## 2012

### Intercity Number Theory Seminar

16 March, CWI Amsterdam. room L016- 12:00–12:45
- Hendrik Lenstra, The transfer map and determinants
In 1970, Cartier defined a map that generalizes the transfer map from group theory. We review his definition, and prove that in certain situations his map coincides with a determinant map. This result has recently been applied to local fields, and there is a second application to groups generated by pseudoreflections.
- 13:45–14:30
- Hendrik Lenstra, Groups generated by pseudoreflections
In the lecture we shall define the notion of a pseudoreflection with respect to a finite collection of one-dimensional subspaces of a finite-dimensional vector space over a field. The group generated by these pseudoreflections can be seen as an algebraic group. Cartier's transfer map plays a role in its study.
- 14:45–15:30
- Maarten van Pruijssen, Multiplicity free systems
A multiplicity free system is a triple (
*G*,*K*,*F*) where*G*is a simple complex reductive group,*K*a connected reductive subgroup of*G*and*F*a face of the positive Weyl chamber of*K*such that the following holds: any irreducible*K*-representation of highest weight μ in*F*occurs at most once in the restriction to*K*of every irreducible*G*-representation. Any Gelfand pair (*G*,*K*) together with the face*F*={0} gives an example. In this talk we give a classification of the multiplicity free systems (*G*,*K*,*F*) where (*G*,*K*) is of rank one and we discuss (very briefly) an application in the branche special functions. - 15:45–16:30
- Jan-Hendrik Evertse, Effective results for unit equations over finitely generated domains
Let
*A*=**Z**[*x*_{1},...,*x*_{r}] be an integral domain which is finitely generated over**Z**. We allow*A*to contain both algebraic and transcendental elements. Denote by*A** the unit group of*A*. We deal with so-called unit equations in two unknowns(1)

*ax*+*by*=*c*in*x*,*y*from*A**where

*a*,*b*,*c*are non-zero elements of*A*. First Siegel, Mahler and Parry in special cases, and finally Lang in 1960 for arbitrary finitely generated domains, proved that Eq. (1) has only finitely many solutions. Their proofs are ineffective in the sense that they do not provide a method to determine the solutions of (1) in principle. In 1979, Györy gave an effective proof of the Siegel-...-Lang theorem, but only in the special case that*A*is contained in the algebraic closure of**Q**. His proof uses estimates of Baker and Coates on lower bounds for linear forms in logarithms. In 1984, Györy extended his effective proof to a restricted class of integral domains which contain also transcendental elements.Recently, Kálmán Gyöy and I gave a general effective finiteness proof for Eq. (1) for arbitrary finitely generated domains which are explicitly given in a well-defined sense. In my lecture I will present our new results.

### Intercity number theory seminar

13 April, Eindhoven. Auditorium building, room 1.This meeting is the day after the NMC 2012 featuring the Beeger lecturer. There will be tea at 15:20.

- 11:00–11:50
- Yuri Bilu, Effective Diophantine analysis on modular curves
I will speak on two effective methods in Diophantine analysis: Baker's method and Runge's method, with a special emphasize to modular curves. The talk is partially based on a joint work with Pierre Parent and Marusia Rebolledo.
- 12:00–12:50
- Mohamed Barakat, Ext-Computability of the category coherent sheaves on a projective scheme.
In this talk I will present an approach to the computability of the category of coherent sheaves on a projective scheme which implies the computability of long exact sequences, several spectral sequences, modules of twisted global sections, higher sheaf cohomology and Ext groups. This approach uses some new categorical tools that came out of an abstract computer implementation of the relevant algorithms.
- 13:30–14:20
- Viktor Levandovskyy, Stratification of affine spaces, arising from Bernstein-Sato polynomials
Over the field of complex numbers, the famous functional equation of J. Bernstein involves an important object, related to the given hypersurface at a given point. This is a monic univariate polynomial, called the local Bernstein-Sato polynomial. There are remarkable connections between singularities and the local/global Bernstein-Sato polynomials of a hypersurface. An algorithm to compute the finite stratification of an affine space into the strata, such that the local Bernstein-Sato polynomial of a given hypersurface is constant on each stratum. Moveover, to each stratum one can naturally attach a holonomic module over Weyl algebra, and the sum of these gives a well-known module, directly related to the hypersurface. Following Budur, Mustata, and Saito, this approach can be generalized to the case of an affine variety.
- 14:30–15:20
- Emil Horobet Universitatea Babes-Bolyai, Basic algebra of a skew group algebra
We give an algorithm for the computation of the basic algebra Morita equivalent to a skew group algebra of a path algebra by obtaining formulas for the number of vertices and arrows of the new quiver.
- 15:40–16:30
- Daniel Robertz, Implicitization of Parametrized Families of Analytic Functions
The correspondence between solution sets of systems of algebraic equations and radical ideals of the affine coordinate ring is fundamental for algebraic geometry. This talk discusses aspects of an analogous correspondence between systems of polynomial differential equations and their analytic solutions. Implicitization problems for certain families of analytic functions are approached in different generality. While the linear case is understood to a large extent, the non-linear case requires new algorithmic methods, e.g., the use of differential inequations, as proposed by J. M. Thomas in the 1930s.

### RISC / Intercity number theory seminar

27 April, CWI Amsterdam. RoomL016. This day is focused on*fully homomorphic encryption*.

- 12:00–12:45
- Vadim Lyubashevsky, Ideal Lattices and FHE, part 1
- 13:45–14:30
- Vadim Lyubashevsky, Ideal Lattices and FHE, part 2
In the first part of the talk, I will cover the Ring-LWE problem (Learning with Error over Rings), its hardness, the equivalence of its search and decision versions, and explain what little is known about the hardness of problems in ideal lattices. In the second part, I will present two (similar) constructions of encryption schemes based on Ring-LWE. Then I will present the NTRU cryptosystem and sketch how it can be easily modified to become a "somewhat-homomorphic" encryption scheme that supports several additions and multiplications, and then finally present the "bootstrapping" technique that converts "somewhat-homomorphic" schemes that meet certain requirements into fully-homomorphic ones. (NB: the NTRU-based scheme that I will present does not meet these requirements, but can be modified to meet them using recent techniques.)
- 14:45–15:30
- Erwin Dassen, Brakerski's scale invariant homomorphic scheme
In a recent pre-print, Brakerski introduced what he called a "scale invariant" homomorphic scheme. The name comes from the fact that, contrary to other schemes, its homomorphic properties depend only on the modulus-to-noise ratio. Furthermore, while in previous works noise would grow quadratically with each multiplication, here it grows linearly. The aim of the talk is to describe this scheme in detail.
- 15:45–16:30
- Alice Silverberg, Some Remarks on Lattice-based Fully Homomorphic Encryption
The talk will include an overview of some lattice-based Fully Homomorphic Encryption schemes such as those proposed by Smart-Vercauteren and Gentry-Halevi. We will also discuss balancing cryptographic security with ease of decryption, for lattice-based FHE schemes.

### Intercity Number Theory Seminar

11 May, Leiden. Room 407- 11:45–12:30
- Michiel Kosters, Futile algebras
Let
*R*be a commutative ring and let*A*be an*R*-algebra. Then*A*is called*R*-futile if it has only finitely many*R*-subalgebras. The problem of classifying futile*R*-algebras for a fixed ring*R*has been studied before by other people, and this turns out to be more complicated than one might expect. In this talk we will discuss this problem in the case where*R*is a finite ring, a field or**Z**. - 13:30–14:15
- Andrea Siviero Leiden, Equidistribution of the Galois module structure of rings of integers with given local behavior
Let
*K*be a number field and let*G*be a finite abelian group. A couple of years ago Melanie Wood studied the probabilities of various splitting types of a fixed prime in a random*G*-extension of*K*. When the number fields are counted by conductor, she proved that the probabilities are as predicted by a heuristic and independent at distinct primes. In the same period Adebisi Agboola studied the asymptotic behavior of the number of tamely ramified*G*-extensions of*K*with ring of integers of fixed realizable class as a Galois module, proving that an equidistribution result exists when the extensions are counted by the absolute norm of the ramified primes of*K*. One may wonder if the two distributions of Wood and Agboola are independent. In this talk I address equidistribution of realizable classes for extensions with a totally split local behavior at one fixed prime. - 14:30–15:15
- Ronald van Luijk, Density of rational points on Del Pezzo surfaces of degree one.
The Segre-Manin Theorem implies that if a Del Pezzo surface of degree at least three, defined over
**Q**, has a rational point, then the rational points are Zariski dense on the surface. A result of Manin yields the same for degree two, as long as the initial point avoids a specific subset. Similar results for Del Pezzo surfaces of degree one are meager: they either depend on conjectures, or they are restricted to small families of surfaces. We will give a sufficient, explicitly computable criterion for the rational points on a general Del Pezzo surface of degree one to be dense. This is joint work with Cecilia Salgado. No prior knowledge about Del Pezzo surfaces will be assumed. - 15:30–16:15
- Alberto Gioia, On a Galois closure for rings
Given a finite separable field extension
*L*/*K*there exists a smallest Galois extension of*K*containing*L*, the Galois closure of*L*/*K*. Bhargava and Satriano generalized this construction to commutative algebras of finite rank over an arbitrary base ring. We will present their construction and see some properties.

### Intercity Number Theory Seminar

25 May, Groningen. Bernoulliborg, room 267- 11:45–12:45
- Anneroos Everts Groningen, From finite automata to power series and back again
Christol's theorem links algebra in an unexpected way with a concept from computer sciences: a power series over a finite field is algebraic if and only if its coefficients are generated by a finite automaton. We examined the proof of Christol's theorem to find answers to the following two questions: Given a finite automaton with m states, what can we say about the algebraic degree of the corresponding power series? Conversely, given an algebraic power series of algebraic degree d and bounded coefficients, can we find a bound on the number of states of an automaton that generates the power series?
In this talk I will explain Christol's theorem and the concept of finite automata, and give answers to the questions above.

- 13:30–14:30
- Paul Helminck Groningen, Tropical elliptic curves and
*j*-invariantsAny elliptic curve over the field of complex Puiseux series has a "tropicalization": an associated tropical curve. We construct, for any elliptic curve over the field of Puiseux series that has a*j*-invariant with negative valuation, a model such that its tropification is a tropical elliptic curve. Moreover, we show that the tropical*j*-invariant of this tropical curve is minus the valuation of the*j*-invariant. Special cases of this were already proven by Markwig, and similar results were obtained quite recently by M. Baker and by Sturmfels and Chan. - 14:45–15:45
- Wilke Trei Carl von Ossietzky University Oldenburg, Elliptic Curve Arithmetic on Vectorized Hardware Platforms
Parallelization of computational intensive algorithms has always been an important task in computational number theory. Modern hardware requires a high on-chip parallelization for gaining maximum possible performance. We discuss several mathematical concepts to implement modular arithmetic with a focus on elliptic curve scalar multiplication on graphic cards and present a new performance record for Lenstra's elliptic curve factoring algorithm on an ordinary personal computer. The resulting implementation is of high cryptographic interest, for instance it can easily be modified to speed up intermediate factorization in the number field sieve algorithm.
- 16:00–17:00
- Arthemy Kiselev Groningen, The deformation quantisation problem for multiplicative structures on noncommutative jet spaces.
We outline the basic notions and concepts from the differential calculus --up to the Schouten bracket-- on a class of noncommutative jet spaces, and then we pose the deformation quantisation problems for the non-associative but commutative multiplications in the two spaces of differential functions (i.e., the noncommutative fields) and integral functionals (i.e, the Hamiltonians), aiming to restore the associative but not commutative star-products. During the entire talk, the constructions and reasonings will appeal to the profound properties of a pair of pants borrowed from the topological closed string theory.

### Intercity Number Theory Seminar

8 June, Utrecht. The lectures are in the Buys Ballot building, the first one in room 001 and the others in room 161- 11:15–12:15
- Peter Stevenhagen, Galois groups as arithmetic invariants
By the work of Neukirch, Uchida and others, we know that number fields
*K*are completely characterized by their absolute Galois group*G*_{K}: if*G*_{K}and*G*_{L}are isomorphic as topological groups, then*K*and*L*are isomorphic number fields. Similar statements hold for the maximal pro-solvable quotient of*G*_{K}, but NOT for the maximal abelian quotient*A*_{K}of*G*_{K}. We focus on this maximal abelian quotient, which admits an explicit class field theoretical description, and show that there are very many imaginary quadratic fields having the same "minimal" absolute abelian Galois group*A*_{K}. This is joint work with my student Athanasios Angelakis. - 13:00–13:30
- Frans Oort, Lifting Galois covers of algebraic curves.
We discuss the question: Does a pair (
*C*,*H*) of an algebraic curve*C*over a field of positive characteristic and a subgroup*H*of*Aut*(*C*) admit a lift to characteristic zero? -- We give motivating (counter)examples and general theory. -- Formulate a conjecture. -- Discuss partial results and possible approaches. This talk serves as an introduction to the talk by Andrew Obus. [notes.pdf] - 13:45–14:45
- Andrew Obus, Proof of the Oort Conjecture
Let
*G*be a cyclic group acting on a smooth, proper curve*X*in characteristic*p*. The Oort conjecture claims that*X*, along with its*G*-action, should lift to characteristic zero. I will discuss a joint result with Wewers, stating that the conjecture is true subject to a certain condition on the higher ramification filtrations of the inertia groups of*G*at the various points of*X*. Pop has recently proved the entire conjecture, by reducing it to our result. - 15:15–16:15
- Eric Delaygue, Integrality of the Taylor coefficients of mirror maps
I will present an effective criterion for the integrality of the Taylor coefficients of power series, called mirror maps, which are of particular interest in Mirror Symmetry Theory. More precisely, I will give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series
*q*_{i}(*z*)=*z*_{i}exp(*G*_{i}(*z*)/*F*(*z*)), with*z*=(*z*_{1},...,*z*_{d}) and where*F*(*z*) and*G*_{i}(*z*)+log(*z*_{i})*F*(*z*),*i*=1,...,*d*are particular solutions of certain A-hypergeometric systems of differential equations. I will also explain how this criterion implies the integrality of the Taylor coefficients of many univariate mirror maps listed in "Tables of Calabi--Yau equations" [arXiv:math/0507430v2, math.AG] by Almkvist, van Enckevort, van Straten and Zudilin.

### Belgian-Dutch algebraic geometry day

15 June, Leuven. The lectures will take place in**Huis Bethlehem, Schapenstraat 34**in the historical centre of Leuven, within walking distance from the train station. There will be coffee at 15:00 and 16:30

- 14:00–15:00
- Ted Chinburg, Small generators for
*S*-arithmetic groupsA surprising discovery of H. W. Lenstra, Jr., was that one can find generators of small height for groups of*S*-units of number fields once*S*is moderately large. I will discuss joint work with Matt Stover on generalizing Lenstra's results from*S*-units to the S-integral points of linear algebraic groups. This has applications to finding presentations for such groups. - 15:30–16:30
- Mathieu Romagny, Models of groups schemes of roots of unity
I will explain the construction of a family of models over
*O*_{K}of the group scheme μ_{pn,K}of*p*^{n}-th roots of unity over a*p*-adic field*K*. This construction is inspired by work of Sekiguchi and Suwa in the late nineties. The contemplation of these models in light of the recent classification of finite flat group schemes by Breuil and Kisin leads to conjecture that they exhaust all possible models of μ_{pn,K}. This is joint work with A. Mézard and D. Tossici. - 16:45–17:45
- Mircea Mustaţă, Adjoint line bundles in positive characteristic
In characteristic zero, adjoint line bundles enjoy many positivity properties that all go back to Kodaira's Vanishing Theorem. I will explain how certain positivity properties can be recovered in positive characteristic by making use of the Frobenius morphism.

### Intercity Number Theory Seminar

28 September, Leiden. Snellius building, first lecture in room B01 and the others in room 405.- 11:30–12:30
- David Holmes Leiden, Explicit Arakelov theory for Néron-Tate heights on the Jacobians of curves
The Néron-Tate height is a positive definite quadratic form on the group of rational points of an abelian variety (the 'Mordell-Weil group'). The existence of such a form is important, for example appearing in the proof of the finite generation of the Mordell-Weil group. However, the exact values of the height are important too; the best known example of this is their appearance in the regulator of an abelian variety, in the conjecture of Birch and Swinnerton-Dyer. Until recently, techniques to compute the Néron-Tate height, or approximate it by a 'naive height', we're restricted to curves of genus at most 3. Using Arakelov theory, we can extend this to give an algorithm for both these problems valid for hyperelliptic curves of arbitrary genus, and effective in some situations in genus up to 10.
- 13:30–14:15
- Michiel Kosters Leiden, Generating the rational points of an elliptic curve over
**F**_{q}by looking at*x*-coordinatesLet*E*/**F**_{q}be an elliptic curve given in Weierstrass form. In this talk we will discuss a theorem which says that the points of*E*(**F**_{q}) with*x*-coordinates in a subgroup of**F**_{q}of size more than 6 √*q*generate*E*(**F**_{q}), unless one is in a very specific case. - 14:45–15:45
- Maarten Derickx Leiden, Torsion points on elliptic curves over number fields of degree 5, 6 and 7.
B. Mazur proved that there are only finitely many groups which can arise as the torsion subgroup of an elliptic curve over
**Q**. Later the uniform boundedness conjecture was stated and proved, generalizing the previous statement to arbitrary number fields in the following way: "Let d be an integer, then there exist a number B such that the torsion subgroup of an elliptic curve over a number field of degree at most d has at most B elements." In particular, Oesterle managed to show that if p is a prime dividing the order of the torsion subgroup of an elliptic curve over a number field of degree d then p is at most (3^{d/2}+1)^{2}. In this talk I will explain how Sage can be used to improve this bound for small d. Slides. - 16:00–17:00
- Marco Streng VU Amsterdam, Smaller class invariants for quartic CM-fields
The theory of complex multiplication allows one to construct elliptic curves with a given number of points. The idea is to construct a curve over a finite field by starting with a special curve E in characteristic 0, and taking the reduction of E modulo a prime number.
Instead of writing down equations for the curve E, one only needs the minimal polynomial of its j-invariant, called a Hilbert class polynomial. The coefficients of these polynomials tend to be very large, so in practice, one replaces the j-invariant by alternative 'class invariants'. Such smaller class invariants can be found and studied using an explicit version of Shimura's reciprocity law.

The theory of complex multiplication has been generalized to curves of higher genus, but up to now, no class invariants were known in this higher-dimensional setting. I will show how to find smaller class invariants using a higher-dimensional version of Shimura's reciprocity law.

### Intercity Number Theory Seminar

19 October, Utrecht. Uithof, Buys Ballot Lab (BBL) room 169.- 13:30–14:30
- Johan Bosman Utrecht, Ranks of elliptic curves with prescribed torsion over number fields
Let d be a positive integer, and let T_d be the set of isomorphism classes of groups that can occur as the torsion subgroup of E(K), where K is a number field of degree d and E is an elliptic curve over K. T_1 is known by Mazur's theorem, T_2 is known as well, and for d equal to 3 or 4, it is known which groups occur infinitely often.
We shall study the following problem: given a d <= 4 and a group T in T_d, what are the possibilities for the Mordell-Weil rank of E, where E is an elliptic curve over a number field K of degree d with the torsion subgroup of E(K) isomorphic to T. For d = 2 and T = Z/13Z or T = Z/18Z, and also for d = 4 and T = Z/22Z, it turns out that the rank is always even. This will be explained by a phenomenon we call "false complex multiplication".

This is joint work with Peter Bruin, Andrej Dujella, and Filip Najman.

- 15:00–16:00
- Sander Dahmen Utrecht, Some generalized Fermat equations of signature (p,p,q)
We discuss how the method of Chabauty-Coleman and the modular method can be combined to attack some cases of the Generalized Fermat equation
*x*^{p}+*y*^{p}=*z*^{q}. In particular, we show how to fully solve this equation (in coprime integers x,y,z) for (p,q) ∈ { (5,7), (5,19), (7,5) }. This is joint work with Samir Siksek. - 16:15–17:15
- Frits Beukers Utrecht, Divisibility sequences among linear recurrent sequences
It is well known, and easy to prove, that the Fibonacci sequence
*u*_{n}starting with*u*_{0}=0,*u*_{1}=1 has the property that*u*_{m}|*u*_{n}if*m*|*n*. Slightly less trivially, the recurrent sequence*u*_{n+4}=*u*_{n+3}-4*u*_{n+2}-2*u*_{n+1}-4*u*_{n}with starting values 0,1,1-1 also satisfies*u*_{m}|*u*_{n}if*m*|*n*. Following a question of Hugh Williams we shall go into the background of such phenomena.

### Springer Day

26 October, Utrecht. A day in memory of Tonny Springer, see the website for programme and registration.### Intercity Number Theory Seminar

2 November, Eindhoven. Room MF 3.119 in the new building called Metaforum.- 11:00–12:00
- Patrik Norén Aalto, Volumes in algebraic statistics
There are many convex sets in algebraic statistics. Some important statistical models form convex sets and convex polytopes are central when studying toric ideals. It is natural to ask what is the volume of a given convex set. I present a surprisingly simple formula for the volumes of convex hulls of polynomially parameterized curves. This formula is then applied to answer a question by Sullivant and Drton about the volume of certain mixture models.
- 13:00–14:00
- Rob Eggermont Eindhoven, Degree bounds on tree models
Tree models are families of probability distributions used in modelling the evolution of a number of extant species from a common ancestor. One method to describe these models is to view a family as a set in an algebraic variety of the form
*V*^{⊗m}, where m is the number of extant species, and to try to ffind polynomial equations that determine its Zariski closure. One important question in this area of research is the following: Can we bound the degree of the equations we need independently of the number of extant species? In this talk, I will tell a bit more about these models and will explain how to prove the existence a bound on the degree of the needed equations by constructing an inffinite limit of models of a specifc form. - 14:20–15:20
- Piotr Zwiernik Eindhoven, Graphical Gaussian models and their groups
Let G be an undirected graph with n nodes. In statistics, given such a graph, we consider the space S(G) of all symmetric positive definite matrices with zeros corresponding to non-edges of G. We call S(G) the Gaussian graphical model. In this talk we describe the stabilizer in
*GL*_{n}(*R*) of the model in the natural action of*GL*_{n}(*R*) on symmetric matrices. This has important consequences for the study of the Gaussian graphical model which I will discuss in the second part of the talk. (joint work with Jan Draisma and Sonja Kuhnt) - 15:20–16:20
- Jan Draisma Eindhoven, Maximum likelihood duality for determinantal varieties
In the recent preprint arXiv:1210.0198 Hauenstein, Rodriguez, and Sturmfels discovered a conjectural bijection between critical points of the likelihood function on the complex manifold of matrices of rank r and cricital points on the complex manifold of matrices of co-rank r-1. I'll discuss a proof of that conjecture for rectangular matrices and symmetric matrices. (Joint work with Jose Rodriguez.)

### Intercity Number Theory Seminar

9 November, Groningen. Bernoulliborg, room 267- 11:45–12:45
- Afzal Soomro Groningen, Maximal curves of genus one and two, and twists
We discuss the data (restricted to genus one and two) that can be found on the website www.ManyPoints.org maintained by Howe, Ritzenthaler, Van der Geer and others. We also describe a construction of Howe, Leprevost, and Poonen which gives an explicit genus two curve over
*F*_{q}having*q*+1+2*t*rational points, given an elliptic curve over*F*_{q}having*q*+1+*t*points. - 13:30–14:30
- Osmanbey Uzunkol Oldenburg, Theta functions and class fields
I am going to talk about the construction of some explicit class fields using special values of quotients of theta functions, theta identities, and the reciprocity law of Shimura together with some applications of this construction.
- 14:45–15:45
- Ane Anema Groningen, Field extensions over which an elliptic curve reaches the Hasse bound
Given an elliptic curve E over a finite field k, we discuss the problem of determining the finite extensions K/k such that #E(K) reaches the Hasse bound on the number of rational points.
- 16:00–17:00
- Max Kronberg Oldenburg, Torsion subgroup of two dimensional abelian varieties with real multiplication
The group of
*K*-rational points of an abelian variety splits into the torsion part and the free part. We are interested in the first of these two parts. While the question of the torsion structures for one dimensional abelian varieties, i.e. elliptic curves, over the rational numbers is completely settled by a result of B. Mazur, for higher dimensional abelian varieties the situation is completely unknown.Since the Siegel moduli space for abelian varieties of dimension two is three dimensional, we restrict to the subspace of abelian varieties with real multiplication (RM), i.e. with an embedding

*O*_{K}→*End*(*A*). These varieties arise as simple factors of the jacobian*J*_{0}(*N*) of the modular curve*X*_{0}(*N*), so this class is of great interest, for example to study the rank of elliptic curves. The moduli space for this class of abelian varieties is given by the Hilbert moduli space which is only two dimensional. Unfortunately the machinery developed by Mazur crucially depends on the fact, that the moduli problem of elliptic curves with a subgroup of order*N*has a one dimensional moduli space. So it is impossible to expand his results to higher dimensions.The actual approach is to restrict to a one dimensional family of hyperelliptic curves of genus two with real multiplication by ζ

_{5}+ζ_{5}^{-1}constructed by Tautz, Top and Verberkmoes. The next step would be to extend this approach to the two parameter family of the same objects constructed by Mestre.

### Intercity Number Theory Seminar

16 November, VU Amsterdam. The first lecture is in F640 of the W&N building, the other lectures are in MF-FG1, in the basement of the medical sciences building.- 11:30–12:30
- Rob de Jeu VU Amsterdam, The syntomic regulator for K_4 of curves
Let C be a curve defined over a discrete valuation field of characteristic zero where the residue field has positive characteristic. Assuming that C has good reduction over the residue field, we compute a certain p-adic ("syntomic") regulator on a certain part of
*K*_{4}^{(}3) of the function field of C. The result can be expressed in terms of p-adic polylogarithms and Coleman integration, or by using a trilinear map ("triple index'') on certain functions. This is joint work with Amnon Besser. - 13:30–14:30
- James Lewis University of Alberta, Regulators of Algebraic Cycles
This talk concerns the cycle class map from the higher Chow groups to Deligne cohomology, for a smooth projective variety defined over a subfield of the complex numbers. One construction of this map, based on the work of Kerr/Lewis/Mueller-Stach, makes use of the cubical description of the Chow groups; however we also explain the map in terms of Bloch's original formulation of his higher Chow groups (joint work with Matt Kerr). We illustrate an example situation where Beilinson rigidity translates to a functional equation for the dilogarithm
- 14:45–15:30
- François-Renaud Escriva VU Amsterdam, Point counting and cup product computations
Kedlaya in 2001 published a point counting algorithm for hyperelliptic curves over finite fields that is based on Monsky-Washnitzer cohomology. His method, and the extensions by others to a larger class of curves, all rely strongly on the particular shape of the equation of the curves. In this talk we present a method that can deal with very general curves. This is joint work with Amnon Besser and Rob de Jeu.
- 16:00–17:00
- Deepam Patel VU Amsterdam, Motives arising from higher homotopy theory
In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points and related them to special values of zeta functions. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we discuss a motivic structure on the (nilpotent completion of) n-th homotopy group of
*P*^{n}minus n+2 hyperplanes in general position.

### DIAMANT symposium, special afternoon on algebra and number theory in cryptography

30 November, Doorn. Part of the Diamant symposium, which runs from Thursday to Friday.### 2nd Belgian-Dutch Algebraic Geometry Day

7 December, UvA Amsterdam. Room A1.10 at Amsterdam Science Park.- 13:30–14:30
- Andrei Caldararu U. of Wisconsin, Derived intersections
In recent years there have been remarkable developments in derived algebraic geometry, a field which lies at the confluence of algebraic geometry and algebraic topology. The main objects of study in derived algebraic geometry are dg schemes, and a typical example of dg schemes arises when studying intersection theory of subvarieties. In my talk I shall try to explain some of the fundamental ideas of derived algebraic geometry, and argue that it allows us to get a more geometric understanding of classical results like the proof by Deligne-Illusie of the Hodge-de Rham degeneration.
- 15:00–16:00
- Eduard Looijenga Utrecht, Cohomological dimension of moduli spaces of curves
We discuss the following theorem and its consequences: the cohomology of a constructible abelian sheaf F on the complex moduli stack
*M*_{g,n}(*C*) (for the Euclidean topology) vanishes in degree greater than*g*-1 plus the dimension of the support of*F*. This implies Harer's bound for the homotopy type of the underlying complex-analytic space as well as the bound of Diaz on the maximal dimension of a complete subvariety of*M*_{g,n}(*C*). We also deduce this type of bound for any open subset of the Deligne-Mumford compactification that is a union of strata. A minor adaptation of the proof shows also that the cohomological dimension of for quasi-coherent sheaves of*M*_{g,n}is at most g-1 in characteristic zero. This implies conjectures of Roth and Vakil. - 16:30–17:30
- Daniel Huybrechts Bonn, Cycles on K3 surfaces
.