Arithmetic and Algebraic Geometry Seminar

Korteweg-de Vries Instituut, University of Amsterdam

Fall Semester 2015

This semester the lectures alternated between a study group on K3 surfaces and their moduli and research talks that are not necessarily related to the study group. The study group is aimed for PhD students and master students. In the first half of the semester we studied the first chapters of Huybrecht's book [1]. In the second half we went more towards moduli of K3 surfaces.

10th of September
Lenny Taelman, Introduction to K3 surfaces.
Jan Van Geel (Ghent): Local-global principle for zero-cycles of degree one.

17th of September
Emma Brakkee, Intersection theory on algebraic surfaces.
Mingmin Shen, Some examples of algebraic K3-surfaces.

24th of September
Wessel Bindt: Classical invariants of K3 surfaces (notes).
Ties Laarakker: Linear systems on K3 surfaces.

1st of October
Maxim Mornev: Hodge theory (of K3 surfaces).

8th of October
Lance Gurney: Analytic and algebraic deformation theory (of K3 surfaces).
Important: This week the seminar is in a different room (Room A1.06, Science Park 904), and starts at a different time (1pm)!

15th of October
Reinier Kramer: Kahler and hyperkahler manifolds.
Juultje Kok: Deformation equivalence of K3 surfaces. (slides)

22nd of October
Reinier Kramer: Kahler and hyperkahler manifolds II. (notes)

29th of October
Hang Xue: A quadratic rational point on a non-hyperelliptic genus four Jacobian.
Abstract: We construct a canonical quadratic rational point on a non-hyperelliptic genus four Jacobian and show that it generates the Mordell--Weil group of the universal Jacobian. We also study how the point degenerates on a hyperelliptic Jacobian.
Zhiyu Tian, Fundamental group of Fano varieties.
Abstract: In this talk will explain a close relation between the fundamental groups of two different objects: the fundamental group of the smooth locus of a Fano variety and the fundamental group of the punctured neighborhood of certain type of singularities and present some finiteness result of both groups. This is based on joint work with Chenyang Xu (BICMR).

5th of November
Lenny Taelman: Global Torelli I (notes)
Judith Ludwig, p-adic Langlands functoriality.
Abstract: In recent years a p-adic version of the Langlands programme has started to emerge. Although the general definition of a p-adic automorphic representation is still missing we have good working definitions of p-adic automorphic forms in many situation. Often these p-adic forms can be organized into reasonable geometric objects, so called eigenvarieties. One can ask which aspects of the classical Langlands programme make sense in the p-adic setting and it is particularly interesting to ask about Langlands functoriality: If G and H are two connected reductive groups defined over a number field together with a classical Langlands transfer from G to H and a construction of eigenvarieties then is there a p-adic Langlands transfer? In this talk we will explain the concept of an eigenvarietiy and construct examples of such p-adic transfers.

12th of November
This week there are no talks in the seminar, due to the Lorenz workshop Moduli spaces and Arithmetic Geometry in Leiden.

19th of November
Mingmin Shen: Global Torelli II
Olivier Taibi, Computing dimensions of spaces of automorphic/modular forms for classical groups using the trace formula
Abstract: Let G be a Chevalley group which is symplectic or special orthogonal. I will explain how to explicitly compute the geometric side of Arthur's trace formula for a function on G(AA) which is a stable pseudo-coefficient of discrete series at the real place and the unit of the unramified Hecke algebra at every finite place. Arthur's recent endoscopic classification of the discrete automorphic spectrum of G allows to analyse the spectral side in detail. For example one can deduce dimension formulae for spaces of vector-valued Siegel modular forms in level one. The computer achieves these computations at least up to genus 7.

20th of November
Announcement from the representation theory seminar (room F3.20, Science Park 105-107)
2:30-3:30 pm
Olivier Taibi, Arthur's multiplicity formula for automorphic representations of certain inner forms of special orthogonal and symplectic groups.
Abstract: I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of certain special orthogonal groups and certain inner forms of symplectic groups G over a number field F. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set S of real places of F such that G has discrete series at places in S and is quasi-split at places outside S, and by restricting to automorphic representations of G(AA_F) which have algebraic regular infinitesimal character at all places in S. In particular, I prove the general multiplicity formula for groups G such that F is totally real, G is compact at all real places of F and quasi-split at all finite places of F. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

26th of November
Wessel Bindt: Monodromy of K3 surfaces
Mingmin Shen: Global Torelli III

3rd of December
Arno Kret: Introduction to Shimura Varieties
Anna Cadoret, specialization of adelic representations of etale fundamental groups.
Abstract: I will discuss the issue of specializing adelic representations of ├ętale fundamental groups, especially those attached to abelian varieties and Shimura varieties. In particular, I will give applications to integral and adelic variants of the Mumford-Tate conjecture for abelian varieties and K3 surfaces. This work is partly joint with Arno Kret and Ben Moonen.

10th of December
Lance Gurney: Fixed point formulas and automorphisms of K3 surfaces
Jonas Bergstrom: Cohomology of a Picard modular surface.
Abstract: In this joint project with Gerard van der Geer we have studied the cohomology of local systems of a Picard modular surface by counting points over small finite fields using that it is a moduli space of tricyclic covers of the projective line.

References for the study group on K3 surfaces:

[1]. D. Huybrechts, Lectures on K3 surfaces.
[2]. A. Beauville, Complex algebraic surfaces.