# Algebra and Number Theory

## Program Description

An area of major interest in this program is the study of the Galois group of the algebraic closure of the rationals. The methods of study of this group involves its representations in other algebraic, geometric or analytic objects. This leads to connections with algebraic groups, analytic varieties (real, complex and p-adic), and Lie theory. The interconnections are deep, and progress in number theory requires a more profound understanding of each of these connections. For example, the conjecture of Shimura-Tanyama-Weil that all elliptic curves over the rationals are modular implies Fermat's Last Theorem.

In recent years, due to the availability of powerful computers and software such as MAPLE, CAYLEY and PARI, large scale computations have proven themselves extremely important in making and verifying conjectures. Computational algebra is rapidly evolving with the production of better and faster algorithms for making computations.

The participating universities contributing to this Institute have many people working in currently active areas of research in number theory, elliptic curves, arithmetic geometry, algebraic groups, group theory and Lie theory, commutative algebra, representation theory of groups and Lie algebras, Galois theory, profinite groups and computational algebra, representation theory of associative algebras, homologic and categoric algebra, ring theory and modules.

Many members of the group are also members of CICMA (Centre interuniversitaire en calcul mathématique algébrique), an inter-university research center that organizes many scientific activities.

## Program Members

This program is designed for strong graduate students with an interest in algebra, group theory, number theory (algebraic and/or analytic) and algebraic geometry. Both theoretical and computational aspects of these topics are covered.

There are no formal prerequisites beyond those required by the departments. However, the following guidelines should be followed and courses selected in consultation with an advisor in the program.

All students are expected to acquire the fundamentals of algebra, such as are contained in beginning level graduate courses at any of the participating universities (e.g., group theory, commutative algebra, Galois theory, number theory).

Students are then expected to take a number of more specialized courses (in their areas of interest and adjacent or complementary areas).

Students are encouraged to participate in advanced seminars and courses in their areas of specialty.

## 2020-21 Course Listings

### Topics in Algebra: p-adic Galois representations

Part 1:

We will discuss the p-adic representations of the absolute Galois group of a finite extension of $Q_\ell$, where $\ell$ is a prime integer different from p.
We will define  Weil-Deligne representations, the notion of monodromy modules and we will see the geometric situations in which they appear.

Part 2:

In the second part, we will study p-adic representation of p-adic local fields. We will start with Tate's theorem on the cohomology of C_p, and then introduce various p-adic period rings that will allow us to understand the geometry behind each Galois representation. Several explicit examples (Tate's curves, Hodge--Tate decomposition of elliptic curves) will be provided.

### Algèbre commutative et théorie de Galois

Théorie de Galois : théorème fondamental; fermeture, normalité, groupe de Galois d'un polynôme; corps finis. Algèbre commutative : idéaux premiers, primaires; anneaux noethériens, de Dedekind; radicaux; anneaux simples, semi-simples, premiers, semi-premiers. Modules libres, projectifs, injectifs. Suites exactes. Foncteurs Hom et produit tensoriel.

### Higher Algebra 1

Review of group theory; free groups and free products of groups. Sylow theorems. The category of R-modules; chain conditions, tensor products, flat, projective and injective modules. Basic commutative algebra; prime ideals and localization, Hilbert Nullstellensatz, integral extensions. Dedekind domains. Part of the material of MATH 571 may be covered as well.Moikael Pichot

### Topics in Algebra and Number Theory: Modularity lifting

This course will be an introduction to modularity lifting and the Taylor-Wiles method. Initially developed by Taylor and Wiles to prove the modularity of semistable elliptic curves over the rationals, their eponymous method has been refined and generalized by many and has become an indispensable tool in the study of Galois representations and automorphic forms.

We will focus on the case of GL(2) over both totally real and CM fields, briefly discussing what happens in higher rank. Along the way, we will also indicate applications of the Taylor-Wiles method beyond modularity lifting.

### Modular forms and the theory of complex multiplication

Singular moduli and their factorisations, following Gross and Zagier. Traces of singular moduli, and modular forms of half integral weight. The theorem of Gross-Kohnen-Zagier. Generalisations via Borcherds' theory of singular theta lifts. p-adic variants, and extensions to real quadratic fields.

### Théorie algébrique des nombres

Nombres et entiers algébriques. Unités. Norme, trace, discriminant et ramification. Base intégrale. Corps quadratiques, cyclotomiques. Groupes de classes. Décomposition en idéaux premiers. Équations diophantiennes.

### Algèbre

Ce cours est une introduction à l'algèbre avancée.

Fondations:

•     Axiome du choix;
•     Lemme de Zorn;
•     applications et particularités

Théorie des catégories:

•     catégories;
•     foncteurs;
•     transformations naturelles;
•     propriétés universelles;
•     (co)produits, produit fibré et somme amalgamée;
•     équivalence des catégories

Théorie des modules:

•     modules de type fini sur un anneau principal;
•     applications aux formes canoniques des matrices
•     (forme normale de Smith et forme canonique de Jordan);
•     modules noethériens et artiniens;
•     modules indécomposables;
•     théorèmes de Jordan-Holder, Krull-Schmidt et Artin-Wedderburn;
•     notion de dimension d'un module;
•     produit tensoriel de modules;
•     algèbre tensorielle, symétrique et extérieure

### Représentations des algèbres

Carquois d'une algèbre, représentations d'algèbres héréditaires, théorie d'Auslander -Reiten, ensembles partiellement ordonnés et catégories d'espaces vectoriels, revêtements d'une algèbre, algèbres auto-injectives, théorie de l'inclinaison.

### Topics in Algebra. Algebraic Geometry I: Schemes

The course will follow chapter II of Hartshorne's book and will develop the basic tools of the theory of schemes: sheaves of abelian groups and rings on topological spaces, the spectrum of a ring with its structure sheaf, the fiber product of schemes, properness and separatedness of morphisms of schemes. A special importance will be given to solving the problems in the text.

### Topics in Algebra: Advanced Elliptic Curves

The course will start recalling what are elliptic curves, from the point of view of Riemann surfaces and algebraic geometry. We will then study the following topics: the Selmer group; complex multiplication and Heegner points; Neron models; elliptic curves over local fields and their formal group. More topics can be treated according to the taste of the audience.

### Higher Algebra 2

Completion of the topics of MATH 570. Rudiments of algebraic number theory. A deeper study of field extensions; Galois theory, separable and regular extensions. Semi-simple rings and modules. Representations of finite groups.

#### Institution: McGill University

Théorème de Freiman-Ruzsa, transformation de Dyson, théorèmes de Van der Waerden et de Roth-Szemeredi-Gowers.

### Sujets spéciaux en théorie des nombres : Anatomie des entiers, des polynômes, et des permutations

Content: the Erdös-Kac theorem; Poissonian distribution of prime factors, of cycles, and of irreducible factors; elements of sieve theory; distribution of divisors of integers, and of invariant sets of permutations; Erdos's multiplication table problem and its generalizations; the Luczak-Pyber theorem; applications to the irreducibility and the Galois group of random polynomials.

### Représentation des groupes

La théorie de la représentation des groupes et une théorie algébrique dont les ramifications s’étendent à de très nombreux domaines des mathématiques ainsi qu’à la Physique te à la Chimie. L’apprentissage de cette théorie permettra entre autre à l’étudiant d’appréhender d’autres théories algébriques de la représentation.

Descripteur : Représentations linéraires des groupes finis. Sous-représentations, théorème de Mashke; représentations irréductibles. Théorie des caractères. Décomposition en composantes isotypiques. Produits tensoriels; représentation induites. Représentations linéaires des groupes compacts. Exemples: groupes cycliques, diédraux, symétriques: tores, groupes de rotations.