MAT 6630


Courbes elliptiques et formes modulaires/Elliptic Curves and Modular Forms


Hiver/Winter 2017

Professeure/Professor:    Matilde Lalín

Échéancier/Dates and times:    January 10 janvier - April 12 avril (pas de cours/no classes February 28 février et/and March 1 mars)

mardi/Tuesdays 11h - 13h, mercredi/Wednesdays 13h30 - 14h30 Pav. A.-AISENSTADT 4186

Disponibilité/Office hours:   mardi/Tuesdays 13-14, mercredi/Wednesdays 11h30-12h30 Pav. A.-AISENSTADT 5145

Tel:   (514) 343-6689

couriel/e-mail:    mlalin at dms . umontreal . ca

Manuels recommandés/Recommended references:    "Elliptic Curves", J. S. Milne, BookSurge Publishers, 2006,

"The Arithmetic of Elliptic Curves", J. Silverman, Second Edition, Springer, 2009.

Information:


Devoir/Homework:

Avis importants/Special Announcements:
  • Teaching evaluations!!! The teaching evaluations are available on-line at the Omnivox system until April 17. You're kindly requested to evaluate your professors, instructors, and teaching assistants. These evaluations are essential to teaching improvement and are strictly confidential.
  • Barème/Grade distribution: Devoir/Homework (100%) (Tous les devoirs seront réparties également.)/(Assignments will have the same weight.) Le devoir le moins bon de chaque étudiant sera ignoré. / The worst of the five assignment marks will be dropped.

Thèmes:
  • April 12 avril : Zeta functions of varieties over Q, of elliptic curves over Q, modularity, Fermat, BSD, Mahler measure and Beilinson(!!!)
  • April 11 avril : Zeta functions: Riemann's, Dedekind's, zeta functions of affine plane curves over finite fields, of projective curves over finite fields, the Weil conjectures.
  • April 5 avril : The parallelogram law for the canonical height. The end of the proof of Mordell's theorem! A few words about elliptic curves over finite fields, the Hasse bound, and Frobenius.
  • April 4 avril : A bound for the height of the image of a point by a morphism, a bound for the height in the Veronese map, heights on E, the canonical height, properties over torsion points, the parallelogram law.
  • March 29 mars : A crash course in Algebraic Number Theory, the weak finite basis theorem without a point of order 2, definition of heights in P^n(Q).
  • March 28 mars : An example of application of the weak finite basis for a point of order 2, bounding the rank when we have full 2-torsion, the weak finite basis theorem without a point of order 2 (everything except for E/2E)
  • March 22 mars : End of the proof of the weak finite basis for a point of order 2.
  • March 21 mars : More on the relationship between E and its lattice, torsion points and endomorphisms. Introduction to Mordell-Weil theorem, beginning of the proof of the weak finite basis for the case of a point of order 2.
  • March 15 mars : Snow day!
  • Jour de Π-day : Quotients of C by lattices and Riemann surfaces, maps between C/Λ and C/Λ', the elliptic curve E(Λ), classification of elliptic curves over C.
  • March 8 mars : The relationship between P and P'. The field of doubly periodic functions.
  • March 7 mars : Periods in an elliptic curve, lattices, doubly periodic functions, the Weierstrass P-function Eisenstein series.
  • March 1 mars : No class (winter break)
  • February 28 février : No class (winter break)
  • February 22 février : Examples of torsion in E(Q), statement of Mazur's theorem. Elliptic curves over C, the invariance of the invariant differential, a motivating discussion about how the integral of omega should be considered over a torus.
  • February 21 février : elliptic curves over Q_p, filtration, E^1(Q_p) is torsion free, torsion points of E(Q) and Lutz-Nagell.
  • February 15 février : singular cubics (continuation), classification of reduction, semistable reduction, elliptic curves over Q_p (introduction).
  • February 14 février ♥ : Isogenies and dual isogenies, Hom(E_1,E_2), End(E), Aut(E), reduction modulo p and singular cubics.
  • February 8 février : j-invariant, addition formulas.
  • February 7 février : Remaining equivalences between the definitions of elliptic curves, the Weierstrass form, discriminant, j-invariant.
  • February 1 février : constant, dominating, and surjective morphisms, degree of rational maps. Statement of the equivalent definitions of elliptic curve.
  • January 31 janvier : Omega (differential for elliptic curves) revisited, statement of Riemann-Roch, the group law in the Picard group, rational and regular maps on curves, ismorphisms.
  • January 25 janvier : Differentials, the group of divisors.
  • January 24 janvier : Group law in a Cubic, rational and regular functions on affine and projective curves, uniformizer, introduction to differentials.
  • January 18 janvier : p-adic numbers and Hensel's lemma.
  • January 11 janvier : Resultants, introduction to intersection numbers.
  • January 10 janvier : Benvenue/Welcome! Pourquoi on est intéressé aux courbes elliptiques, courbes algébriques planes, singularités/ Why do we care about elliptic curves, Plane algebraic curves, Singularities

Ouvrages complémentaires:

  • "Lectures on Elliptic Curves", J. Cassels
  • "Introduction to Elliptic Cuves and Modular Forms", N. Koblitz
  • "Elliptic Curves", A. Knapp
  • "Rational Points on Elliptic Curves", J. Silverman and J. Tate,


Dernière mise à jour: le 4 janvier 2017 (ou plus tard)