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.
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.
We will start the course with the basic theory of modular functions and modular forms, and study the space of cusp forms, the Hecke operators and the zeta functions of modular forms.
Using this material, we will cover some topics of complex multiplication elliptic curves, and complex multiplication theory.
The following textbooks are good references for the course:
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms
H. Iwaniec, Topics in Classical Automorphic Forms
The evaluation of the course will consist of assignments and presentations by the students.
We will develop local and global class field theory through the lenses of Galois cohomology. We will spend a substantial amount of time developing from scratch the cohomology of discrete modules for a profinite group and only then move towards the intended arithmetic target. As a background, some earlier exposure to algebraic number theory and Galois theory would be helpful.
p-Adic Hodge-theory is a theory which allows us to classify p-adic representations of the absolute Galois group of a p-adic field, i.e. a finite extension of Qp. This classification is done using p-adic period rings.
Further on, this theory allows us to understand geometrically p-adic Galois representations coming from étale cohomology of smooth proper algebraic varieties over a p-adic field by "comparing" it with de Rham cohomology of the rigid analytic space associated to the variety.
The grade in the course will be assigned as follows: 50% based on the homework assignments during the semester, and 50% based on the final exam, at the end of the course.
Groupes et algèbres de Lie : groupes de Lie, espaces tangents et champs de vecteurs, algèbres de Lie, application exponentielle, représentations adjointes et coadjointes. Structure et classification des algèbres de Lie : algèbres résolubles et nilpotentes, décomposition en espaces de racines, groupes de Weyl, matrices de Cartan, esquisse de la classification, théorème de Serre. Théorie des représentations : théorème de Weyl, décompositions en espaces de poids, algèbres enveloppantes, modules de Verma. Sujets à option : introduction à l'analyse harmonique non commutative, algèbres de Lie de dimension infinie, théorie géométrique des représentations, formules des caractères et formes modulaires.
Affine varieties. Radical ideals and Hilbert's Nullstellensatz. The Zariski topology. Irreducible decomposition. Dimension. Tangent spaces, smoothness and singularities. Projective spaces and projective varieties. Regular functions and morphisms. Rational maps and indeterminacy. Blowing up. Divisors and linear systems. Projective curves.
Prerequisites: MATH 456 or permission of the instructor. Some familiarity with rings, ideals, and multivariable calculus is expected.
• Categories and functors, adjoint and equivalence, tensor products, localization of rings and module, limits.
• Affine schemes. Integral extensions.
• Noetherian and artinian rings and modules. Hilbert’s basis theorem, Noether’s normalization lemma and Hilbert’s Nullstellensatz. The affine space.
• Representations of finite groups.
Algèbre de Lie d’un groupe de Lie. Formes de Maurer-Cartan. Théorèmes de Lie. Application exponentielle, coordonnées canoniques. Sous-groupes fermés. Sous-groupes connexes par arcs. Formes de Killing et les groupes semi-simples.
Anneaux commutatifs et leurs modules. Localisation : idéaux premiers, racine d'un idéal, anneaux et modules de fractions, anneaux locaux. Dépendance entière: clôture intégrale, théorème de montée. Anneaux et modules noethériens, anneaux de polynômes sur un anneau noethérien. Ensembles algébriques affines, théorème des zéros de Hilbert, ensembles algébriques irréductibles et idéaux premiers, propriétés des courbes planes, dimension des variétés. Applications.
Lemme de Zorn. Catégories et foncteurs: notions et exemples de base: catégories de structures mathématiques, monoïde, catégorie des ensembles; section, rétraction, exemples géométriques et algébriques. Foncteurs et transformations naturelles: exemples de base, catégories de foncteurs. Équivalence de catégories: exemples de base. Modules. Théorèmes d'homomorphisme et d'isomorphisme. Sommes et produits directs, modules libres. Modules de type fini sur un anneau principal et applications aux formes canoniques des matrices. Modules noethériens et artiniens: exemples et propriétés de base. Modules indécomposables, théorème de Krull-Schmidt. Anneaux et polynômes: nilradical et localisation; élimination classique, ensembles algébriques, théorème des zéros de Hilbert. Théorie des corps: groupe de Galois, résolution par radicaux; indépendance algébrique, degré de transcendance, dimension des ensembles algébriques irréductibles; corps ordonnables, 17e problème de Hilbert.
The course will introduce modular forms geometrically, as section of line bundles and as cohomology classes in de Rham cohomology. We will present the Eichler–Shimura isomorphism comparing de Rham cohomology of the modular curve with coherent cohomology of automorphic coherent sheaves. We will describe applications to Galois representations and L-function of modular forms, and study congruences between (p-adic) eigenforms forms, introducing the notion of p-adic families, following approaches of Katz, Hida, and Coleman.
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
Anneaux commutatifs, idéaux premiers, rudiments de géométrie algébrique, Nullstellensatz de Hilbert, localisation, complétion, théorie de la dimension.
Groupe des points d’une courbe elliptique. Théorème de Mordell-Weil. Groupes de Selmer et de Tate-Shafarevich. Les expansions de Fourier des formes modulaires et l’idée de modularité. Applications aux équations diophantiennes.
L’objectif de ce cours est d’approfondir la théorie des groupes et la théorie des anneaux tout en présentant certaines applications de ces structures algébriques en mathématiques, en physique ou en informatique. Les sujets suivants seront présentés :
Révision de la théorie des groupes et de la théorie des anneaux de base, homomorphismes, théorèmes fondamentaux, théorème de Jordan-Hölder, théorème de Sylow, idéaux spéciaux, anneau des polynômes, groupe linéaire général et ses sous-groupes, groupes de Lie et leurs représentations, algèbres de Lie.