
Intercity number theory seminarFebruary 25, Utrecht. The lectures are in room BBL205 on the second floor of the Buys Ballot
building. There will be tea and coffee at 15:2011:3012:20  Tom Ward, Counting fixed points of group automorphism Abstract. Compact group automorphisms form a simple family of dynamical systems, and a
natural dynamical invariant is their periodic point data. Understanding this amounts to
knowing about how many fixed points an automorphism has, and this is a surprisingly subtle
question. This talk will give an overview of some recent approaches to this problem, and
some of the numbertheoretic issues that arise.  13:3014:20  Soroosh Yazdani, Local Szpiro's conjecture Abstract. Given an elliptic curve over the rationals, its conductor and minimal
discriminant are two integers that keep track of its primes of bad
reduction. A conjecture of Szpiro states (in a fairly explicit manner) that
the minimal discriminant can't be much larger than the conductor of a given
elliptic curve. It is known that this conjecture is equivalent to the ABC
conjecture, and has many applications to Diophantine equations. In this
talk, we present a similar conjecture which we call Local Szpiro's
Conjecture. Specifically, we conjecture that given any semi stable elliptic
curve with minimal discriminant Δ, one can always find a prime of bad
reduction such that v_{p}(Δ) ≤6. In this talk we present some
evidence and application for this conjecture. This is joint work with Mike
Bennett.
 14:3015:20  Sander Dahmen, Klein forms and the generalized superelliptic equation Abstract. Let F be a binary form over the integers and consider the exponential
Diophantine equation F(x,y)=z^{n} with x and y coprime. For general F it seems very difficult to solve this equation, but as we will explain in this talk, for socalled Klein forms F, the modular method can provide a good starting point. Together with solving certain infinite families of Thue equations, we can show in particular, that there exist infinitely many (essentially different) cubic forms F for which the equation above has no solutions for large enough exponent n. This is joint work with Mike Bennett.  15:4016:30  Jonathan Reynolds, Modular methods for perfect powers in elliptic divisibility sequences Abstract. For Mordell curves, I will explain why there are finitely many perfect
powers in an elliptic divisibility sequence whose first term is greater than 1.
The proof uses recent modular methods by Bennett, Billerey, Dahmen, Vatsal and
Yazdani. 

Intercity number theory seminarMarch 11, VU Amsterdam. room M623 (W&N building)11:1512:15  James Lewis, An Archimedean height pairing on the equivalence relation defining Bloch's higher Chow groups Abstract. The existence of a height pairing on
the equivalence relation defining Bloch's higher Chow
groups is a surprising consequence of some
recent joint work by myself and Xi Chen on
a nontrivial K_{1}class on a selfproduct
of a general K3 surface. I will explain how this
pairing comes about.  13:1515:15  JeanLouis ColliotThelene, Lectures on the Hasse principle, I Abstract. Hasse principle and weak approximation :
elementary fibration method, BrauerManin obstruction for
rational and integral points, examples.
To which extent can we compute the Brauer group
and the BrauerManin set?  15:3016:30  Ronald van Luijk, Computability of Picard numbers Abstract. The NéronSeveri group of a variety is the group of its
divisor classes modulo algebraic equivalence. The rank of this group
is called the Picard number of the variety. After giving a short
review of ad hoc methods that compute the Picard number in certain
cases, I will sketch an idea of Bjorn Poonen to prove that the Picard
number is computable in general. This is joint work in progress with
Damiano Testa.


Intercity number theory seminarMarch 25, Leiden. room 41211:1512:15  Damiano Testa, The surface of cuboids and Siegel modular threefolds Abstract. A perfect cuboid is a parallelepiped with rectangular faces all of
whose edges, face diagonals and long diagonal have integer length. A
question going back to Euler asks for the existence of a perfect
cuboid. No perfect cuboid has been found, nor it is known that they
do not exist. In this talk I will show that the space of cuboids is a
divisor in a Siegel modular threefold, thus allowing to translate the
existence of a perfect cuboid to the existence of special torsion
structures in abelian surfaces defined over number fields.
 13:1515:15  JeanLouis ColliotThelene, Lectures on the Hasse principle, II Abstract. Hasse principle and weak approximation :
classes of varieties for which they hold
(in particular, some intersections of two quadrics).
Some classes for which they do no hold.
Interpretation by means of the BrauerManin obstruction.  15:3016:30  Martin Bright, Some computational aspects of the BrauerManin obstruction Abstract. Given a surface X over a number field, how far can we go towards
computing the BrauerManin obstruction on X? I have been implementing
algorithms in this direction with the Magma group in Sydney, and several
interesting problems in computational arithmetic geometry have come up
along the way. I will discuss a selection of these problems. 

Intercity number theory seminarApril 8, Leiden. room 412At 17:00 there will be drinks offered at the Foobar to celebrate that
Martin Lübke has been at the department for 25 years. 11:4512:45  Christine Berkesch, The rank of a hypergeometric system Abstract. Gelfand, Kapranov, and Zelevinsky's introduction of a torus action to
the study of hypergeometric systems brought a wealth of combinatorial
and geometric tools to the theory. They showed that the dimension, or
rank, of the solution space of such a system is constant for generic
parameters. I will discuss homological tools developed by Matusevich,
Miller, and Walther, which can be used to obtain a combinatorial
formula for the rank of a hypergeometric system at any parameter.  13:4515:45  JeanLouis ColliotThelene, Lectures on the Hasse principle, III Abstract. Continuation of Lecture III. How to compute the Brauer group.
Harari's formal lemma, applications.
Torsors under algebraic groups and descent.
Beyond the BrauerManin obstruction.
 16:0017:00  Andrei Teleman, Holomorphic bundles and holomorphic curves on class VII surfaces. The classification problem for class VII surfaces. Abstract. The classification of complex surfaces is not finished yet.
The most important gap in the KodairaEnriques classification table
concerns the Kodaira class VII, e.g. the class of surfaces X having
kod(X)=∞, b_{1}(X)=1. These surfaces are interesting from a
differential topological point of view, because they are nonsimply
connected 4manifolds with definite intersection form. The main
conjecture which (if true) would complete the classification of class
VII surfaces, states that any minimal class VII surface with b_{2}>0
contains b_{2} holomorphic curves. We explain a new approach, based on
ideas from Donaldson theory, which gives existence of holomorphic
curves on class VII surfaces with small b_{2}. In particular, for
b_{2}=1 we obtain a proof of the conjecture, and for b_{2}=2 we prove
the existence of a cycle of curves.


Intercity number theory seminarApril 29, Groningen. room 105 Bernoulliborg11:4512:45  Karl Rökaeus (Amsterdam), Global function fields with many rational places Abstract. The Riemann hypothesis for curves (proved by Weil) give an upper bound on how many rational places there can be in a global function field of genus g with constant field of cardinality q. This bound is rarely met; when g/q is big it can be improved substantially and in general it is unknown how far it is from being achieved. For small q (mostly powers of 2 and 3) and g<50 much work has been done on constructing fields with many places, and this combined with work improving the upper bounds has determined the maximum possible number of places in such a fields, or a small interval in which it lies.These intervals are recorded on the webpage manYPoints.org.
In this talk we discuss some (classical) methods from class fields theory that can be used for constructing fields with many places. We then apply them in a systematic computer search. This yields many new lower bounds in cases that had not been much investigated earlier; it also gives some new fields in characteristic 2 and 3.  13:3015:45  JeanLouis ColliotThelene, Lectures on the Hasse principle, IV Abstract. The Brauer group may be used to define a BrauerManin
obstruction to the existence
of a zerocycle of degree 1. For arbitrary smooth projective
varieties over a global field,
is this the only obstruction?  16:0017:00  Remke Kloosterman, Calculating the MordellWeil group for a class of elliptic threefolds Abstract. Let K be a subfield of the complex numbers.
In the first part of the talk I will give a heuristic argument why the calculation of the MordellWeil rank of a "general" elliptic curve over K(s,t) is easier than the calculation of the MordellWeil rank of a "general" elliptic curve over K(t).
I will illustrate this by discussing a particular class of elliptic curves over K(s,t) with constant jinvariant 0. For this class of elliptic curves we will give an upper bound for the MordellWeil rank in terms of the degree of the discriminant of the elliptic curve that is roughly half a naive upper bound for the rank.
We get this bound by considering the syzygies of the ideal of the singular locus of the discriminant curve.
These consideration lead to a lot of interesting byproducts, i.e., we obtain an example of a "Zariski triple", and we obtain examples of triples (g,k,n) such that the locus {[C] ∈M_{g}  C admits a g^{2}_{6k} and the image of C has at least n cusps } ⊂M_{g} has much bigger dimension than expected. 

Intercity Number Theory SeminarMay 13, Utrecht. room 211, Minnaert buildingToday's theme: applications of measure theory in number theory 11:3012:20  Janne Kool (Utrecht), Measure theoretic rigidity for Mumford curves Abstract. Measure theoretic rigidity (in the style of Mostow) states that two
compact hyperbolic Riemann surfaces of the same genus are isomorphic if and only
if the "boundary map" associated to their uniformizations is absolutely
continuous. In this talk, we will formulate the analog of this result for
"padic Riemann surfaces", i.e., for Mumford curves. In this case, the mere
absolute continuity of the boundary map implies only isomorphism of the special
fibers of the Mumford curves, and needs to be enhanced by a finite list of
conditions on the harmonic measures (in the sense of Schneider and Teitelbaum)
on the boundary to guarantee an isomorphism of the Mumford curves.  13:3014:20  Tom Ward, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture Abstract. This will be a short introduction to some of the ideas used in the
work of Einsiedler, Katok and Lindenstrauss on the Littlewood problem. This will
include the connection between the modular surface and continued fractions and
an introduction to measure rigidity.  14:4015:30  Tanja Eisner, Arithmetic progressions via ergodic theory Abstract. We sketch the development from van der Waerden's theorem on
arithmetic progressions to the recent GreenTao theorem and show how methods
from ergodic theory have been decisive in this field.  15:4016:30  Karl Petersen, Some combinatorial and numbertheoretic questions related to certain adic dynamical systems Abstract. Trying to figure out dynamical properties of particular systems
often leads to interesting (and sometimes very difficult) questions in
combinatorics or number theory. This has certainly been the case with the
Pascal, Euler, faux multidimensional Euler, and Delannoy adic systems, which
have led to results, questions, and conjectures about binomial coefficients,
Eulerian numbers, Stirling numbers, and Delannoy numbers. We review how these
matters arise and some of what we know and would like to know about them. 

Intercity Number Theory SeminarJune 10, Gent. Lecture room "Emmy Noether", campus Sterre S25, Galglaan 2, Gent. This is a 10
minute walk or 5 minute tram ride (tram 21 or 22) from train station "Gent SintPieters",
see directions and maps.
If you want to participate in the (free) lunch buffet at 1pm, please register by sending an email with your name to Jan Van Geel (jvg@cage.ugent.be) before 7 June. You can also contact him for information on staying overnight in Gent.
13:3014:50  Gunther Cornelissen, Reconstructing number fields using quantum statistical mechanics Abstract. I will start with an overview of the history of (not) reconstructing
number field isomorphism from equality/isomorphism of invariants such as zeta
functions, adele rings and abelian/absolute Galois groups. Then I will discuss
joint work with Matilde Marcolli that reconstructs isomorphism of global fields
from isomorphism of associated quantum statistical mechanical systems (which are
certain dynamical systems derived from class field theory), and how this implies
that abelian Lseries determine the isomorphism type of a global field.  15:0015:50  Bart de Smit, Characterizing number fields with abelian Lfunctions Abstract. The main result of this lecture is the following. Every number field K has a cubic character whose Lfunction occurs only as an abelian Lfunction over fields that are isomorphic to K. This does not hold with "cubic" replaced by "quadratic": there is a number field K so that every quadratic Lfunction over K is equal to a quadratic Lfunction over a field that is not isomorphic to K.  16:1017:30  Frits Beukers, Ahypergeometric functions Abstract. Hypergeometric functions are among the most familiar classical
functions in mathematics. They play an important role in
parts of analysis, geometry, number theory and of course
mathematical physics. This is an introductory lecture starting
with some elementary aspects of Gauss's classical
hypergeometric function. We then extend these aspects to
the case of several variable hypergeometric functions.
We do this in the extremely elegant setting
provided by the socalled Ahypergeometric functions.
These were introduced Gel'fand, Kapranov and Zelevinsky
at the end of the 1980's. In particular we shall pay attention
to analytic continuation and monodromy of these functions.


Intercity Number Theory SeminarSeptember 16, Leiden. Room 40313:3014:30  Alberto Facchini, KrullSchmidt Theorem: the case two Abstract. I will mainly present the content of a joint paper with Pavel
Prihoda (The KrullSchmidt Theorem in the case two, Algebr. Represent. Theory
14(3) (2011), 545570), but also other results obtained jointly with A. Amini,
B. Amini, S. Ecevit, M. T. Kosan and N. Perone. Essentially, the
KrullSchmidtAzumaya Theorem says that if M_{1},...,M_{m},N_{1},...,N_{n} are
Rmodules with local endomorphism rings and M_{1}⊕...⊕M_{m}≅N_{1}⊕... ⊕N_{n}, then n=m and there exists a permutation σ
of {1,...,n} such that M_{i}≅N_{σ(i)} for every i=1,...,n.
I will present what happens if the endomorphism rings of the modules M_{i} and
N_{j} have two maximal ideals instead of only one. Several examples of these
modules will be given.
 14:4515:45  Antonella Perucca, Radical characterizations of elliptic curves Abstract. Let K be a number field, and let E be an elliptic curve over K. A
famous result by Faltings of 1983 can be reformulated for elliptic
curves as follows: if S is a set of primes of good reduction for E
having density one, then the Kisogeny class of E is determined by the
function which maps a prime p in S to the size #E(k_p) of the group of
points over the residue field. We prove that it suffices to look at
the radical of the size. We also replace E(k_p) by the image of the
MordellWeil group via the reduction modulo p, and solve this
analogous problem for a large class of abelian varieties. This is a
joint work with Chris Hall.
 16:0017:00  René Schoof, Integral points on a modular curve of level 11 Abstract. Using lower bounds for linear forms in elliptic logarithms we determine the
integral points of the modular curve associated to the normalizer
of a nonsplit Cartan group of level~11.
As an application we obtain a new solution of the class number one problem
for complex quadratic fields. This is joint work with Nikos Tzanakis. 

Intercity Number Theory SeminarSeptember 30, Delft. Snijderszaal (zaal LB 01.010, 1e verdieping EWI gebouw)
13:0014:00  Valérie Berthé, Adic constructions for fundamental domains for Kronecker sequences Abstract. The aim of this lecture is to provide explicit constructions for fundamental domains for Kronecker sequences
in order to get arithmetic codings of the corresponding dynamical system (the underlying toral translation)
that preserve their arithmetic properties.
It is wellknown how to associate such fundamental domains with algebraic (and more precisely Pisot) parameters,
when the Kronecker sequence is generated by a substitution (a substitution is a symbolic action that replaces letters by words).
These fundamental domains are called Rauzy fractals or central tiles. It is more delicate to consider nonalgebraic parameters.
To this end, substitutive symbolic dynamical systems can be extended to socalled Sadic systems governed by multidimensional
continued fraction algorithms. These latter systems are obtained by iterating not only one substitution, but a finite number of them.
The hierarchies thus produced have not necessarily the same structure at each level, but there are only finitely many possible
structures.
Our aim here is under suitable convergence properties that play the role of the Pisot assumption
to define generalized Rauzy fractals with tiling properties in this framework.
This a joint work with W. Steiner and J. Thuswaldner.
 14:1014:55  Milan Lopuhaä (Leiden), Field topologies on countable fields Abstract. In 1972, it was proven that any field K admits exactly 2^(2^K) field
topologies on it, i.e. topologies such that all field operations are
continuous. In my Bachelor Thesis, I modified the proof of the countable
case to show that in the case of an algebraic closure of a finite field,
these topologies can be made in such a way that all field automorphisms are
continuous, and the proof of this will be shown in this seminar.
 14:5515:40  Henk Don, Polygons in billiard orbits Abstract. We study the geometry of billiard orbits on rectangular billiards. A truncated
billiard orbit induces a partition of the rectangle into polygons. We prove that
thirteen is a sharp upper bound for the number of different areas of these polygons.


Intercity Number Theory SeminarOctober 28, Leuven. Building 200C, 01.0010 (first floor); see this campus map [PDF] for directions.13:3014:30  Wouter Castryck, Frobenius statistics for varying curves over fixed finite fields Abstract. Fix a large prime power q and an integer g>0. Let C be a "random" genus g
curve over the finite field GF(q). Let Jac(C) be the group of rational
points on its Jacobian. For various reasons, one can wonder about

the probability that #Jac(C) is divisible by N (for a given integer N)
 the probability that #Jac(C) is prime,
 the probability that Jac(C) is cyclic,
 ...
 the asymptotic behavior of these probabilities for g going to infinity.
We will show how all of this can be estimated (sometimes heuristically,
sometimes provably) using a random matrix framework. This is joint work with
Amanda Folsom, Hendrik Hubrechts, and Andrew Sutherland.  14:4515:45  Gabriele Dalla Torre, Lfunctionpreserving isomorphisms of groups of quadratic characters Abstract. Let Mer(C) be the set of meromorphic functions on C.
Given a number field K, we denote by L_{K}: K^{*}/K^{*2} →Mer(C) the function that maps (d mod K^{*2}) to the quadratic Lfunction associated to the quadratic extension K(√d) of K.
The main result of this lecture is the
following theorem. Let K and K' be number fields. Then the natural
map from the set of field isomorphisms K →K' to the set of
group isomorphisms β: K^{*}/K^{*2} →K'^{*}/K'^{*2} with the
property that L_{K'} o β= L_{K}, is bijective.  16:1517:15  Peter Jossen, The unipotent part of the MumfordTate conjecture Abstract. The MumfordTate conjecture is a statement which compares singular
cohomology with ladic cohomology of algebraic varieties, classically
abelian varieties, defined over finitely generated fields of characteristic
zero. I will show that a small but already useful part of this conjecture is
true, and can be formulated in positive characteristic as well. 

RISC/Intercity Number Theory SeminarNovember 11, CWI Amsterdam. Last minute room change: the lectures are in room *L120* (first floor, new wing). See also the RISC page.11:4512:30  Ariel Gabizon (Israel Institute of Technology  Technion / CWI), Extractors : Background, Applications and Recent Constructions Abstract. Randomness extractors are functions whose output is guaranteed to be uniformly distributed, given some assumption on the distribution of the input. The first instance of a randomness extraction problem comes from vonNeumann who gave an elegant solution to the following problem: How can a biased coin with unknown bias be used to generate `fair' coin tosses? In this case the input distribution consists of independent identically distributed bits. Since then many families of more complex input distributions have been studied. Also, the concept of randomness extraction has proven to be useful for various applications. The talk will give some background on extractors and review applications and techniques used in recent constructions of extractors.  14:0014:45  Gil Cohen (Weizmann Institute of Science), NonMalleable Extractors with Short Seeds and Applications to Privacy Amplification Abstract. Motivated by the classical problem of privacy amplification, Dodis and Wichs (STOC '09) introduced the notion of a nonmalleable extractor, significantly strengthening the notion of a strong extractor. A nonmalleable extractor is a function nmExt that takes two inputs: a weak source W and a uniform (independent) seed S, and outputs a string nmExt(W,S) that is nearly uniform given the seed S *as well* as the value nmExt(W,S') for any seed S' \neq S that may be determined as an arbitrary function of S. In this work we present the first unconditional construction of a nonmalleable extractor with short seeds. By instantiating the framework of Dodis and Wichs with our nonmalleable extractor, we obtain the first 2round privacy amplification protocol for minentropy rate 1/2 + delta with asymptotically optimal entropy loss and polylogarithmic communication complexity. This improves the previously known 2round privacy amplification protocols: the protocol of Dodis and Wichs whose entropy loss is not asymptotically optimal, and the protocol of Dodis, Li, Wooley and Zuckerman whose communication complexity is linear and relies on a numbertheoretic assumption. Joint work with Ran Raz and Gil Segev.  15:0015:45  Stefan Dziembowski (Rome), LeakageResilient Cryptography From the InnerProduct Extractor  16:0016:45  Christian Schaffner (UVA/CWI), Randomness extraction and expansion in the quantum world Abstract. Randomness extraction is a fundamental task in cryptography, where it is intimately connected with the problem of privacy amplification. In this talk we will survey the specific challenges posed by this task in the setting where an adversary may hold *quantum* information about the source and give an overview over the known results in the area. In the last part, we touch on recent joint work with Fehr and Gelles. We demonstrate that quantum mechanics allows to expand some initial randomness in a secure way even if the used devices are manufactured by the adversary. 

Intercity Number Theory SeminarDecember 9, Leiden. Room 17413:3014:30  Pascal Autissier, On the nondensity of integral points Abstract. We give nondensity results for integral points on affine varieties,
in the spirit of
the LangVojta conjecture. In particular, le tX be a projective
variety of dimension
d>1 over a number field K (resp., over C). Let H be the sum of 2d properly
intersecting ample divisors on X. We show that any set of
quasiintegral points
(resp., any integral curve) on XH is not Zariski dense.  14:4515:45  Fabrizio Andreatta, On padic families of elliptic overconvergent modular forms Abstract. I will report on a joint project with A. Iovita and G. Stevens.
I will show how to construct families of overconvergent elliptic
modular forms as global sections of suitable
"modular sheaves" even for nonintegral weights. These are defined by
correcting the HodgeTate map using the theory of the canonical
subgroup.
I will show how we can explain/recover the theory of Coleman in a way
which can be generalized to other groups beside GL_{2,Q}.
 16:0017:00  JanHendrik Evertse, Multiply monogenic orders Abstract. An order in an algebraic number field K is a subring of the ring of
integers of K which has quotient field K.
An order generated over Z by one element, i.e. of the shape Z[w],
is called monogenic. Given an order O, the set of w with Z[w]=O
falls naturally into equivalence classes, where two elements v, w
of O are called equivalent if vw or v+w is a rational integer.
In 1976, Györy proved that for any number field K, and any order
O in K, there are only finitely many equivalence
classes of w in O such that O=Z[w].
An order O for which there are at least k equivalence classes of w with
Z[w]=O is called k times monogenic; if there are precisely/at most
k such equivalence classes, it is called precisely/at most
k times monogenic. For instance any order in a quadratic number field
is precisely one time monogenic.
In this talk I fix a number field of degree at least 3 and consider
varying orders in this field. The first main result is, that every
number field K of degree at least 3 has at most finitely many
orders which are three times monogenic.
This bound 3 is best possible.
The second main result is, that
under some additional constraints imposed on K,
there are only finitely many two times monogenic orders in K
which are not 'of a special type.'
This is joint work with Attila Bérczes and
Kálmán Györy. 

Intercity Number Theory SeminarDecember 16, Groningen. The first two lectures will be in the Heymanszaal in the Academiegebouw, Broerstraat 5, in Groningen.
The PhD defense will be in the Aula in the same building.11:0012:00  Matthias Schuett, Arithmetic of quintic surfaces Abstract. The Picard number is a nontrivial invariant of an algebraic
surface which captures much of its inner structure. We will discuss the
fundamental problem which Picard numbers occur on surfaces on general
type for the prototype example of quintics in IP^3. We review what
seems to be known and explain a new technique based on arithmetic
deformations.
 13:0014:00  Tetsuji Shioda, Cubic surfaces via MordellWeil lattices, rerevisited  14:3015:30  Bas Heijne, PhD defense: Elliptic Delsarte Surfaces. 


