## 2008

### Intercity Number Theory Seminar

15 February, Leiden. Room 407- 12:15–13:00
- Hendrik Lenstra, Degrees of field automorphisms
Consider a field extension where the top field is algebraically closed and of finite non-zero transcendence degree over the bottom field. The lecture describes a large discrete quotient of the relative automorphism group.
- 14:00–14:45
- Jonathan Hanke, The 290-Theorem and Representing Numbers by Quadratic Forms
This talk will describe several finiteness theorems for quadratic forms, and progress on the question: "Which positive definite integer-valued quadratic forms represent all positive integers?". The answer to this question depends on settling the related question "Which integers are represented by a given quadratic form?" for finitely many forms. The answer to this question can involve both arithmetic and analytic techniques, though only recently has the analytic approach become practical.
We will describe the theory of quadratic forms as it relates to answering these questions, its connections with the theory of modular forms, and give an idea of how one can obtain explicit bounds to describe which numbers are represented by a given quadratic form.

- 15:00–15:45
- Chia-Fu Yu, On geometric mass formulas
In this talk we will start by discussing the Eichler-Deuring mass formula concerning supersingular elliptic curves. We then discuss a result on geometric mass for superspecial principally polarized abelian varieties due to Ekedahl and some others, and its function field analogue for supersingular Drinfeld modules. We shall also describe a generalization to the quaternion unitary cases.
- 16:00–16:45
- Lenny Taelman, Cubic rings, cubic forms
After Gan, Gross, Savin and Deligne: see Deligne's letter to Gan, Gross and Savin and Deligne's letter to Edixhoven

### L-functions and friends

29 February, VU Amsterdam. Room M129. Today's program is part of Rob de Jeu's Algebra seminar. The seminar finishes early because of the inaugural lecture of Rob van der Vorst at 15:45.- 11:00–12:00
- Tejaswi Navilarekallu, Galois actions and L-values
Let
*K*/*Q*be a finite Galois extension with Galois group*G*. For every character χ of*G*, the special value*L*(χ,0) of the Artin*L*-function carries arithmetic information about the extension. The equivariant Tamagawa number conjecture predicts the precise relation between the*L*-values and certain arithmetic invariants. In this talk, we shall give a formulation of the conjecture and indicate some techniques of verification. - 13:00–14:00
- Don Zagier, Modular Green's functions
The functions of the title arose many years ago in connection with heights of Heegner points and special values of
*L*-functions, but turned out to have further interesting properties, including a conjectural algebraicity statement for their special values. This conjecture has recently been proved in many cases by Anton Mellit. We will discuss this and related work. - 14:15–15:15
- Xavier-François Roblot, Computing values of
*p*-adic*L*-functions over a real quadratic fieldFollowing the works of P. Cassou-Noguès, D. Barsky, N. Katz and P. Colmez, I will give an explicit construction of a continuous*p*-adic function interpolating (some of) the values at negative integers of the Hecke*L*-function of a real quadratic field, the so-called*p*-adic Hecke*L*-function. I will show how this construction allows one to compute (approximations of) the values of this function.

### Intercity Number Theory Seminar: Van der Kallen Celebration

14 March, Utrecht. room BBL 160. Today's seminar is dedicated to the 61st birthday of Wilberd van der Kallen. After the talks at 4PM there will be a reception in the library.- 10:30–11:30
- Jan Draisma, Phylogenetic tree models and classical invariant theory
Tree models are families of probability distributions that are used in reconstructing evolutionary trees from genetic data. They are given by a parameterisation, whereas an implicit description, by means of polynomial equations, would facilitate testing whether a given empirical distribution lies in the family. These equations are hard to find, but I will explain how to use classical invariant theory to reduce the quest for them to the case of simple trees.
- 11:45–12:45
- Wim Hesselink, Euclidean skeletons of digital images in linear time by the integer medial axis transform
Roughly speaking, a skeleton of a two-dimensional image is a line drawing that represents the form of the image. The aim of the talk is to resolve the vagueness of this description. Digital image and volume data are given as grey values associated to grid points (pixels or voxels). There can be millions of grid points. Therefore efficient algorithms are needed. One of the efficient algorithms available is the feature transform that, for a given set of background points in a rectangular box, determines in linear time for every grid point in the box a closest background point. The integer medial axis IMA is a way to use this for skeletonization. We discuss theorems and conjectures about IMA, one of which was both refuted and proved by Wilberd van der Kallen.
- 13:45–14:45
- Johnny Edwards, The chocolate bar conjecture
In a recent paper, Wilberd van der Kallen proves an inequality satisfied by quadruples of points in the lattice
**Z**^{2}. This inequality is at the heart of an algorithm in image processing. See the last lecture. In his paper, van der Kallen conjectures a similar inequality for points in the 3 dimensional lattice**Z**^{3}. I will explain the chocolate bar conjecture and show that it is true in dimension 2 and 3. From this I will deduce that van der Kallen's conjecture is true, modulo at most a finite set of exceptional quadruples of lattice points. - 15:00–16:00
- Vincent Franjou, Finite generation for (higher) invariants
A classical problem in invariant theory, often referred to as Hilbert's 14th problem, asks, when a group acts on a finitely generated algebra by algebra automorphisms, whether the ring of invariants is still finitely generated. Nagata's work on this problem led to the notion of geometric reductivity. I shall present a variant of this notion, "power reductivity", which is better adapted to working over a general commutative ring. I shall then present progress on the corresponding conjecture of van der Kallen for higher invariants. This is work in progress with Wilberd van der Kallen.

### Intercity Number Theory Seminar

4 April, Groningen. room 105 of the new Bernoulliborg building, Nijenborgh 9: the big blue building on your right when bus 15 enters the campus.- 11:45–12:45
- Cécile Poirier, Stacks, vector bundles and `geometric Langlands'.
- 13:30–14:30
- Fai-Lung Tsang, Skew rings and geometric method in convolutional codes
There is a natural way to assign a degree structure on the module
*F*[*z*]⊗*V*(*V*a finite dimensional*F*-space), namely the one induced by the polynomial ring*F*[*z*]. One has the question, for any given sequence of positive numbers*d*_{1},dots,*d*_{r}, can one find a submodule such that (1) there is a basis with degrees*d*_{i}and (2) the sum of the degrees of this basis is minimal. The answer is affirmative for*F*[*z*]^{n}.In this talk, we study the case where

*F*[*z*]⊗*V*is a skew (polynomial) ring. We will construct a free vector bundle on**P**^{1}and the subbundle associated to the submodule. It turns out that we can give a positive answer when ∑*d*_{i}≤*n*. The same technique is applicable to problems concerning polynomial matrices, I will talk about a question posed by A. Dijksma and B. Ćurgus, and give an answer to the problem using the geometric method developed. This is joint work with M. van der Put. - 14:45–15:45
- Steve Meagher, Freiburg), Genus three curves revisited
- 16:00–17:00
- Evgeny Verbitskiy, Marius van der Put, Abelian sandpiles and discrete difference equations

### Intercity Number Theory Seminar: genus 2 day

18 April, Utrecht. room BBL 160 (Buys-Ballot-Lab)- 11:00–11:50
- Marco Streng, Igusa class polynomials
Igusa class polynomials are the genus 2 analogue of the classical Hilbert class polynomial. We explain both notions and discuss the differences between the classical (elliptic) case and the genus 2 case, mostly from a computational perspective.
- 12:00–12:50
- Jeroen Sijsling, Humbert Surfaces and Shimura Curves
Humbert surfaces are special subvarieties of the modular threefold of genus 2 curves over the complex field, parametrizing the curves whose Jacobian has an endomorphism algebra containing a real quadratic extension of the rationals. Certain Shimura curves, moduli curves parametrizing curves for which the endomorphism algebra of the Jacobian contains a quaternion algebra over the rationals, can be obtained as a curve in the modular threefold as the intersection of Humbert surfaces. We will explain these notions in more detail, then work towards some concrete results by Hashimoto/Murabayashi and Lange/Wilhelm on the equations defining Humbert surfaces in terms of Igusa invariants.
- 14:30–15:20
- Andrew Hone, Somos Sequences and genus two addition formulae
Somos sequences are nonlinear recurrence sequences defined by a quadratic relation. They arise in number theory (Morgan Ward's elliptic divisibility sequences), combinatorics (Fomin & Zelevisnky's cluster algebras) and integrable systems (discrete Hirota equations, Quispel-Roberts-Thompson maps). In the fourth order (Somos-4) and fifth order case (Somos-5) they correspond to sequences of points on elliptic curve. After reviewing the elliptic case, we present some results in genus 2, mainly concentrating on Somos-6, for which we give a formula in terms of Kleinian sigma functions.
- 15:30–16:20
- Dan Bernstein, Hyperelliptic-curve cryptography
The only public-key cryptographic systems currently recommended by the United States National Security Agency are elliptic-curve systems. I'll explain how elliptic curves are used in cryptography and how genus-2 hyperelliptic curves can do better; in particular, I'll discuss recent progress in genus-2 scalar multiplication and in constructing secure genus-2 curves. To balance the picture I'll also discuss recent progress in elliptic-curve computations.

### Intercity Number Theory Seminar

9 May, Leiden. room 207 Huygens building- 12:00–13:00
- Hugo Chapdelaine, An introduction to the 12th Hilbert problem
The 12th Hilbert problem consists in finding a way of constructing explicitly the maximal abelian extension of a given number field
*K*. In the first half of the talk we will illustrate fragmentary results which are known on this problem, namely in the case where*K*is the field of rational numbers or an imaginary quadratic number field. For the second half of the talk we will mention some recent*p*-adic constructions of conjectural*p*-units in abelian extensions of real quadratic fields. Hopefully this should bring some new insight towards the 12th Hilbert problem. - 14:00–16:00
- Cristian Popescu, On the Coates-Sinnott Conjectures
The conjectures in the title were formulated in the late 1970s as vast generalizations of the classical theorem of Stickelberger. They make a subtle connection between the
**Z**[*G*(*L*/*k*)]-module structure of the Quillen*K*-groups*K*_{*}(*O*_{L}) in an abelian extension*L*/*k*of number fields and the values at negative integers of the associated*G*(*L*/*k*)-equivariant*L*-functions Θ_{L/k}(*s*). These conjectures are known to hold true if the base field*k*is**Q**, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields*k*. - 16:15–17:15
- Remke Kloosterman, Computing the Mordell-Weil group of elliptic threefolds
In this talk we discuss a method to compute the rank of
*E*(**Q**(*s*,*t*)) for a class of elliptic curves*E*defined over**Q**(*s*,*t*).This method relies on an explicit method to compute

*H*^{4}(*Y*,*C*) for a class of singular threefolds*Y*. This is joint work with Klaus Hulek.

### Intercity number theory seminar

12 September, Leiden. Room 412- 14:00–14:45
- Gabriel Chênevert, The quartic fields method
In this talk I want to explain Serre's quartic fields method, which provides a way to decide whether two 2-dimensional, 2-adic absolutely irreducible Galois representations are equivalent or not. As an example, the method will be used to determine the modular form corresponding to a part of the cohomology of a certain smooth cubic fourfold admitting an action by the symmetric group
*S*_{7}. - 15:00–15:45
- Sylvain Brochard, Picard functor and algebraic stack
The Picard functor of a scheme, classifying invertible sheaves on it, has been studied extensively in the 60's. However, the work of Giraud, Deligne, Mumford and Artin gave birth in the 70's to the notion of an algebraic stack, which generalizes that of a scheme. The following question arises then: does the Picard functor of an algebraic stack behave like that of a scheme ? In this talk I will briefly recall what the Picard functor of a scheme is and what it is designed for. Then I will explain in few words what an algebraic stack is and try to answer the preceding question.
- 16:00–16:45
- Ronald van Luijk, Density of rational points on diagonal quartic surfaces
It is a wide open question whether the set of rational points on a smooth quartic surface in projective three-space can be nonempty, yet finite. In this talk I will treat the case of diagonal quartics
*V*, which are given by*ax*^{4}+*by*^{4}+*cz*^{4}+*dw*^{4}=0 for some nonzero rational*a*,*b*,*c*,*d*. I will assume that the product abcd is a square and that*V*contains at least one rational point*P*. I will prove that if none of the coordinates of*P*is zero, and*P*is not contained in one of the 48 lines on*V*, then the set of rational points on*V*is dense. This is based on joint work with Adam Logan and David McKinnon.

### Standard models of finite fields

26 September, Nijmegen.- 11:30–12:15
- Frank Lübeck, Conway polynomials
I will give the definition of the Conway polynomials which define finite fields, mention some cases where they are used, and explain how they can be computed. Then I will address the problem that the Conway polynomials which are not yet known are very difficult to compute. On the other hand one would like to know them for any field GF(
*q*) for which the factorization of (*q*-1) is known (these are the fields in which elements can be tested for primitivity). I will propose a modification of the definition such that the modified polynomials can be computed in reasonable time. - 13:30–14:15
- Wieb Bosma, Dealing with finite fields in Magma
In computations with finite fields it is essential to maintain subfield relations in a consistent way. In this talk I will describe the different representations for finite fields in the computer algebra system Magma, and the mechanism used for ensuring that subfield diagrams commute.
- 14:45–15:30
- Bart de Smit, Consistent isomorphisms between finite fields
We give a deterministic polynomial time algorithm that on input two finite fields of the same cardinality produces an isomorphism between the two. Moreover, if for three finite fields of the same cardinality one applies the algorithm to the three pairs of fields then one obtains a commutative triangle. The algorithm depends on the definition given in the next talk.
- 16:00–16:45
- Hendrik Lenstra, Defining
**F**_{q}The lecture provides a definition of**F**_{q}as an actual field of cardinality*q*, as opposed to a field just defined up to isomorphism. The definition is complicated enough that it occupies most of the lecture. No easier definition is known that has the attractive algorithmic properties needed in the previous talk.

Notes: Standard models of finite fields: the definition [PDF].

*Intercity number theory seminar*

*10 October, Leiden. The first talk is in room 402 and the others in room 174.*

- 12:00–12:45
- Karen Aardal, Integer programming and some connections to number theory
We introduce integer programming, some aspects of polyhedral combinatorics, and links to number theory.
- 13:45–14:30
- Andrea Montanari, Torus based cryptography
We give a brief description of recent developed techniques for public-key cryptography over finite fields relying on algebraic torus. In fact in some cases it is possible to get a compressed representation of elements in the torus. We discuss an explicit compression/decompression map for elements of the torus in quadratic finite field extensions.
- 14:45–15:30
- Willemien Ekkelkamp, Predicting the sieving effort for the number field sieve
In order to estimate the most time-consuming step of the number field sieve, namely the sieving step in which the so-called relations are generated, we present and discuss a new method for predicting the sieving effort. Our method takes relations from a short sieving test as input and simulates relations according to this test. After removing so-called singletons, we decide how many relations we need for the factorization according to the simulation and this gives a good estimate (within 2%) for the real sieving.

*Intercity number theory seminar*

*14 November, Groningen. room 267 of the mathematics building Bernoulliborg*

- 11:30–12:30
- Andrey Timofeev, Index-calculus in the Brauer groups with arithmetic applications
The Brauer group is an important invariant of field. In this talk the definitions and some basic properties will be given and considered how can we apply Index Calculus algorithm for computing Euler's totient function as well as for solving the discrete logarithm problem in finite fields with approach of Brauer groups.
- 13:30–14:30
- Felix Fontein, A Concise Interpretation of the Infrastructure of a Global Field
In this talk, we will present an interpretation of the infrastructure of a global field. We will describe explicit arithmetic and relate the infrastructure to the (Arakelov) divisor class group. This extends work by D. Shanks, H. Lenstra, R. Schoof and others.
- 14:45–15:45
- Marius van der Put, A geometric approach to Painlevé differential equations
The singularities of the solutions of a linear differential equation (in one variable) coincide with the singularities of the equation. This is not the case for non linear differential equations. Painlevé introduced the notion `no moving singularities' to obtain interesting equations. The ones with order one were classified in a more or less satisfactory way. The chaos of the order two equation with the above `Painlevé's property' have been put into the cages PI-PVI. They remained there until about 1980 when the subject obtained new impulses from physics. The research on Painlevé equations is ever growing. In this lecture I will present new results, obtained in collaboration with Masa-Hiko Saito, on the `moduli' and the `monodromy spaces' associated to the Painlevé equations.
- 16:00–17:00
- Jaap Top, Maximal curves over finite fields
A (smooth, complete, geometrically irreducible) curve
*C*defined over a finite field**F**_{q}is called maximal, if the cardinality of the set of**F**_{q}-rational points on*C*equals*q*+1+*gm*. Here*g*denotes the genus of the curve, and*m*is the largest integer not exceeding 2√*q*. We consider the following question: given a curve over**Z**[1/*N*], for which prime powers*q*coprime with*N*, is*C*⊗**F**_{q}maximal over**F**_{q}? Although in general this is a difficult problem (already for genus 1!), it turns out that in certain examples one can give a complete answer. In the talk this is done for a particular hyperelliptic curve of genus 3.

*Intercity number theory seminar*

*12 December, UvA Amsterdam. Morning program in room P0.19 of the Euclides building.*

Tom Koornwinder gives his farewell address in the afternoon at 15:00.

Tom Koornwinder gives his farewell address in the afternoon at 15:00.

- 10:30–11:15
- Arjen Stolk, Fast group operations on Jacobians
Kamal Khuri-Makdisi has described a construction which reduces the problem of computing with points on the Jacobian of a general curve to linear algebra in certain Riemann-Roch spaces. These algorithms are elegant and fast. In this talk I will explain the basic ideas behind Kuhri-Makdisi's approach.
- 11:30–12:15
- Peter Bruin, Finding random points on curves over finite fields
Consider a smooth, complete, absolutely irreducible curve
*X*of genus*g*over a finite field of*q*elements, given by a projective embedding as described by K. Khuri-Makdisi (*Math. Comp.*73 (2004) and 76 (2007)). We give an algorithm for picking uniformly random elements of the set of rational points on*X*, with expected running time polynomial in*g*and log*q*. - 12:30–13:15
- Sylvain Brochard, On De Smit's conjecture on flatness on Artin rings
Let
*f*:*A*→*B*be a flat morphism of Artin local rings with the same embedding dimension (the embedding dimension is the dimension of the tangent space). Bart de Smit conjectured that any finite B-module that is*A*-flat is*B*-flat. We will give some partial results in this direction, and explain the proof for a particular example.