
Intercity Number Theory SeminarMarch 16, Utrecht. Room 202 Minnaert11:3012:30  Jaap Top, Elliptic surfaces with a high Picard number Abstract. In characteristic zero an elliptic K3 surface has Picard number
at most 20. We will present many examples with high Picard number and discuss
reduction modulo primes.  13:1513:45  Nguyen Khuong An (Groningen), The algebraic subgroups of GL_{2}(C). Abstract. The algebraic subgroups of PGL_{2}(C) and SL_{2}(C)
are well known. In contrast to this, complete and dependable
literature on the algebraic subgroups of GL_{2}(C) seems to be
missing.  14:0015:00  Marius van der Put, Solving linear differential equations Abstract. The theme of solving a linear differential equation in terms
of equations of lower order goes back to L. Fuchs around 1880 and
G. Fano 1900. We will explain how the representation theory of semisimple
Lie algebras can be used for this problem.  15:1516:15  Steve Meagher (Freiburg), When is the twist of a Jacobian a Jacobian? Abstract. Over a non algebraically closed field a Jacobian variety
may have twists that are not Jacobians. How can one characterize these?  16:3017:00  Lenny Taelman, An exercise in algebraic topology Abstract. I shall formulate and solve a little exercise.
Familiarity with only the most basic algebraic
topology will be assumed. 

DIAMANT Intercity Number Theory SeminarMarch 30, Leiden. room 405. Joint session with RISC seminar (CWI)11:0011:45  Peter Montgomery, Parallel block Lanczos Abstract. Some factorization and discrete logarithm algorithms have a linear
algebra phase, where a huge sparse system must be solved over a finite
field. One avoids a memory explosion by using iterative methods, but run
time can remain high. We describe how to parallelize the linear algebra
and relate our experiences.
 12:0012:45  Ramarathnam Venkatesan (Microsoft Research), Cryptographic applications involving Spectral analysis of Rapid mixing Abstract. We survey some applications that involve spectral analysis in various domains.
First one is the analysis of classic Pollard Rho. Second one stems from the question if all
elliptic curves of the same order over a finite field have the same difficulty of discrete log.
The third one involves the design of a stream cipher called MV3.
This is joint work with Steve Miller (Rutgers), David Jao (Waterloo).
 13:4514:30  Corentin Pontreau, Bogomolov's problem, small points on varieties Abstract. Height functions describe in a certain sense the arithmetic complexity
of an algebraic number or more generally of an algebraic variety.
We will present which kind of lower bounds for the height of algebraic
varieties (Bogomolov's problem) and points of such varieties one can expect.
Even if most of the results can be stated for semiabelian varieties, we
will mainly deal with the torus case, roughly speaking
G_{m}×... ×G_{m} over the algebraic closure of Q.  14:4515:30  Sierk Rosema (Leiden), Sturmian substitutions, cutting paths and their projections Abstract. From a string of zeros and ones of finite length we construct a stepped
line that we call a cutting path. By projecting the integer points on
this path onto the yaxis, we form a new string of zeros and ones. If
σ is a Sturmian substitution, we apply this process to
u_{n}=σ^{n}(0) to define a sequence of words v_{n}. We will show that
if σ has an incidence matrix with determinant 1, then there
exists a Sturmian substitution τ such that v_{n}=τ(v_{n1}) for
every n>1.


Intercity Number Theory SeminarMay 25, Leiden. The first talk will be in the Sitter room (ground floor of the Oort building), and the other talks will be in room 204 of the Huygens building (directions)12:0013:00  David Kohel, Complex multiplication and canonical lifts Abstract. The jinvariant of an elliptic curve with complex multiplication by
K is wellknown to generate the Hilbert class field of K.
Such jinvariants, or rather their minimal polynomials in Z[x],
can be determined by means of complex analytic methods from a given
CM lattice in C. A construction of CM moduli by padic lifting
techniques was introduced by Couveignes and Henocq. Efficient versions
of onedimensional padic lifting were developed by Bröker.
These methods provide an alternative application of padic canonical
lifts, as introduced by Satoh for determining the zeta function of an
elliptic curves E/F_{pn}.
Construction of such defining polynomials for CM curves is an area of
active interest for use in cryptographic constructions. Together with
Gaudry, Houtmann, Ritzenthaler, and Weng, we generalised the elliptic
curve CM construction to genus 2 curves using 2adic canonical lifts.
The output of this algorithm is data specifying a defining ideal for
the CM Igusa invariants (j_{1},j_{2},j_{3}) in Z[x_{1},x_{2},x_{3}].
In contrast to Mestre's AGM algorithm for determining zeta functions
of genus 2 curves C/F_{2n}, this construction pursues the
alternative application of canonical lifts to CM constructions.
With Carls and Lubicz, I developed an analogous 3adic CM construction
using theta functions. In this talk I will report on recent progress
and challenges in extending and improving these algorithms.
 14:0015:45  Johan Bosman, A polynomial with Galois group SL_{2}(F_{16}) Abstract. An interesting computation challenge is to calculate polynomials P in Q[x]
that have a prescribed Galois group. By this we mean the Galois group of
the splitting field together with its action on the roots of P. Jürgen
Klüners and Gunther Malle developed methods that work for many groups,
including all transitive permutation groups of degree up to 15.
In this talk we will present a polynomial whose Galois group is isomorphic
to SL_{2}(F_{16}), a group that Klüners and Malle could not handle with their
approach. The computation makes use of Galois representations of modular
forms.
 16:0017:00  Reinier Bröker, Lifting supersingular curves Abstract. In this talk we present a padic algorithm to compute the Hilbert
class polynomial corresponding to an imaginary quadratic order O. This
polynomial has integer coefficients and it's roots, say in the complex numbers,
are jinvariants of elliptic curves with endomorphism ring O.
The prime p that we use in this padic algorithm is inert in O, and is
therefore quite small. The main step in the algorithm is computing
the `canonical lift' of a supersingular elliptic curve over the finite
field F_{p}. Many examples will be given.


Intercity Number Theory SeminarJune 8, Utrecht. Room 211 Minnaert building Tea with cookies will be served at 15:10.11:0012:00  Harm Voskuil, pAdic uniformisation: introduction and examples Abstract. I briefly define and explain affinoid domains and rigid analytic spaces.
Then the subject of padic uniformisation of (analytic) varieties is discussed.
The main examples treated are those of abelian varieties and algebraic curves.
Moreover, I will compare the padic and the real uniformisations of these
examples.
 13:0014:00  Fumiharu Kato, Topological rings in rigid geometry Abstract. This is a jointwork with Kazuhiro Fujiwara (Nagoya). While
classical algebraic geometry only deals with finite type rings over a
field, scheme theory involves arbitrary rings; of course, fields and
finite type rings over them are still important in scheme theory,
because fields are `point objects', and finite type rings over a
field are 'fiber objects' for locally of finite type maps between
schemes. Rigid geometry a la TateRaynaud, on the other hand, has
been developed over aadically complete valuation rings of height 1.
It has become recognized by experts that, in order to detect all
'points' of rigid spaces, one has to consider aadically complete
valuation rings of arbitrary height. This leads one to the quest for
a reasonable class of topological rings that allows a generalization
of `classical' rigid geometry, something compared with the scheme
theory as a generalization of classical algebraic geometry, in such a
way that aadically complete valuation rings are `point objects', and
that complete rings topologically of finite type over an aadically
complete valuation rings are 'fiber objects' for locally of finite
type morphisms. In this talk, we would like to propose a candidate of
such a class of topological rings, the socalled, topologically
universally adhesive (t.u.a.) rings. This class is closed under
topologically finite type extensions, contains any Noetherian ring
complete with respect to an ideal, and has several nice topological,
ringtheoretical, and homological properties, such as, ArtinRees
property, coherency, etc. Moreover, a deep theorem, which we
attribute to Gabber, says that any aadically complete valuation ring
is contained in this class. We would like to indicate that, by means
of this notion, one can develop a reasonable theory of formal schemes
that admits one to generalize theorems in EGA III, and, based on
these foundations, one has a generalization of the notion of rigid
spaces.
 14:1015:10  Francis Brown, Multiple zeta values and periods of moduli spaces of genus zero curves Abstract. Let n≥4, and let M_{0,n} denote the moduli space of curves
of genus 0 with n marked points. In a recent paper, Goncharov and
Manin showed how a pair of boundary divisors on the compactification
M_{0,n} defines a mixed Tate motive unramified over the
integers. They conjectured that the periods one obtains in this way
are multiple zeta values.
In this talk I will outline a proof of this conjecture. I will give
an explicit construction of the compactification
M_{0,n}, and recall some of its geometric and
combinatorial properties. I will then explain how to compute the
periods by iterated application of Stokes' formula in a suitable
algebra of polylogarithm functions on M_{0,n}.
 15:3016:30  Gautam Chinta, The theory of Weyl group multiple Dirichlet series Abstract. A Weyl group multiple Dirichlet series associated to a finte root system
Φ of rank r is a Dirichlet series in r complex variables having an
analytic continuation to r copies of C and satisfying a group of
functional equations isomorphic to the Weyl group of Φ. These series
have been the topic of intense study in recent years. I will discuss
the history of the subject, describe how to construct these series and
indicate some applications.


GTEM daySeptember 21, Leiden. Special day of lectures at the GTEM workshop at the Lorentz Center (room 201 Huygens)11:0011:45  Dieter Geyer, Higher dimensional class field theory Abstract. Higher dimensional class field theory, i.e. the theory of abelian coverings of
higher dimensional arithmetical schemes including varieties over finite fields,
was started in case of regular schemes in the 1980's by Bloch, Kato and Saito
in several papers using higher dimensional Milnor Ktheory. Subsequent papers
by Jannsen, Stevenhagen, Spiess, A. Schmidt and others followed. I will speak
on a new approach by Goetz Wiesend, using only K_{0} and K_{1} groups, and a
thesis of Walter Hofmann generalizing Wiesend's results from regular schemes to
singular schemes.  12:0012:45  Andrea Surroca Ortiz, On the MordellWeil and the TateShafarevich groups of abelian varieties Abstract. This talk is about some conjectures relating the height, the conductor, the regulator and the TateShafarevich group of abelian varieties over number fields. We will see that a result of GoldfeldSzpiro relating the order of the TateShafarevich group to the conductor of an elliptic curve over Q can be extended to arbitrary abelian varieties over number fields. On the other hand, we will also mention an application to the abcconjecture, which is a work in collaboration with V. Bosser.  14:0014:45  Christian Wuthrich, Computations about the TateShafarevich group using Iwasawa theory Abstract. In analogy to the zeta function for varieties over finite
fields, the padic Lseries of an elliptic curve E over the rational
field can provide us with interesting arithmetical information via
Iwasawa theory. I will present an algorithm that can compute upper
bounds on the order of the pprimary part of the TateShafarevich
group E. This is joint work with William Stein.  15:0015:45  Gabor Wiese, Modular Forms in Inverse Galois Theory Abstract. Modular forms which are eigenfunctions for all Hecke operators give rise to
2dimensional mod p representations of the absolute Galois group of the
rationals. In the talk we will show how these representations, and hence
modular forms, can be used to derive results on the occurrence of groups of
the type PSL_{2}(F_{pr}) as Galois groups over the rationals.


Intercity Number Theory SeminarOctober 5, Delft. Snijderszaal op de 1e etage EWI gebouw, Mekelweg 4, Delft Bereikbaar via bus 129 vanaf Delft CS
Parkeerplaats achter het gebouwThere is a group lunch for the seminar participants, and there will be tea and coffee at 14:30 11:3012:30  Graham Everest, Elliptic Curves and Hilbert's tenth problem Abstract. Hilbert's Tenth Problem asks if an algorithm can be constructed
which will determine if a finite system of Diophantine equations,
with rational integer coefficients, has an integral solution. This
was answered negatively in 1970 by Yuri Matiyasevic, building on
work of Davis, Putnam and Robinson. The same question, except now to
determine of there is a rational solution, has not been resolved.Recent work of Poonen has shown the same negative answer for some
large subrings of the rationals using the arithmetic of elliptic
divisibility sequences. In my talk I will report on Poonen's work as
well as give some new results about which subrings of Q are covered
by Poonen's methods.
 13:3014:30  Karma Dajani, Ergodic properties of signed binary expansions Abstract. Signed separated binary expansions of integers are
expansions of the form n=∑_{i=0}^{k1} a_{i}2^{i}, where a_{i}∈{
1,0,1} and a_{i} a_{i+1}=0. Identifying an integer n with
its corresponding sequence of SSBdigits a_{0},a_{1},...,a_{k1}, we
consider an SSBcompactification K of Z, namely the set
K={ (x_{0},x_{1},...) ∈{ 1,0,1}^{N} : x_{i} x_{i+1}=0
for all i∈N}. On K there are two natural
transformations, the shift σ and the odometer τ (the
latter is analogous to adding 1 mod 2 with carry). In this talk,
we discuss the ergodic properties of these transformations.
 14:4515:45  Fritz Schweiger, Multidimensional continued fractions  new results and old problems Abstract. Regular continued fractions exhibit a number of remarkable
properties.

If one puts p_{n}/q_{n} := 1/(a_{1}+1/(a_{2}+1/...(a_{n1}+1/a_{n})...)),
then one obtains "good" approximations to x.

The related map Tx=1/xa_{1}(x) is ergodic and admits an
absolutely continuous invariant measure.

The algorithm becomes eventually periodic, i.e., T^{n+m}x=T^{m}x for
some ngeq0, mgeq1, if and only if x is a quadratic
irrational number ("Theorem of Lagrange").
Since the days of C.G.J. Jacobi (18041851) who invented an
algorithm for pairs of cubic irrational numbers, numerous
multidimensional continued fraction algorithms have been proposed.
In this talk the following related topics will be addressed.

Convergence results and Diophantine properties of multidimensional
continued fractions

Invariant measures for multidimensional continued fractions

Algebraic properties of multidimensional continued fractions.


RISC / INTS: Computational Number TheoryOctober 19, CWI Amsterdam. Lectures in room Z009 (ground floor)12:0013:00  Ronald van Luijk, Explicit twisting of Jacobians of dimension 2 Abstract. In order to bound the rank of the group of rational points A(k) on an abelian
variety A over a number field k, one often does a 2descent to bound the order
of the finite group A(k)/2A(k). This group injects into H^{1}(k,A[2]), where A[2]
denotes the group of 2torsion points. The elements
of this galois cohomological group correspond to certain twists of A, which are
varieties over k that become isomorphic to A over the algebraic closure of k.
Such a twist is in the image of A(k)/2A(k) if and only if it contains
a rational
point. In this talk I will first explain how twists correspond to
cocyles. Then I
will show how to find explicit equations of these twists as the intersection of
72 quadrics in P^{15} in the case that A is the Jacobian of a curve of genus 2.
 14:0015:00  Alexander May (Bochum), Using LLLReduction for Solving RSA and Factorization Problems: A survey Abstract. The talk addresses the problem of inverting the RSA function and the
problem of factorizing integers. We relax these problems in several
ways and show that the relaxations lead to polynomial time solvable
problems. In this approach, we model the relaxed problems as
polynomial equations which have roots of small size. The roots are
then found by a method originally introduced by D. Coppersmith in
1996, which in turn is based on the famous LLL lattice reduction
algorithm.We also present a novel application for RSA with socalled small CRT
exponents. Namely, we show that the factorization of an RSA modulus
N=pq can be found in polynomial time provided that RSA is used with a
secret exponent d such that both d (mod p1) and d (mod q1) are
smaller than N^{0.073}. The existence of such a polynomial time attack
answers a longstanding open problem by Wiener.  15:1516:15  David Freeman, Constructing abelian varieties for pairingbased cryptography Abstract. In recent years, the Weil and Tate pairings on abelian varieties over
finite fields have been used to construct a vast number of new and
useful cryptosystems. The abelian varieties used in these systems
must have small embedding degree with respect to a large primeorder
subgroup. Such "pairingfriendly" abelian varieties are rare and
thus require specific constructions.In this talk we describe our two recent contributions to the
catalogue of pairingfriendly abelian varieties: ordinary elliptic
curves of prime order with embedding degree 10, and ordinary abelian
surfaces over F_{q} having arbitrary embedding degree with
respect to a prime subgroup of order roughly q^{1/4}. Both results
require finding curves with complex multiplication by a specified CM
field; making this step feasible while maintaining the
pairingfriendly property is the difficult part of such constructions. 

Intercity Number Theory SeminarNovember 9, VU Amsterdam. Room change: the first lecture will take place in room Q105, the second in room C147, and the third in Q112. All rooms are in the "W&N gebouw", number 1081 on this map.12:0012:45  Rob de Jeu, Part I: introduction to K_{0}, K_{1}, K_{2}  13:3014:15  Rob de Jeu, Part II: K_{2} of curves over number fields Abstract. We start by giving an introduction to K_{0}, K_{1} and K_{2} of rings, with an
emphasis on arithmetic. The second talk will concentrate on a special case
of a conjecture by Beilinson, relating K_{2} of a curve over a number field
with the value of its Lfunction at 2.
 14:3016:15  Hendrik Lenstra, Algorithms for ordered fields Abstract. The lectures address the following problem from an
algorithmic perspective. Given a finite extension of an
ordered field, how can one, in a reasonably explicit manner,
write down all orderings of the extension field that extend
the given ordering of the base field?


Special day on discrete tomographyNovember 23, Leiden. Room 403. Tentative program:11:4512:30  Rob Tijdeman, Words with any periods Abstract. Let be given positive integers n, p_{1}, ..., p_{k}.
In the lecture an algorithm is presented which computes a word of length n
with periods p_{1}, ...,p_{k} and among such words one with the maximum number
of distinct letters. The algorithm has linear runtime. Some further
properties of such words are mentioned. This concerns joint work with Luca
Zamboni.
 13:3014:15  Sierk Rosema (Leiden), Betasubstitutions, cutting paths and their projections Abstract. A βsubstitution σ is a particular type of substitution over an
alphabet of k+1 letters.
By projecting the integer points on the cutting path that corresponds to
σ^{n}(0), we form a kdimensional rotation word v^{(n)}.
We define a function φ for which we prove that v^{(n+1)}=φ(v^{(n)})
for every n∈N.
 14:3015:30  Birgit van Dalen, Dependencies between line sums Abstract. In discrete tomography an integer matrix is reconstructed from the line
sums in several directions. Between those line sums exist linear relations.
We consider the problem of finding these relations. We can distinguish
between socalled global and local dependencies. In this talk the local
dependencies will be constructed for any set of directions.
 15:4516:45  Arjen Stolk, An algebraic approach to line sum dependencies Abstract. We rephrase the problem of dependencies between line sums in an algebraic way.
Using this point of view, we have a good handle on the global dependencies
(those which do not depend on the shape of the table). We compute the number of
independent global dependencies and indicate how one could construct a basis.


Intercity Number Theory SeminarDecember 7, Utrecht. The first talk will be in room BBL 160 (Buys Ballot Lab, Uithof), the others in BBL 105.11:0012:00  Victor Abrashkin (Durham), An elementary approach to Breuil's classification of finite flat pgroup schemes Abstract. In 2000 Breuil obtained a classification of finite flat commutative pgroup
schemes over valuation rings of complete discrete valuation fields of characteristic 0 with perfect residue field of characteristic p. This classification is based on a very extensive use of crystalline technique, in particular, it uses the BreenBerthelotMessing crystalline Dieudonne
theory. In the talk it will be explained a direct way how to establish
the main ingredient of this classification  the case of group schemes
killed by p.  13:0014:00  Cameron Stewart, On a refinement of the ABC conjecture Abstract. We shall present a refinement of the original ABC
conjecture of Masser and Oesterle together with a heuristic
justification for the refinement.  14:1015:00  Frits Beukers, Nearby perfect powers Abstract. Let m, n be coprime integers >1. According to
the ABCconjecture distinct mth and nth
powers of integers have a distance of order at least
X^{(11/m1/nε)}, where X is the size of the two
powers. We consider the question for which exponents c we can find
an infinite number of nth and mth powers with distance O(X^{c}).  15:3016:30  Attila Bérczes, On pairs of polynomials and binary forms with given resultant Abstract. Let F and G be two polynomials or binary forms and
denote by R(F,G) their resultant. Let S be a set of primes and
denote by Z_{S} the ring of Sintegers. In the talk we
consider resultant equations of the type
R(F,G) = c to be solved in polynomials F,G with coefficients from
Z_{S}, where c is a positive integer. 

Special day on cryptologyDecember 21, Leiden. The first two talks will be in room 174. The inaugural lecture of Ronald Cramer will take place in the Poortgebouw, Rijsburgerweg 10, Leiden (see map). To attend this lecture, please register here and be present 10 minutes in advance. This day is organized jointly with the RISC seminar.11:3012:30  Ivan Damgaard, The Past, Present and Future of Secure MultiParty Computation  13:3014:30  Yuval Ishai, Secure MultiParty Computation in the Head  16:1517:15  Ronald Cramer, Inaugural lecture (in Dutch) 


