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 4186Disponibilité/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:

- due/date limite April 11 avril Devoir 5/Homework 5 (Solutions)

- due/date limite March 21 mars Devoir 4/Homework 4 (Solutions)

- due/date limite March 7 mars Devoir 3/Homework 3 (Solutions)

- due/date limite February 14 février Devoir 2/Homework 2 (Solutions)

- due/date limite January 31 janvier Devoir 1/Homework 1 (Solutions)

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.

- The books of Silverman and Cassels books are available at the librairie de la Université de Montréal (Librairie Scientifique, L-315), and here is the book of Milne.

- 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 17 janvier : Intersection numbers, projective plane curves, Bezout's theorem, p-adic numbers. Selmer's example of the failure of Hasse--Minkowski by K. Conrad

- 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:

- "Algebraic Curves", Fulton

- "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)