameter. The second one, due to Cramer, Damgard and Fehr (2001), has close to optimal share size O(k + n) but is computationally inefficient. Finally, I will present a more recent solution by Cevallos, Fehr, Ostrovsky and Rabani (2012) that combines the advantages of the two: it has short shares of size O(k + n log(n)) and runs in polynomial time. Whether the linear dependency of the share size on n is necessary remains an open question.
Aner Moshe Ben Efraim Ben-Gurion, Multi-Linear Secret-Sharing Schemes
We study the power of multi-linear secret-sharing schemes. On the one hand, we prove that ideal multi-linear secret-sharing schemes, in which the secret is composed of p field elements, are more powerful than schemes in which the secret is composed of less than p field elements (for every prime p). On the other hand, we prove super-polynomial lower bounds on the share size in multi-linear secret-sharing schemes. Previously, such lower bounds were known only for linear schemes. This is joint work with Amos Beimel, Carles Padro, and Ilya Tyomkin.
Ronald Cramer CWI/Leiden, Optimal Algebraic Manipulation Detection Codes
Algebraic manipulation detection (AMD) codes, introduced at EUROCRYPT 2008, may, in some sense, be viewed as {\em keyless} combinatorial authentication codes that provide security in the presence of an {\em oblivious}, {\em algebraic} attacker. Its original applications included robust fuzzy extractors, secure message transmission and robust secret sharing. In recent years, however, a rather diverse array of additional applications in cryptography has emerged. In this paper we consider, for the first time, the natural regime of arbitrary positive constant error probability ε in combination with messages of unbounded binary length ell. Adapting a known bound to this regime, it follows that the binary length ρ of the tag satisfies ρ> logl+ Ωε(1). We shall call AMD codes meeting this lower bound {\em optimal}. Owing to our refinement of the mathematical perspective on AMD codes, which focuses on symmetries of codes, we propose novel constructive principles. Our main result is an efficient randomized construction of optimal AMD codes based on a careful adaptation of certain asymptotically good quasi-cyclic codes. Joint work with Carles Padró and Chaoping Xing.
Daniele Venturi Roma 1, Non-Malleable Codes and Applications
Non-malleable codes (Dziembowski et al., ICS 2010) encode a message in such a way that the decoded value corresponding to a modified codeword (w.r.t. some class of modifications) is either the original encoded value or a completely independent one. Compared to error correction/detection, non-malleability is a weaker guarantee that can be achieved for much richer classes of modifications. In this talk I will survey recent results on the construction of non-malleable codes, both in the computational and in the information theoretic setting. In the last part of the talk I will focus on applications relevant for the field of cryptography, mainly in the context of tamper resistance and chosen-ciphertext security.

Diamant Symposium

6 June, Arnhem. Two days, see this page.

RISC Seminar/Intercity Number Theory Seminar

19 September, KNAW Amsterdam. This day is part of the RISC Seminar on the Theory of Cryptography (September 18-19), which takes place in the Tinbergenzaal of the Trippenhuis. Registering is free, but required, also for those coming only on Friday (deadline September 12). Lunch will be provided on Friday. See the webpage for more information, including the program.

Intercity Seminar Number Theory, "Algebraic groups and number theory"

17 October, Utrecht. Buys Ballot Building, room 061 (note: due to construction works, you can no longer get to this building by walking through the Minnaert building, you have to walk around it and enter from Princetonplein).
Jan Draisma, Euclidean distance degrees of homogeneous varieties
The topic of this talk is motivated by the following optimisation problem: given a real algebraic variety X in a vector space V with inner product, and given a u in V, compute the point x in X nearest to u. This x will have the property that u-x is perpendicular to the tangent space to X at x. The ED degree of X counts the number of complex x with the latter property, for general u. I will discuss many examples of varieties X with a transitive algebraic group action for which this ED degree is known. (Joint work with Emil Horobet, Giorgio Ottaviani, Rekha Thomas, Bernd Sturmfels, and with Jasmijn Baaijens.)
Valentijn Karemaker, Hecke algebra isomorphisms and adelic points on algebraic groups
Let G denote an algebraic group over Q and K and L two number fields. Assume that there is a group isomorphism of points on G over the adeles of K and L, respectively. We establish conditions on the group G, related to the structure and the splitting field of its Borel groups, under which K and L have isomorphic adele rings. Under these conditions, if K or L is a Galois extension of Q and G(AK) and G(AL) are isomorphic topological groups, then K and L are isomorphic as fields.

As a corollary, we show that an isomorphism of Hecke algebras for GL(n) (for fixed n>1), which is an isometry in the L1-norm over two number fields K and L that are Galois over Q, implies that the fields K and L are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over Q. (joint work with Gunther Cornelissen)

Roland Lötscher, Essential dimension of gerbes
The essential dimension of an algebraic group G is roughly speaking the number of algebraically independent parameters needed to describe G-torsors. Torsors can be considered as objects of the classifying stack BG of G. The latter is a prototypical example of a gerbe. In this talk I will give an introduction into essential dimension of algebraic groups, gerbes and functors and then discuss results on the essential dimension of gerbes banded by groups of multipliactive type which I recently obtained.
Nikita Karpenko, A numerical invariant for linear representations of finite groups
We study the notion of essential dimension for a linear representation of a finite group. We relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type.

Intercity Seminar Number Theory

31 October, Leiden. Morning in room Snellius B1, afternoon in 312
Fabien Pazuki, Bad reduction of curves with CM jacobians
An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However the converse statement is false already in genus 2. In this joint work with Philipp Habegger, we prove the following result:

Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over Q with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere.

Marco Streng, Modular units and elliptic divisibility sequences
The modular curve Y1(N) parametrizes pairs (E,P), where E is an elliptic curve and P is a point of order N on E, up to isomorphism. A unit on the affine curve Y1(N) is a holomorphic function that is nowhere zero, and I will mention some applications of the group of units in the talk. The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining equations of Y1(N) for n<(N+3)/2. This result proves a conjecture of Maarten Derickx and Mark van Hoeij if N is a prime power.

Benjamin Matschke, Solving S-unit and Mordell equations via Shimura-Taniyama conjecture
We present two types of algorithms that practically solve S-unit and Mordell equations. The first type builds on Cremona's algorithm, and the second one combines explicit height bounds with enumeration algorithms. In particular we refine de Weger's sieve for S-unit equations and solve a large class of such. Additionally our new results on Mordell's equation implies an improved version of a theorem of Coates on the difference of coprime squares and cubes. Our results and algorithms crucially rely on a method of Faltings (Arakelov, Parsin, Szpiro) combined with the Shimura-Taniyama conjecture, and they do not use the theory of logarithmic forms. This is joint work with Rafael von Känel.
Martin Bright, The Brauer-Manin obstruction and reduction mod p
The Brauer-Manin obstruction plays an important role in the study of rational points on varieties. Indeed, for rational varieties (such as cubic surfaces) it is conjectured that this obstruction determines whether or not a variety has a rational point. I will give a brief introduction to the Brauer-Manin obstruction and then look at recent results relating it to the geometry of the variety at primes of bad reduction.

7th Belgian-Dutch Algebraic Geometry Day

14 November, UvA Amsterdam. Science Park 904. First lecture in C1.110, second and third in C0.110. See also this page.
Minhyong Kim Oxford, Non-abelian reciprocity laws and Diophantine geometry
We formulate a generalisation of Artin reciprocity to allow non-abelian coefficients. We outline applications to Diophantine geometry.
Mingmin Shen UvA, Multiplicative Chow-Künneth decompositions
This is joint work with Charles Vial. We will define what a multiplicative Chow-Künneth decomposition is and explain how this induces a bigrading on the Chow ring. I will discuss several examples of varieties with multiplicative CK decomposition.
Jakob Stix Frankfurt, Gorenstein orders and abelian varieties over finite fields
Thanks to a result of Deligne, the category of ordinary abelian varieties over a fixed finite field can be described in terms of finite free Z-modules equipped with a linear operator F (playing the role of Frobenius) satisfying certain axioms. Deligne's result is based on Serre-Tate canonical lifting. We prove a similar result for the full subcategory of all abelian varieties over the prime field supported on non-real Weil numbers, thereby obtaining a description of non-ordinary isogeny classes in Deligne's spirit. However, we must substitute lifting to characteristic zero by the commutative algebra of Gorenstein orders, because thanks to work of Chai, Conrad and Oort functorial lifting is not always possible. This is joint work with Tommaso Centeleghe (Heidelberg).

DIAMANT Symposium

28 November, TBA. The Symposium also includes November 27.