Arithmetic Geometry Year

Poncelet French-Russian Laboratory,

Moscow, Russia

2012 - 2013

Lectures Program

There are several lectures given at the Higher School of Economics over at the Independant University along the year. Please feel free to contact us to know the exact schedule.

Tue-Wed 12/11/13 -> 10/12/13 (1001),

17:30 - 19:00 Philippe Lebacque (Université de Besançon, France)
Applications of global fields to error correcting codes and sphere packings

Error correcting codes are at the heart of the digital world and therefore play a central role in modern societies. In the late 70's, V.D. Goppa constructed the first error correcting codes from algebraic geometry. The problem of construction of good algebraic geometric codes reduces to the question of finding curves of arbitrary large genus defined over finite fields with many rational points.
After recalling basic definitions and properties concerning codes, we will focus on algebraic geometric codes and on their relations to global fields. We will discuss several constructions of asymptotically good families of curves over finite fields that lead to good codes. Namely,
* explicit equations (after Garcia and Stichtenoth),
* class field theory (after Serre),
* reduction of modular curves (after Ihara and Tsfasman-Vladuts-Zink).
If time permits, we will also cover the constructions of sphere packings (continuous analogues of codes) that use number fields. In the course we will deal with algebraic curves, the Riemann-Roch and Riemann--Hurwitz theorems, the ramification theory, the class field theory, Galois cohomology. All the necessary notions and results will be recalled.
The last lecture will be devoted to the most recent advances concerning invariants of infinite global fields.

Tue 29/10 (311-312), Wed 30/10 (311-312), Thur. 31/10 (311-312)

17:00 - 17:30 Arnaud Durand (Université de Paris-Sud, France)
Introduction to the metric theory of Diophantine approximation:

Diophantine approximation originally aims at describing the quality with which the real numbers may be approached by the rationals. In this series of lectures, we will adopt a metric point of view on this question: we will study the size of the set of reals that are approximated at a given rate by the rationals; these size properties will be expressed in terms of Lebesgue measure or Hausdorff dimension, or using finer tools from geometric measure theory. We will review the most classical results, due to Dirichlet, Khintchine, JarnÌk and Besicovitch, and we will give a flavor of the more recent advances on the topic.

Mon 9/9 (1001), Wed 11/9 (311), Fri 13/9 (1001)

15:30 - 17:00 Florian Breuer (University of Stellenbosch, South Africa)
Introduction to Drinfeld Modules:

Drinfeld modules are the natural function-field analogue of elliptic curves. In this mini-course I intend to give an introduction to this topic, starting with motivations from cyclotomy, elliptic curves as well as modular forms. I will cover the basic properties as well as some selected topics like Drinfeld modular forms, moduli spaces of Drinfeld modules and an analogue of the Andre-Oort Conjecture.

Tue 3/9 (1001), Wed 4/9 (1001), Fri 6/9 (15:30, 1001), Mon 9/9 (1001)

Tue-Wed-Mon 17:00 - 18:30
Fri 15:30 - 17:00 Francesco Lemma (Paris 7)
Introduction to Shimura varieties:

Shimura varieties are algebraic varieties defined over number fields which play a central role in modern arithmetic geometry. One reason for this is that their geometry and cohomology is related to automorphic forms. The most studied Shimura varieties are the ones associated to GL(2). In this case, the set of their complex points is a finite disjoint union of quotients of the upper half-plane by arithmetic subgroups of GL(2). In these lectures we will introduce the axioms defining a Shimura variety and explain in some detail why the Shimura variety associated to the symplectic group) GSp(2n)) can be seen as a moduli space of abelian varieties.

Wed 19/6 (311), Fri 21/6 (317), Mon 24/6 (311), Wed 26/6 (1001)

15:30 - 17:00 Adrien Deloro (Paris 6)
Basic model theory of algebraically closed fields:

The course will introduce the very first notions and methods of model theory: first-order logic, ultraproducts, completeness, quantifier elimination. The emphasis will be on algebraically closed fields and theorems which can be proved by model-theoretic means. Lefschetz' transfer principle will serve as a guideline. As far as logic is concerned the course will be self-contained but prospective students should know a little field theory (algebraic closures, finite fields, algebraic elements).

Tue 21/5 - Wed 22/5 - Thu 23/5 - Mon 27/5 - Tue 28/5 - Wed 29/5

17:00 - 18:30 Antoine Ducros (Paris 6) and Jérôme Poineau (Strasbourg)
Introduction to Berkovich analytic spaces:

At the end of the eighties, Vladimir Berkovich introduced a new way to define p-adic analytic spaces. A surprising feature is that, although p-adic fields are totally discontinuous, the resulting spaces enjoy many nice topological properties: local compactness, local path-connectedness, etc. On the whole, those spaces are very similar to complex analytic spaces. They already have found numerous applications in several domains: arithmetic geometry, dynamics, motivic integration, etc.
In this course, we will introduce Berkovich spaces and study their basic properties. The program will cover the following topics: - non-Archimedean fields, absolute values - Tate algebras, affinoid algebras and their properties - affinoid spaces - Berkovich spaces - analytification of algebraic varieties - analytic curves (local structure, homotopy type)
In the last lecture, we plan to give an overview of some applications.

Tue 2/4 - Wed 3/4 - Thu 4/4

Tue-Wed 17:00
Thu 15.30,17.00 Gabor Wiese (Université du Luxembourg)
Modular Galois Representations and Applications:

Lecture 1: Representations of profinite groups. Basic definitions. Basic properties of representations of profinite groups.
Lecture 2: Galois representations. Basic definitions (ramification, Frobenius elements, etc.). Examples (cyclotomic character, Galois representations attached to elliptic curves). Chebotarev's density theorem.
Lecture 3: Galois representations attached to modular forms. Explanation of Galois representations attached to Hecke eigenforms. Statement of Serre's modularity conjecture, i.e. the theorem of Khare and Wintenberger, including level lowering and level raising.
Lecture 4: Applications. Sketch of the proof of Fermat's Last Theorem via Serre's modularity conjecture. An application of modular Galois representations to the inverse Galois problem.

Tue26/3-Wed27/3-Thur28/3-Mon1/4

17:00-18:30 Alexander Schmidt (University of Heidelberg)
Higher dimensional class field theory following Wiesend

In this series of lectures I will give an introduction to Wiesend's class field theory for higher dimensional arithmetic schemes. It is more elementary than the class field theory of Bloch-Kato-Saito and therefore better suited for applications in some situations. The setting is as follows: Given a regular scheme X, flat and of finite type over Spec Z, there exists a higher idèle class group C_X which is build out of data attached to all curves on X. There is a natural reciprocity homomorphism
rec : C_X → π₁^ét(X)^ab
such that for every finite Ètale Galois covering Y → X$ the induced homomorphism
C_X / N_Y|XC_Y → Gal(Y|X)^ab
is an isomorphism. The higher idÕle class group carries a natural topology and, as in classical (i.e. one-dimensional) class field theory, the norm groups are exactly the open subgroups. Similar results hold for smooth varieties over finite fields, however, in this case the theory describes the tamely ramified coverings only.

The central idea is to consider covering data, which are compatible systems of finite (not necessarily abelian) Ètale Galois coverings of all curves on X and to investigate the question whether they are induced by coverings of X.

Thursday 28/2/13 -> 21/3/13

15:30-17:00 Philippe Lebacque (Laboratoire de Mathématiques de Besançon)
Introduction to Class Field Theory:

Class field theory is the description of abelian extensions of global and local fields. It started in the 19th century with reciprocity laws and with the Kronecker-Weber theorem which states that every abelian extension of Q is contained in a cyclotomic one. During the 20th century, it has been shown that every abelian extension of a global or local field could be described in terms of the field itself.
We aim at stating and proving the main results of the local and global class field theory in our series of lectures. If time permits, we will also give some concrete applications to the information theory.

Mon - Wed 31/10/12 -> 28/11/12

15:30-17:00 Aurélien Galateau (Laboratoire de Mathématiques de Besançon)
Serre's "big image" theorem:

The course will be an introduction to the theory of elliptic curves, with a view toward Serre's work on the Galois representations attached to their torsion. We will start with a geometric description (projective embedding, group law, isogenies, endomorphism ring) and then focus on the case of elliptic curves defined over finite fields (Hasse's estimate, ordinary and supersingular curves) and local fields (reduction, NÈron-Ogg-Shafarevich criterion). Next, we will explain the link between complex multiplication (CM) on elliptic curves and class field theory of imaginary quadratic fields. The last part of the course will be devoted to non-CM elliptic curves and the Galois properties of their torsion points.

Poncelet Laboratory home page

Sponsors: ANR, Campus France, CNRS, french ambassy, HSE Lab. of Alg. Geom., IUM

Tue-Wed 12/11/13 -> 10/12/13 (1001),
17:30 - 19:00	Philippe Lebacque (Université de Besançon, France)
	Applications of global fields to error correcting codes and sphere packings
	Error correcting codes are at the heart of the digital world and therefore play a central role in modern societies. In the late 70's, V.D. Goppa constructed the first error correcting codes from algebraic geometry. The problem of construction of good algebraic geometric codes reduces to the question of finding curves of arbitrary large genus defined over finite fields with many rational points. After recalling basic definitions and properties concerning codes, we will focus on algebraic geometric codes and on their relations to global fields. We will discuss several constructions of asymptotically good families of curves over finite fields that lead to good codes. Namely, * explicit equations (after Garcia and Stichtenoth), * class field theory (after Serre), * reduction of modular curves (after Ihara and Tsfasman-Vladuts-Zink). If time permits, we will also cover the constructions of sphere packings (continuous analogues of codes) that use number fields. In the course we will deal with algebraic curves, the Riemann-Roch and Riemann--Hurwitz theorems, the ramification theory, the class field theory, Galois cohomology. All the necessary notions and results will be recalled. The last lecture will be devoted to the most recent advances concerning invariants of infinite global fields.
Tue 29/10 (311-312), Wed 30/10 (311-312), Thur. 31/10 (311-312)
17:00 - 17:30	Arnaud Durand (Université de Paris-Sud, France)
	Introduction to the metric theory of Diophantine approximation:
	Diophantine approximation originally aims at describing the quality with which the real numbers may be approached by the rationals. In this series of lectures, we will adopt a metric point of view on this question: we will study the size of the set of reals that are approximated at a given rate by the rationals; these size properties will be expressed in terms of Lebesgue measure or Hausdorff dimension, or using finer tools from geometric measure theory. We will review the most classical results, due to Dirichlet, Khintchine, JarnÌk and Besicovitch, and we will give a flavor of the more recent advances on the topic.
Mon 9/9 (1001), Wed 11/9 (311), Fri 13/9 (1001)
15:30 - 17:00	Florian Breuer (University of Stellenbosch, South Africa)
	Introduction to Drinfeld Modules:
	Drinfeld modules are the natural function-field analogue of elliptic curves. In this mini-course I intend to give an introduction to this topic, starting with motivations from cyclotomy, elliptic curves as well as modular forms. I will cover the basic properties as well as some selected topics like Drinfeld modular forms, moduli spaces of Drinfeld modules and an analogue of the Andre-Oort Conjecture.
Tue 3/9 (1001), Wed 4/9 (1001), Fri 6/9 (15:30, 1001), Mon 9/9 (1001)
Tue-Wed-Mon 17:00 - 18:30 Fri 15:30 - 17:00	Francesco Lemma (Paris 7)
	Introduction to Shimura varieties:
	Shimura varieties are algebraic varieties defined over number fields which play a central role in modern arithmetic geometry. One reason for this is that their geometry and cohomology is related to automorphic forms. The most studied Shimura varieties are the ones associated to GL(2). In this case, the set of their complex points is a finite disjoint union of quotients of the upper half-plane by arithmetic subgroups of GL(2). In these lectures we will introduce the axioms defining a Shimura variety and explain in some detail why the Shimura variety associated to the symplectic group) GSp(2n)) can be seen as a moduli space of abelian varieties.
Wed 19/6 (311), Fri 21/6 (317), Mon 24/6 (311), Wed 26/6 (1001)
15:30 - 17:00	Adrien Deloro (Paris 6)
	Basic model theory of algebraically closed fields:
	The course will introduce the very first notions and methods of model theory: first-order logic, ultraproducts, completeness, quantifier elimination. The emphasis will be on algebraically closed fields and theorems which can be proved by model-theoretic means. Lefschetz' transfer principle will serve as a guideline. As far as logic is concerned the course will be self-contained but prospective students should know a little field theory (algebraic closures, finite fields, algebraic elements).
Tue 21/5 - Wed 22/5 - Thu 23/5 - Mon 27/5 - Tue 28/5 - Wed 29/5
17:00 - 18:30	Antoine Ducros (Paris 6) and Jérôme Poineau (Strasbourg)
	Introduction to Berkovich analytic spaces:
	At the end of the eighties, Vladimir Berkovich introduced a new way to define p-adic analytic spaces. A surprising feature is that, although p-adic fields are totally discontinuous, the resulting spaces enjoy many nice topological properties: local compactness, local path-connectedness, etc. On the whole, those spaces are very similar to complex analytic spaces. They already have found numerous applications in several domains: arithmetic geometry, dynamics, motivic integration, etc. In this course, we will introduce Berkovich spaces and study their basic properties. The program will cover the following topics: - non-Archimedean fields, absolute values - Tate algebras, affinoid algebras and their properties - affinoid spaces - Berkovich spaces - analytification of algebraic varieties - analytic curves (local structure, homotopy type) In the last lecture, we plan to give an overview of some applications.
Tue 2/4 - Wed 3/4 - Thu 4/4
Tue-Wed 17:00 Thu 15.30,17.00	Gabor Wiese (Université du Luxembourg)
	Modular Galois Representations and Applications:
	Lecture 1: Representations of profinite groups. Basic definitions. Basic properties of representations of profinite groups. Lecture 2: Galois representations. Basic definitions (ramification, Frobenius elements, etc.). Examples (cyclotomic character, Galois representations attached to elliptic curves). Chebotarev's density theorem. Lecture 3: Galois representations attached to modular forms. Explanation of Galois representations attached to Hecke eigenforms. Statement of Serre's modularity conjecture, i.e. the theorem of Khare and Wintenberger, including level lowering and level raising. Lecture 4: Applications. Sketch of the proof of Fermat's Last Theorem via Serre's modularity conjecture. An application of modular Galois representations to the inverse Galois problem.
Tue26/3-Wed27/3-Thur28/3-Mon1/4
17:00-18:30	Alexander Schmidt (University of Heidelberg)
	Higher dimensional class field theory following Wiesend
	In this series of lectures I will give an introduction to Wiesend's class field theory for higher dimensional arithmetic schemes. It is more elementary than the class field theory of Bloch-Kato-Saito and therefore better suited for applications in some situations. The setting is as follows: Given a regular scheme X, flat and of finite type over Spec Z, there exists a higher idèle class group C_X which is build out of data attached to all curves on X. There is a natural reciprocity homomorphism rec : C_X → π₁^ét(X)^ab such that for every finite Ètale Galois covering Y → X$ the induced homomorphism C_X / N_Y\|XC_Y → Gal(Y\|X)^ab is an isomorphism. The higher idÕle class group carries a natural topology and, as in classical (i.e. one-dimensional) class field theory, the norm groups are exactly the open subgroups. Similar results hold for smooth varieties over finite fields, however, in this case the theory describes the tamely ramified coverings only. The central idea is to consider covering data, which are compatible systems of finite (not necessarily abelian) Ètale Galois coverings of all curves on X and to investigate the question whether they are induced by coverings of X.
Thursday 28/2/13 -> 21/3/13
15:30-17:00	Philippe Lebacque (Laboratoire de Mathématiques de Besançon)
	Introduction to Class Field Theory:
	Class field theory is the description of abelian extensions of global and local fields. It started in the 19th century with reciprocity laws and with the Kronecker-Weber theorem which states that every abelian extension of Q is contained in a cyclotomic one. During the 20th century, it has been shown that every abelian extension of a global or local field could be described in terms of the field itself. We aim at stating and proving the main results of the local and global class field theory in our series of lectures. If time permits, we will also give some concrete applications to the information theory.
Mon - Wed 31/10/12 -> 28/11/12
15:30-17:00	Aurélien Galateau (Laboratoire de Mathématiques de Besançon)
	Serre's "big image" theorem:
	The course will be an introduction to the theory of elliptic curves, with a view toward Serre's work on the Galois representations attached to their torsion. We will start with a geometric description (projective embedding, group law, isogenies, endomorphism ring) and then focus on the case of elliptic curves defined over finite fields (Hasse's estimate, ordinary and supersingular curves) and local fields (reduction, NÈron-Ogg-Shafarevich criterion). Next, we will explain the link between complex multiplication (CM) on elliptic curves and class field theory of imaginary quadratic fields. The last part of the course will be devoted to non-CM elliptic curves and the Galois properties of their torsion points.