Semester | Mand. | Option | Filières |
MA1, MA3 | x |
Lecture | 2 weekly |
Exercises | 2 weekly |
Study plan | Semester | Mand. | Option | Filières |
Ingénierie Mathématiques 2019-20 | MA1 | x | ||
Ingénierie Mathématiques 2019-20 | MA3 | x | ||
Mathématiques Master 2019-20 | MA1 | x | ||
Mathématiques Master 2019-20 | MA3 | x |
Cursus | Sem. | Type |
---|---|---|
Ing.-math | MA1, MA3 | Opt. |
Mathématicien | MA1, MA3 | Opt. |
This year we will discuss the theory of adéles and idèles.
Idèles were invented by Claude Chevalley to provide an compact reformulation of Class Field Theory and Artin's reciprocity law.
However the group of idèles together with its associated ring of adèles are powerful tools to encode all sorts of local-global principles in number theory and arithmetic geometry; the associated terminology is by now the lingua franca of the Langlands program.
The course will introduce the language and we will use it to revisit various aspects of classical algebraic number theory. For instance we will give new proofs of classical results like the finiteness of the class group, Dirichlet's units theorem or the class number formula.
-completions in number fields. Ostrowski's Theorem.
-Local-global principles : the case of the space of lattices.
- Topology and harmonic analysis on adeles and ideles.
-The ring of adeles and the group of ideles associated to a number field. Finitness of the class group and Dirichlet's unit theorem all in one.
-Tate's thesis. Analytic properties of Dedekind and Dirichlet L-functions.
- The adelic formulation of class field theory (without proofs)
- Modular forms in the adelic language.
Anbalysis III & IV
Galois Theory.
Rings and Modules.
Introduction to Algebraic Number Theory.
Not required but possibly useful
Introduction to Analytic Number Theory.
It will be good to have had some exposure to the theory of Riemann's zeta function.
Automorphic forms and L-functions.
Some exposure to the classical theory of modular forms migh be used at the end of the course to compare with the adelic viewpoint.
-Good knowledge of Galois theory, Algebraic Number Theory and the associated commutative algebra.
Ex-Cathedra Course
We expect a proactive attitude during the courses and the exercises sessions (possibly with individual presentation of the solution of various problems).
Exam Oral
Dans le cas de l’art. 3 al. 5 du Règlement de section, l’enseignant décide de la forme de l’examen qu’il communique aux étudiants concernés.
Current research in number theory
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
HIV
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
HIV
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
HIV
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
HIV
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