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.
Number Fields and Ideals. Dedekind domains, unique factorization of ideals, ideal class groups. Geometry of numbers, finiteness of the class number and the unit theorem. Special Fields (quadratic, cyclotomic, etc), applications to Fermat's last theorem. Analytic Methods, Zeta and L-functions, analytic continuation, density theorems.
TEXTBOOK: Number Fields by Daniel A. Marcus, Universitext, Springer-Verlag.
The course will be an unorthodox introduction to analytic number theory for people with some exposure to/liking of hard analysis. The emphasis will be on harmonic analysis as used in analytic number theory. The unorthodoxy will consist in that we will follow the natural development and inter-relation of various techniques rather than focus on the primes, which are typically given prominence in introductory courses.
Time. Tuesday, Friday, 12:05pm to 13:25pm.
First meeting on September12, 2017 at noon.
Le groupe modulaire et les sous-groupes de congruence, les formes modulaires et ses propriétes de base, séries de Eisenstein, séries theta, formule des valences, opérateurs de Hecke, théorie de Atkin-Lehner, fonctions L, courbes modulaires, modularité.
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.
The course is devoted to the study of formal groups and their applications. More precisely, we will mostly focus on one-dimensional commutative formal groups. Our goals are :
1. Understanding of the basic theory of formal groups (how they arise, universal group laws, deformation theory).
2. Applications to elliptic curves, class field theory, topology (perhaps…), p-adic dynamics.
3. Period maps on the moduli space of formal groups.
The main references are
1. Hazewinkel: formal groups and applications.
2. Lazard: commutative formal groups, LNM 443.
3. Lubin-Tate: “Formal moduli for one-parameter formal Lie groups” Bull. SMF, 94 (1996), 49-59.
and “Formal complex multiplication in local fields” Annals of Math., 81 (1965), 380-387.
4. Hopkins-Gross: Equivariant vector bundles on the Lubin-Tate moduli space, Contemp. Math. 158 (1994), 23-88.
The grade will be given on the basis of submitted exercise sets, which some students will be able to substitute for by giving presentations. There will also be assigned reading to cover additional background material.
The prerequisites are: MATH 456 and 457 (can be taken in parallel to the course).
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.
Ce cours se veut une introduction aux surfaces de Riemann compactes (aussi appelées courbes algébriques sur les complexes). La théorie sera développée en faisant appel simultanément à des notions d'algèbre, d'analyse complexe et de topologie. Voici les thèmes qui seront traités: courbes projectives, courbes affines, applications holomorphes et méromorphes, singularités (noeud, cusp), formule d'Hurwitz, courbes hyperelliptiques, revêtements, représentations de monodromie pour les revêtements et les EDO, formes différentielles, diviseurs et Théorème de Riemann-Roch, problème de Mittag-Leffler, différentielles de première espèce et théorème d'Abel, faisceaux, classification des fibrés en droite holomorphes sur des courbes projectives et lisses.
The goal of this course will be to study algebraic methods for the estimation of exponential sums over finite fields, and the applications of the latter in number theory. A particular emphasis will be put on ideas around the Weil conjectures for curves and varieties, and Deligne's generalization to weights of l-adic sheaves.