This is an Autumn School of the SFB/TRR 45 Bonn-Essen-Mainz financed by the DFG (Deutsche Forschungsgemeinschaft). It will take place from October 9th until October 13th, 2017 at the University of Mainz (Germany). To register, please scroll down.
This summer school is intended for advanced master students, PhD students, and young researchers in algebra, number theory and geometry.
Martin Bright, University of Leiden
Rachel Newton, University of Reading
Fabien Pazuki, University of Copenhagen
Alessandra Sarti, University of Poitiers
Alexei Skorobogatov, Imperial College of London
Arne Smeets, Radboud University Nijmegen
Olivier Wittenberg, ENS Paris
Abstracts and titles: mini-courses
Title: Rational points in families of varieties
Given a polynomial equation in several variables, a fundamental number-theoretical question one can ask is whether the equation has any solutions in rational numbers. If the coefficients of the equation are allowed to vary, then this binary question becomes a quantitative one: how often does the equation have rational solutions? This corresponds to looking at a family of algebraic varieties, and asking how many varieties in the family admit a rational point.
A necessary condition for a variety to have a rational point is that it be locally soluble, that is, soluble over every completion of the rational numbers, real and p-adic. A natural approach to the above question is therefore first to ask about the proportion of varieties in the family that are locally soluble. Whether local solubility implies solubility in Q is the much-studied subject of the Hasse principle. For example, quadrics (varieties defined by a single equation of degree 2) always satisfy the Hasse principle, so studying local solubility is sufficient to completely understand solubility in Q. On the other hand, there are obstructions to the Hasse principle, such as the Brauer-Manin obstruction, and so we are led to study how these obstructions behave in families of varieties.
I will give a survey of past work on this topic and present recent results on the Brauer-Manin obstruction to the Hasse principle in certain families of varieties.
Title: Automorphisms of Hyperkähler Manifolds
The aim of the lectures is to show how to use lattice theory in the study of automorphisms of Hyperkähler manifolds, in particular of dimension four. I will show how lattice theory is a powerful tool to describe moduli spaces. Time permitting I will discuss some well known families of hyperkähler fourfolds in detail, as the Fano variety of lines on a cubic fourfold and double EPW sextics.
Title: Finiteness and uniformity for abelian varieties and K3 surfaces with complex multiplication
Recent analytic work by many authors culminating in a result of Jacob Tsimerman allows one to prove that abelian varieties of CM type defined over number fields of fixed degree fall into finitely many isomorphism classes over an algebraic closure of Q. The same result can be proved for K3 surfaces of CM type, defined as K3 surfaces with commutative Mumford-Tate group. The link is provided by the Kuga-Satake construction and the Torelli theorem, leading to an interpretation of the moduli space of polarised K3 surfaces in terms of Shimura varieties. Quite generally, one has the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford-Tate conjecture (which is known for K3 surfaces). Combining these results one confirms V\'arilly-Alvarado's uniform boundedness conjecture for the Brauer group of K3 surfaces in the CM case. (Based on joint work with Martin Orr)
Title: Some topics around algebraic cycles on real varieties
If -1 can be written as a sum of squares in the function field of a real threefold, can it be written as a sum of 7 squares? Do real quartic threefolds contain rational curves? These two questions are open. In these lectures, taking such questions as a starting point, I will give an introduction to some topics in the study of algebraic cycles on real algebraic varieties, with an emphasis on connections with Hodge theory.
Abstracts and titles: talks
Title: Potential density on Calabi-Yau threefolds.
Let X be a projective variety defined over a number field k. We say that X satisfy the potential density property if there exists a finite extension k'/k such that the rational points X(k') are dense in X for the Zariski topology. We report on a joint work with Bogomolov, Halle and Tanimoto where we prove that Calabi-Yau threefolds defined over number fields and admitting double fibrations in abelian surfaces satisfy the potential density property. We will also give explicit examples of such threefolds.
Title: Transcendental Brauer-Manin obstructions on Kummer surfaces
Title: Pseudo-split fibres and arithmetic surjectivity
Abstract: The celebrated Ax-Kochen theorem states that for every positive integer d, there is a finite set S_d of primes such that if p is a prime not in S_d, then every homogeneous polynomial of degree d over Q_p in more than d^2 variables has a non-trivial zero. This classical result was originally proven using model theory. I will present an optimal, geometric theorem of Ax-Kochen type, obtained in joint work with Dan Loughran and Alexei Skorobogatov. Our work uses some birational and toroidal geometry, and builds on earlier work of Colliot-Thélène and Denef.
|9:00-10:00||Registration||Bright II||Skorobogatov II||Skorobogatov III||Wittenberg IV|
||Coffee break||Coffee break||Coffee break||Coffee break|
|10:30-11:30||Sarti I||Sarti II||Wittenberg III||Bright III||Sarti IV|
|11:45-12:45||Wittenberg I||Wittenberg II||Pazuki||Sarti III||Bright IV
|15:00-16:00||Bright I||Newton||Free afternoon||Smeets|
|16:00-16:30||Coffee break||Coffee break||Free afternoon
|16:30-17:30||Skorobogatov I (by Newton)||Questions||Free afternoon||Skorobogatov IV|
Registration will be on Monday October 9th from 9h-10h in the Hilbertraum Room 05-432 (5th floor)
All lectures take place in Room 05-426, the question session on Tuesday at 16:30 will take place in Rooms 05-136, 05-522, 04-224 and 04-230.
Dino Festi (Mainz)
Ariyan Javanpeykar (Mainz)
Davide Cesare Veniani (Mainz)
Registration and financial support
Full financial support is available for members of the SFB/TRR 45.
To register, please click here.
Closest airport is located at Frankfurt/Main. From there, there are frequent trains to Mainz central station (Hauptbahnhof). One possibility would be to take the S8 from Frankfurt airport to Mainz.
The following lines serve the university from the main station (get off at the stop "Friedrich-von-Pfeiffer-Weg"):
- tram 51 (towards Lerchenberg)
- tram 53 (towards Lerchenberg)
- bus 54 (towards Klein-Winternheim)
- bus 55 (towards Finthen)
- bus 56 (towards Finthen/Wackernheim)
- tram 59 (towards Hochschule Mainz)
- bus 75 (towards Schwabenheim/Ingelheim)
- bus 650 (towards Sprendlingen)
From the Friedrich-von-Pfeifferweg stop, walk over the pedestrian bridge to the university campus and follow the street to the right. After about 100 meters there is a left curve. After passing the Mensa (campus cafeteria) you see the Mathematics building right in front of you. (The walk is about 10 minutes.)
All lectures will take place at the Institute of Mathematics of the University of Mainz, Staudinger Weg 9, in room 05-426 (next to the Hilbertraum).
Suggestions for the evenings and Wednesday afternoon:
- There will be a dinner in a restaurant organized on Wednesday (probably at Heiliggeist, to be confirmed). We will meet at 19.00 inside the restaurant (so that we do not block the street waiting altogether outside). If you are interested to come, then please inform the organizers by Tuesday evening.
- For Wednesday afternoon, if you want to visit the city we recommend you visit the Gutenberg museum, the Mainz cathedral, and the St. Stephan church. However, if you prefer to walk outside the city, you can either hike along the Rhein (see here for more information) or else here is a pleasant walk that you can do in three/four hours.
- Here is a list of some nice places where you can eat in Mainz: Heiliggeist, Hintz und Kuntz, Mosch Mosch, Lomo, Bully's burgers, Pomp, Sausalitos, Santiago, Buddha, Niko Niko Tei, Burgerladen, Thai Express.