Masters thesis presentations Niels uit de Bos: Deriving general (non-additive) functors Raymond van Bommel: Almost all hyperelliptic Jacobians have a bad semi-abelian prime Giulio Orecchia: Torsion-free rank one sheaves on a semi-stable curve Rosa Winter: Concurrent exceptional curves on del Pezzo surfaces of degree one Andrea Pasquali: Representations of SL_{2} and GL_{2} in defining characteristic Riccardo Ferrario: Galois closures for monogenic degree-4 extensions of rings
2/6/2014 (Mon)
16:00-17:00
Dijana Kreso (Graz) Rational function decomposition and Diophantine equations
For any rational function f(x) with coefficients in a field, we examine the structure of an expression of f(x) as the functional composition of rational functions that have coefficients in the same field, are of degree at least 2 and cannot be written as the composition of lower-degree rational functions with coefficients in the same field. Under certain hypotheses, we exhibit several invariants of any such decomposition, by means of Galois theory and group theory. We further explain the consequences of our general results for some well-studied classes of rational functions. Results on polynomial decomposition, obtained by Ritt in 1920's, have applications to various areas of mathematics. These include Bilu and Tichy's classification of all polynomials f(x) and g(x) with rational coefficients such that the equation f(x)=g(y) has infinitely many integer solutions. We show how their result applies to Diophantine equations. These results come from a joint work with Michael Zieve.
26/5/2014 (Mon)
16:00-17:00
Ziyang Gao (Orsay/Leiden) The Ax-Lindemann theorem for the universal family of abelian varieties
The Ax-Lindemann theorem is a functional algebraic independence statement,
which generalizes (the analog of) the classical Lindemann-Weierstrass
theorem. This theorem plays a key role in the Pila-Zannier method of
proving the mixed Andre-Oort conjecture. Klingler-Ullmo-Yafaev have
recently proved the Ax-Lindemann theorem for all pure Shimura varieties.
Using their result, I proved it for all mixed Shimura varieties. The
o-minimal theory, in particular the counting theorems of Pila-Wilkie, is
very important for all the proofs.
In this talk, I will focus on the universal family of abelian varieties. I
will explain how to view this family as a mixed Shimura variety, give the
statement of Ax-Lindemann (and Ax of log type) and then present the
strategy of the proof. The differences between my proof (for the mixed
case) and the proof of KUY (for the pure case), as well as the use of
Pila-Wilkie, will also be explained.
22/5/2014 (Thu)
11:00-12:00 Sn 412?
Daniel Loughran (Hannover) Good reduction of complete intersections
In 1983, Faltings proved the famous Mordell conjecture on the finiteness of the set of rational points on curves of higher genus. Along the way he proved numerous other finiteness statements, including the Shafarevich conjecture, which states that there are only finitely many curves of higher genus over a number field which have good reduction outside any given set of prime ideals. In this talk we shall consider analogues of the Shafarevich conjecture for certain classes of Fano varieties given as complete intersections in projective space. This is joint work with Ariyan Javanpeykar.
19/5/2014 (Mon)
17:00-18:00
Jon Gonzalez Sanchez (Bilbao) From p-groups to pro-p groups and back
A finite p-group is a group whose order is a power of a prime. Pro-p groups are constructed as inverse limits of finite p-groups. In this talk we discuss the following two problems:
A) How to construct a finite p-group with "small" automorphism group?
B) Is it true that a torsion-free p-adic pro-p group G satisfying that the number of generators G equals the dimension of G has a natural structure of Lie algebra?
19/5/2014 (Mon)
15:45-16:45
Mima Stanojkovski (Leiden) Evolving groups and intensity
Let G be a finite group. We say that G is evolving if there are two nilpotent groups N and T of coprime orders such that G equals their semidirect product and every subgroup of N has a T-stable conjugate. It follows that nilpotent groups are evolving, but can we construct non-nilpotent examples? During the talk we will see how to build such examples starting from a suitable p-group N of class at most 3.
14/4/2014 (Mon)
16:00-17:00
Rostislav Devyatov (Berlin-Leiden) Equivariant infinitesimal deformations of a special class of
three-dimensional T-varieties
A T-variety is a normal and (in this talk) affine variety
with an action of an algebraic torus. Unlike toric varieties, the
action of the torus on the T-varieties does not have to have an open
orbit, the dimension of the torus may be smaller.
The set of deformations of any affine variety X over the double point
admits a "natural" structure of a vector space. The word "natural"
will not be clarified in the talk, we will "just use" this vector
space structure. If a torus acts on X, it also acts on this vector
space, and the fixed points of this action are exactly the
deformations that admit torus action on the fibers, i. e. equivariant
deformations. In the talk, we will study this vector space
corresponding to infinitesimal deformations and find its dimension.
8/4/2014 (Tue)
14:15-16:00 Sn 412
Jan Bouw (Leiden) An algorithm to compute norm residue symbols
This talk is about the computation of the norm residue symbol in the context of local class field theory. There is a polynomial time algorithm that computes this symbol. I will give an overview of the methods used to determine its exact value.
24/3/2014 (Mon)
16:00-17:00
Niels Lindner (Berlin-Leiden) Density of quasismooth hypersurfaces in simplicial toric varieties
Given a smooth projective variety X and a divisor D, one can ask for the proportion of smooth elements in the linear system corresponding to D. For algebraically closed fields, the answer is known by Bertini's theorem. In 2004, B. Poonen proved an analogue for finite fields, which relates the density of smooth hypersurfaces in projective space intersecting a fixed subvariety X smoothly with the Hasse-Weil zeta function of X. In this talk, it will be explained how Poonen's result can be extended to nice enough toric varieties over a finite field, relaxing "smooth" to "quasismooth".
17/3/2014 (Mon)
16:00-17:00
Giulia Battiston (Berlin-Leiden) The monodromy of stratified bundles in positive characteristic
In positive characteristic stratified bundles play the role that in characteristic zero is played by flat connections. The topological fundamental group of a complex manifold acts on the fibers of the flat connection via parallel transport giving the monodromy action and it is still possible to define a monodromy action in positive characteristic via the Tannakian formalism. We will in particular see how the monodromy group varies in smooth families.
10/3/2014 (Mon)
16:00-17:00
Owen Biesel (Leiden) A New Definition of Discriminant Algebra
For a finite separable field extension in characteristic other than 2, its discriminant field controls whether its Galois closure has Galois group contained in the alternating group. In the last decade, Rost, Deligne, and Loos have suggested generalizations of the discriminant field to a "discriminant algebra" defined for locally free constant-rank algebras over a base ring. We present a new construction of a discriminant algebra, show that it satisfies the properties outlined by Deligne, and discuss its relation to other constructions by Rost and Loos. This is joint work with Alberto Gioia.
6/3/2014 (Thu)
16:00-17:00
Elmar Grosse Klönne (Berlin) A partial generalization of Colmez' functor
Abstract: In Colmez' work on the mod-p local Langlands correspondence
for GL_{2}(Q_{p}), an important ingredient is a certain
functor from smooth admissible mod p representations of
GL_{2}(Q_{p}) to mod p representations of
Gal(Q_{p}/Q_{p}). I want to discuss a partial
generalization of this functor for more general split reductive groups G
over Q_{p}. This functor is defined for smooth admissible mod
p representations V of G(Q_{p}), but (in its present state)
it is sensitive only to the space of invariants of V under a
pro-p-Iwahori subgroup. For G=GL_{n} it induces a bijection
between the set of isomorphism classes of simple supersingular
n-dimensional modules over the mod p pro-p-Iwahori Hecke algebra for
G(Q_{p}), and the set of isomorphism classes of
irreducible n-dimensional mod p-representations of
Gal(Q_{p}/Q_{p}).
3/3/2014 (Mon)
16:00-17:00
Dirk Basson (Leiden) Drinfeld modular forms of higher rank
Drinfeld modular forms are functions that closely correspond to classical
(elliptic) modular forms. However, the domain and codomain of these
functions have positive characteristic. The rank 2 (or 1-dimensional) case
has been studied since the 1980s and analogues for many properties of
classical modular forms are known. Due to the non-triviality of
compactifying the higher dimensional moduli space of rank r ≥ 3
Drinfeld modules, modular forms of higher rank have only been defined
recently and little is known about them. This talk will be about
elementary properties of these functions, like the coefficients in their
Fourier expansions "at infinity".
10/2/2014 (Mon)
16:00-17:00
José Ignacio Burgos Gil (Madrid) The singularities of the invariant metric of the sheaf of Jacobi forms on the universal elliptic curve.
A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, provided with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key to being able to compute arithmetic intersection numbers from these line bundles.
Hence it is natural to ask whether Mumford's result remains valid for line bundles on mixed Shimura varieties.
In this talk we will examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We will show that Mumford's result can not be extended to this case and that a new interesting kind of singularities appear that are related to the phenomenon of Height jumping introduced by Hain.
We will discuss some preliminary results. This is joint work with G. Freixas, J. Kramer and U. Kühn.
Fall 2013
25/11/2013 (Mon)
16:00-17:00
Rachel Newton (Bonn) Computing transcendental Brauer groups of products of CM elliptic curves
In 1971, Manin showed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the existence of points everywhere locally on X despite the lack of a global point is sometimes explained by non-trivial elements in Br(X).
Since Manin's observation, the Brauer group has been the subject of a great deal of research. The 'algebraic' part of the Brauer group is the part which becomes trivial upon base change to an algebraic closure of K. It is generally easier to handle than the remaining 'transcendental' part and a substantial portion of the literature is devoted to its study. On the other hand, there are comparatively few results concerning the transcendental part of the Brauer group.
The transcendental part of the Brauer group is known to have arithmetic importance – it can give non-trivial obstructions to the Hasse principle and weak approximation. I will use class field theory together with results of Ieronymou, Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the product ExE for an elliptic curve E with complex multiplication.
21/11/2013 (Thu)
16:00-17:00 Sn 312
Guillermo Mantilla (Lausanne) Weak arithmetic equivalence of number fields
Inspired by the invariant of a number field given by its Dedekind zeta function we define the notion of weak arithmetic equivalence, and we show that under certain ramification hypothesis this equivalence determines the local root numbers of the number field. This is analogous to a result Rohrlich on the local root numbers of a rational elliptic curve. Additionally we prove that for tame non-totally real number fields the integral trace form is invariant under weak arithmetic equivalence.
18/11/2013 (Mon)
16:00-17:00 B03
Ekaterina Amerik (Moscow, Paris) Some remarks about the dynamical Hasse principle
Inspired by Scharaschkin’s approach to the Brauer–Manin obstruction
for curves, Hsia, Silverman and Voloch have recently initiated the
study of the “dynamical Brauer-Manin obstruction to the dynamical Hasse
principle”. We shall explain what it is and make some remarks on this
in the particular case of etale self-maps.
This is a joint work with P. Kurlberg, K. Nguyen, A. Towsley, B. Viray and
F. Voloch.
11/11/2013 (Mon)
16:00-17:00
Sorina Ionica (Paris) Isogeny graphs with real multiplication
4/11/2013 (Mon)
16:00-17:00
Marco Streng (Leiden) Computing endomorphism rings of abelian varieties using class groups (joint work with Gaetan Bisson)
24/10/2013 (Thu)
15:45-16:45 Sn 312
Gabor Wiese (Luxembourg) On Galois representations and the Inverse Galois Problem
We give an account of joint work with Sara Arias-de-Reyna, Luis
Dieulefait and Sug Woo Shin, in which we realise, for every even n and
every d, the group PGSp_{n}(F_{pd}) or PSp_{n}(F_{pd}) as Galois group
over the rationals, for p in a set of primes of positive density. The
proof relies on compatible systems of automorphic Galois representations
with special local properties.
14/10/2013 (Mon)
16:00-17:00
Markus Perling (Bielefeld) Exceptional collections on (rational) surfaces
To any variety one can associate its derived category of
coherent sheaves. These categories are difficult to handle
in general, but understanding their rich structure brings
a big reward and their study has been subject of extensive
research in recent years. Among the most interesting aspects
is that one can use derived equivalences in order to translate
between (commutative) algebraic geometry and (noncommutative)
representation theory. This has been applied to study
non-commutative resolutions and the classification of
vector bundles.
An essential tool to study derived categories are so-called
semi-orthogonal decompositions. By now, there are a number
of results which show that such decompositions are deeply
tied to the underlying geometry. Exceptional collections
represent the simplest type of semi-orthogonal decompositions,
where each semi-orthogonal component is generated by just
one object. Though many examples are known and the existence
of exceptional collections imposes strong constraints on the
underlying geometry, their existence is an open problem in
general. In this talk we will consider exceptional collections
on smooth algebraic surfaces, and in particular on rational
surfaces. We will show that here the problem of constructing
exceptional collections is intrinsically related to toric
geometry and singularity theory.
7/10/2013 (Mon)
16:00-17:00
David Holmes (Leiden) Rational points on Kummer varieties and ranks of elliptic curves
19/9/2013 (Thu)
11:00-12:00 (!) Sn B3
Martin Bright (Beirut) Brauer groups of singular del Pezzo surfaces
16/9/2013 (Mon)
16:00-17:00
Tzanko Matev (Bayreuth) Algebraic dependencies of the multiplicative group
Spring 2013
18/2/2013 (Mon)
16:00-17:00
David Holmes (Leiden) Distribution of rational points on Kummer varieties
4/3/2013 (Mon)
16:00-17:00
Owen Biesel (Princeton) G-closures for ring extensions
7/3/2013
16:00-17:00
Cecília Salgado (Universidade Federal do Rio de Janeiro) Unirationality of del Pezzo surfaces of degree two
14/3/2013
16:00-17:00
Alex Bartel (Warwick University) The difference between rational representations and permutation representations of a finite group
18/3/2013 (Mon)
16:00-17:00
Hendrik Lenstra (Leiden) Lattices with symmetry, I
21/3/2013
15:45-17:30
Hendrik Lenstra (Leiden) Lattices with symmetry, II
25/3/2013 (Mon)
16:00-17:00
Jan Steffen Müller (Hamburg) Canonical heights on Jacobians of hyperelliptic curves
Algant pre-defense (Duong Hoang Dung defends his PhD thesis on May 14) Andrea Lucchini (Padova): Boundedly generated subgroups of finite groups Christopher Voll (Bielefeld): Zeta functions of groups and rings
16/5/2013
16:00-17:00
Paloma Bengoechea (Collège de France) From quadratic polynomials and continued fractions to modular forms
25/6/2013 (Tue) Sn 402
13:45-14:45 15:15-16:15 16:30-17:30
Olivier Wittenberg (ENS Paris, CNRS) On the cycle class map for zero-cycles over local fields Jinbi Jin (Leiden): Classification of torsors over curves David Madore (ENST Paris): Church-Turing computability of étale cohomology
26/6/2013 (Wed) Sn 402
15:00-17:15
Bhargav Bhatt (IAS Princeton) Crystalline and de Rham cohomology 1 & 2
3/7/2013 (Wed) Sn 402
15:00-17:15
Bhargav Bhatt (IAS Princeton) Crystalline and de Rham cohomology 3 & 4
4/7/2013
16:00-17:00
Wei Ho (Columbia) Families of lattice-polarized K3 surfaces