Courses taught in previous years

2022-23

Fall 2022

Advanced elliptic curves (Concordia, rosso)

The course will focus on the study of elliptic curves over the complex and p-adic numbers. It will cover topics such as: complex uniformisation, Weistrass P-functions, the periods of an elliptic curve, the formal group of an elliptic curve, ordinary and supersingular elliptic curves, integral model of elliptic curves, the local Galois representation.

algebraic geometry (concordia, Iovita)

The main objective of the course is to study geometrically algebraic objects, for example commutative rings with identity. To such a ring we will attach a topological space and a sheaf of rings on it, making it into a geometric object called "affine scheme". We will see that affine schemes can be glued together to give other (non-affine) schemes.

Theorie algebrique des nombres (U de M, Lalín)

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

Winter 2023

Arithmetic Galois Groups (Concordia, Pagano)

This course will revolve around qualitative and quantitative aspects of the Galois inverse problem (GIP), asking which finite groups are realizable over the rationals. Conjecturally all of them do and, furthermore, Malle has proposed a conjectural asymptotic formula for the number of finite extensions of the rationals with a given Galois group and bounded discriminant.

After a warm-up, establishing the GIP in several examples, we will rapidly prove it for the class of (odd) nilpotent groups and discuss the status of Malle's conjecture for nilpotent groups. We next focus on proving the considerably stronger conclusion where GIP is established for all solvable groups (Shafarevich).

After that we will focus on Galois groups with prescribed ramification and establish Shafarevich's theorem on their number of generators and relations (after pro-l completion), prove the Golod--Shafarevich inequality and give examples of infinite class field towers. Finally, we will focus on the theory of random profinite groups developed by Liu and Wood, conjecturally giving a statistical description of such Galois groups in natural families of number fields.

Advanced Topics in Algebra 1: Class Field Theory (McGill, Allen)

Class Field Theory describes the abelian Galois extensions of local or global fields (for example, finite extensions of the p-adic numbers or the rational numbers) in terms of the internal arithmetic of the field. We aim to cover aspects of both the local and global theory, and along the way learn a little bit of Lubin-Tate theory and Galois cohomology.

Distribution des nombres premiers (U de M, granville)

Distribution des nombres premiers. Fonction zêta de Riemann et fonctions-L de Dirichlet. Le théorème des nombres premiers, et de Bombieri-Vinogradov. La répartition des nombres premiers consécutifs.

2021-22

Fall 2021

modular forms (Concordia, rosso)

This course will be an introduction to the theory of modular forms over the complex number. We shall cover the following topics: the modular group and the upper half-plane, Eisenstein series, Hecke operators, L-functions, modular curves, geometric interpretation of modular forms. If time allows it, further topics (Galois representations or Eichler--Shimura relations) will be considered. Knowledge of complex analysis, Riemann surfaces, and sheaves is useful but not necessary. Further reading on p-adic or Drinfeld modular forms will be available for motivated students.

algebraic number theory (concordia, david)

This is a first course in the study of algebraic number fields. In the first part of the course, we will concentrate on proving the two main basic results in the subject: the ideal class group is finite and the unit group is finitely generated. Other topics will include: the distribution of ideals, the Dedekind zeta function and the class number formula.

Topics in Algebra and Number Theory: Modular forms and orthogonal groups (McGill, Darmon)

This course will give an introduction to modular forms with emphasis on its connections with quadratic forms. Topics may include representations of integers by quadratic forms. theta functions, the Weil representation and the theta correspondence, as well as Hilbert and Siegel modular forms as forms on orthogonal groups.

Topics in Geometry and Topology: Compact Lie groups and their representations (MCGill, Lipnowski)

Winter 2022

Algebraic Geometry: scheme (Concordia, Iovita)

The main reference is chapter 2 of Hartshorne's book: Algebraic Geometry.We will study the category of sheaves of abelian groups and rings on a topological space. Then, we will attach to every commutative ring such a sheaf of rings and the pair consisting of the commutative ring and its sheaf of rings will be called an affine scheme. A scheme is a ringed space which is locally an affine scheme. Then we will study the main properties of schemes: open and closed immersions, fiber product of schemes, separated and proper schemes. An important part of the course will be to solve the many exercises at the end of each section in Hartshorne's book.

Advanced topics in Algebraic geometry (McGill, Goren)

Topics in the theory of Shimura varieties. The exact syllabus depends on the audience. Some topics I am considering are:
(1) A tour of Shimura varieties of low dimension - modular curves, Hilbert-Blumenthal surfaces, Picard modular surfaces, quaternionic modular surfaces, Siegel modular threefolds.
(2) Toric varieties and toroidal compactifications of Shimura varieties of low dimension.
(3) Group schemes, Dieudonné modules and stratification of moduli spaces of abelian varieties.
(4) Dimension formula for modular forms.
(5) Deformation theory of abelian varieties and local models.

Advanced topics in number Theory: Automorphic representations (McGill, Allen)

Automorphic representations generalize the theory of classical modular forms and their Hecke operators, and have become both a central object of study and an indispensable tool in number theory. This course will be an introduction to automorphic representations. We will emphasize examples, and some structural theorems will be taken as black boxes. Some familiarity with algebraic number theory, modular forms, and Lie algebras will be assumed.

Courbes elliptiques et formes modulaire (U de M, Granville)

We will study the integer and rational points on linear, quadratic and cubic curves, culminating in the proof of Mordell's Theorem, which describes a surprising structure amongst the rational points on elliptic curves.

2020-21

Fall 2020

p-adic Galois representations (Concordia, Iovita & Rosso)

Part 1: We will discuss the p-adic representations of the absolute Galois group of a finite extension of Q_l, where l 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.

Theorie algebrique des nombres (U de M, Lalín)

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

Modular forms and the theory of complex multiplication (McGill, Darmon)

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.

Modularity Lifting (McGill, Allen)

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.

Winter 2021

Advanced Elliptic curves (Concordia, Rosso)

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.

Additive Combinatorics (U de M, Granville)

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

Anatomy of integers, polynomials and permutations (U de M, Koukoulopoulos)

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.

2019-20

  Fall 2019

Modular Forms (Concordia, Rosso)

This course will be an introduction to the theory of modular forms over the complex number. We shall cover the following topics: the modular group and the upper half-plane, Eisenstein series, Hecke operators, L-functions, modular curves, geometric interpretation of modular forms. If time allows it, further topics (Galois representations or Eichler--Shimura relations) will be considered.

Introduction to Elliptic curves I  (U de M, Granville)

We will study the integer and rational points on linear, quadratic and cubic curves, culminating in the proof of Mordell's Theorem, which describes a surprising structure amongst the rational points on elliptic curves.

Theorie algebriques des nombres (U de M, Lalín)

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

Sheaf Cohomology (Concordia, Iovita)

This is a second course in Algebraic Geometry and it will follow chapter III of R. Hartshorne's book: Algebraic Geometry. We will present the general theory of derived functors with applications to the sheaf cohomology of schemes. As main application we will present the Riemann-Roch theorem for smooth proper algebraic curves.

Winter 2020

Analytic Number Theory and the Distribution of Prime Numbers (Concordia, David)

This course is an introduction to the subject of analytic number theory. Our main goal will be the proof of the prime number theorem, proving an asymptotic for the number of primes up to x with an explicit error term, and explain the links with the Riemann Hypothesis and zero free regions of the Riemann Zeta function. This will include generalization to counting primes in arithmetic progressions, and the study of Dirichlet L-functions.

2018-19

Fall 2018

Elliptic curves (Concordia, David)

This course will cover the basic theory of elliptic curves and algebraic curves. The prerequisites for this course is the standard background in abstract algebra (groups, rings, field extensions, Galois theory etc). A first course in algebraic number theory is recommended, but not mandatory. The course is open to all graduate students, either at the master or the Ph.D. level.

Introduction to Class Field Theory (McGill, Darmon)

The course will describe the statement and proofs of the main results of class field theory, both local and global, following the treatment given in the textbook of Cassels-Frolich, which shall be followed fairly closely.

Galois theory for schemes (Concordia, Iovita)

The course will develop the theory of the etale fundamental group of a connected scheme in parallel to the Galois theory of a field and the theory of the fundamental group of a topological space. Some knowledge of Algebraic Geometry would be helpful but not necessary as the course will be self  contained.

Sieve methods and application (UdeM, Koukoulopoulos)

This is an introductory course to sieve methods and their applications. After reviewing some background material in probabilistic number theory, we will introduce and study the main objects of sieve theory. In particular, we will develop the combinatorial sieve and Selberg's sieve and use them to prove various estimates about prime numbers. We will then use sieve theory to study L-functions and establish Linnik's theorem. The proof of this result provides a natural entry point to the theory of bilinear sum estimates and the development of the Large Sieve. As an application of this circle of ideas, we will prove the Bombieri-Vinogradov theorem. The course will conclude with a discussion of the recent spectacular developments about bounded and large gaps between primes.

Compact Lie groups and their representations (McGill, Lipnowski)

Compacts groups and the Peter-Weyl theorem.  Weyl's unitary trick.  Lie groups, Lie algebras, the exponential map, Ad, and ad.  Maximal tori and conjugacy of maximal tori.  Root systems.  Theorem of highest weight.  Weyl character formula.

Theorie algebrique des nombres (UdeM, Lalín)

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

Winter 2019

p-adic modular forms (Concordia, Iovita)

This course will present the theory of p-adic and overconvergent modular forms as they appear in the work of J.-P. Serre, N. Katz and R. Coleman. Basic knowledge of classical modular forms (for example as they appear in "A course in Arithmetic" by J.-P. Serre) and of algebraic geometry is required.

Elliptic curves II (McGill, Goren)

This course will cover advanced topics in the theory of elliptic curves. It is intended as a continuation of the course “Elliptic Curves” to be taught by Prof. David in the Fall at Concordia University. Although it is not required to take David’s course, I will assume that students know the material covered in David’s course. The exact selection of topics will be determined once a more precise syllabus for David’s course becomes available, but they will mostly be chosen from Silverman’s books on elliptic curves.

Analytic Number Theory (U de M, Granville)

The goal of this course is to give a complete proof of the prime number theorem, a proof that will help the student appreciate many of the important theorems in the subject. We will review in detail the motivation for the prime number theorem (and other conjectures and theorems about prime numbers), and focus on the background needed in both number theory and analysis (so that students feel comfortable with the techniques used). Then we will prove the prime number theorem and begin to appreciate the importance of the Riemann Hypothesis.  Having gone slowly over this we will be ready to use these ideas in many different directions. Our only scheduled goal will be to prove Dirichlet's theorem, that there are infinitely many primes in each reduced residue class a mod q, though we will at least sketch how to estimate how many primes like in each such class.  Moreover, if things go well, we will apply these ideas to primes in short intervals, in short arithmetic progressions, study least quadratic non-residues, ...