Autumn school: Topics in arithmetic and algebraic geometry

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.

 

List of participants: click here.

Notes below!

 

Invited speakers

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

Martin Bright

Title: Rational points in families of varieties

Abstract:

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.

NOTES Bright click here

Alessandra Sarti

Title: Automorphisms of Hyperkähler Manifolds

Abstract:

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.

 

Alexei Skorobogatov

Title: Finiteness and uniformity for abelian varieties and K3 surfaces with complex multiplication

Abstract:

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)

 

NOTES Skorobogatov Click here

 

Olivier Wittenberg

Title: Some topics around algebraic cycles on real varieties

Abstract:

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.

 

NOTES Wittenberg Click here

 

Abstracts and titles: talks

Fabien Pazuki

Title: Potential density on Calabi-Yau threefolds.

Abstract:

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.

 

Rachel Newton

Title: Transcendental Brauer-Manin obstructions on Kummer surfaces

Abstract:

In 1970, Manin observed 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 lack of a K-point on X despite the existence of points over every completion of K is sometimes explained by non-trivial elements in Br(X). The 'algebraic' part of Br(X) 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 has been widely studied. Until recently, very little was known about the transcendental part of the Brauer group.
Results of Skorobogatov and Zarhin allow one to compute the transcendental Brauer group of a product of elliptic curves. Ieronymou and Skorobogatov used these results to compute the odd order torsion in the transcendental Brauer group of diagonal quartic surfaces. The first step in their approach is to relate a diagonal quartic surface to a product of elliptic curves with complex multiplication by the Gaussian integers. I will show how to extend their methods to compute transcendental Brauer groups of products of other elliptic curves with complex multiplication. Using these results, I will give examples of Kummer surfaces where there is no Brauer-Manin obstruction coming from the algebraic part of the Brauer group but a transcendental Brauer class causes a failure of weak approximation.

 

 

Arne Smeets

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.

 

Schedule

Monday Tuesday Wednesday Thursday Friday
9:00-10:00 Registration  Bright II Skorobogatov  II Skorobogatov  III Wittenberg IV
10:00-10:30 Coffee break
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
13:00-15:00 Lunch Lunch Lunch Lunch Lunch 
15:00-16:00 Bright I Newton Free afternoon Smeets
16:00-16:30 Coffee break Coffee break Free afternoon
Coffee break
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.

Organizers

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.

 

Travel information:

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.

Miscellaneous: