Intercity Number Theory SeminarMarch 1, Groningen. The first talk takes place in room 165 of the Bernoulliborg and the
three other talks take place in room 253.
|12:00-13:00||Dino Festi (Mainz), A method to compute the geometric Picard lattice of a K3-surface of degree 2|
Abstract. K3 surfaces are surfaces of intermediate type, i.e., they
are in between surfaces whose arithmetic and geometry is fairly well
understood (rational and ruled surfaces) and surfaces that are still
largely mysterious (surfaces of general type). The Picard lattice of a
K3 surface contains much information about the surface, both from a
geometric and an arithmetic point of view. For example, it tells about
the existence of elliptic fibrations on the surface; if the surface is
over a number field, then by looking at the Picard lattice one can
have information about the Brauer group, and the potential density of
rational points. Although much effort, there is not yet a practical
algorithm that, given an explicit K3 surface, returns the Picard
lattice of the K3 surface. In this talk we are going to give an
overview on how practically compute the geometric Picard lattice of a
K3 surface of degree two over a field of characteristic zero.
|13:45-14:45||Florian Hess (Oldenburg), Partially euclidean global fields|
|15:00-16:00||Marc Paul Noordman (Groningen), Algebraic first order differential equations|
Abstract. Autonomous algebraic first order differential equations (i.e.
differential equations of the form P(u, u') = 0 for P a polynomial
with constant coefficients) can be interpreted as rational
differential forms on an algebraic curve. In this talk, based on joint
work with Jaap Top and Marius van der Put, I will explain how this
perspective clarifies the possible algebraic relations between
solutions of such differential equations. In particular, we will see
that there are almost no algebraic relations between distinct
non-constant solutions of the same differential equation, unless that
differential equation comes from a one-dimensional group variety.
|16:15-17:15||Jan Vonk (Oxford), Singular moduli for real quadratic fields|
Abstract. The theory of complex multiplication describes finite
abelian extensions of imaginary quadratic number fields using singular
moduli, which are special values of modular functions at CM points. I
will describe joint work with Henri Darmon in the setting of real
quadratic fields, where we construct p-adic analogues of singular
moduli through classes of rigid meromorphic cocycles. I will discuss
p-adic counterparts for our proposed RM invariants of classical
relations between singular moduli and the theory of weak harmonic
Intercity Number Theory SeminarMarch 15, Utrecht.
|11:00-12:00||Efrat Bank (Technion, Israel), Primes in short intervals on curves over finite fields|
Abstract. We prove an analogue of the Prime Number Theorem for short intervals on a smooth proper curve of arbitrary genus over a finite field. Our main result gives a uniform asymptotic count of those rational functions, inside short intervals defined by a very ample effective divisor E, whose principal divisors are prime away from E.
In this talk, I will discuss the setting and definitions we use in order to make sense of such count, and will give a rough sketch of the proof.
This is a joint work with Tyler Foster.
|13:15-14:15|| Francesca Balestrieri (MPIM Bonn, Germany), Arithmetic of zero-cycles on products of Kummer varieties and K3 surfaces|
Abstract. The following is joint work with Rachel Newton. In the spirit of work by Yongqi Liang, we relate the arithmetic of rational points to that of zero-cycles for the class of Kummer varieties over number fields. In particular, if X is any Kummer variety over a number field k, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on X over all finite extensions of k, then the Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of any odd degree on X. Building on this result and on some other recent results by Ieronymou, Skorobogatov and Zarhin, we further prove a similar Liang-type result for products of Kummer varieties and K3 surfaces over k.
|14:15-15:15||Kęstutis Česnavičius (Orsay, France), Macaulayfication of Noetherian schemes|
Abstract. To reduce to resolving Cohen-Macaulay singularities, Faltings initiated the program of "Macaulayfying" a given Noetherian scheme X. Under various assumptions Faltings, Brodmann, and Kawasaki built the sought Cohen-Macaulay modifications without preserving the locus where X is already Cohen-Macaulay. We will discuss an approach that overcomes this difficulty and hence completes Faltings' program.
|15:30-16:30||Judith Ludwig (Heidelberg, Germany), Perfectoid Shimura varieties and applications|
Abstract. Given a tower of Shimura varieties (where the level at a fixed prime p grows) one may ask whether one can equip the inverse limit with a geometric structure.
As I will explain in the talk, this is possible in many cases. The geometric structure is that of a perfectoid space.
I will then show you the impact of this results by explaining some applications.
Intercity Number Theory SeminarMarch 28, Leiden. This is a Thursday. All talks will be held in the Pieter de la Courtgebouw, room A5-47, which is walking distance from the train station. The PhD defense of Anna Somoza takes place in the Academiegebouw.
|11:00-12:00||Christophe Ritzenthaler (Rennes), Reduction of plane quartics|
Abstract. Given a smooth plane quartic over a discrete valuation field K, we give a characterization of its reduction type (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over K in terms of the existence of a special plane quartic model and over the algebraic closure in terms of the valuations of the Dixmier-Ohno invariants of C. Joint work with Qing Liu, Elisa Lorenzo García and Reynald Lercier.
|12:45-13:45||Pınar Kılıçer (Groningen), TBA|
|14:00-15:00||Elisa Lorenzo-García (Rennes), TBA|
|16:15-17:15||Anna Somoza (Leiden), PhD defense|
Seminarium Computer Algebra Nederland / Intercity Number Theory SeminarApril 12, UvA Amsterdam. The first three lectures will be in room F3.20 in Science Park 107.
Tim Dokchitser's inaugural lecture will be in the Aula.
|10:15-11:15||Steffen Löbrich (UvA), On cycle integrals of meromorphic modular forms|
Abstract. In joint work in progress with Markus Schwagenscheidt, we study cycle integrals of meromorphic modular forms associated to quadratic forms of negative discriminant. In particular, we relate them to evaluations of locally harmonic Maass forms at CM-points. This allows us to generalize a recent theorem by Alfes-Neumann, Bringmann, and Schwagenscheidt on the rationality of these cycle integrals in several directions.
|11:30-12:30||Alex Bartel (Glasgow), Torsion homology and regulators of isospectral manifolds|
Abstract. It is well known by now that two drums that sound the same
need not look the same. But how different can they really look? There
are two general constructions of such pairs of "drums", a representation
theoretic one due to Sunada, and a number theoretic one due to Vigneras.
After introducing the general setting and recalling the two
constructions, I will discuss what can be said about the homology groups
of such isospectral manifolds, e.g. for what primes p the p-torsion in
the homology can differ. The answers in the two settings look very
different... until one looks more closely! This is joint work with Aurel
Page, and, fittingly for the occasion, it was originally inspired by the
work of Tim and Vladimir Dokchitser on the Birch and Swinnerton-Dyer
|13:30-14:30||Maarten Derickx (MIT), Modularity of elliptic curves over totally real cubic fields|
|16:00-17:00||Tim Dokchitser (Bristol / UvA), Inaugural lecture|
Intercity Number Theory SeminarMay 10, Leiden.