MAT 6684w


Sujets spéciaux en théorie des nombres - formes modulaires/ Special Topics in Number Theory - Modular Forms


Automne/Fall 2017

Professeure/Professor:    Matilde Lalín

Échéancier/Dates and times:    September 6 septembre - December 12 décembre (pas de cours/no classes October 23 et/and 25 octobre)

lundi/Mondays 11h - 12h Pav. A.-AISENSTADT 5183, mercredi/Wednesdays 13h - 15h Pav. A.-AISENSTADT 5448

Disponibilité/Office hours:   mardi/Tuesdays 10h30-12h30, mercredi/Wednesdays 15-16 Pav. A.-AISENSTADT 5145

Tel:   (514) 343-6689

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

Manuels recommandés/Recommended references:    ''Modular forms'', Lecture notes, P Bruin & S. Dahmen

''A First Course in Modular Forms'', GTM 228, Springer-Verlag 2005, F. Diamond & J Shurman

''Introduction to Elliptic Curves and Modular Forms'',GTM 97, Springer-Verlag 1993, N. Koblitz

''A Course in Arithmetic''(Chap. 7), GTM 7, Springer-Verlag, 1996, J.P. Serre


Information:



Devoir/Homework:

  • due/date limite November 27 novembre
  • due/date limite November 13 novembre
  • due/date limite October 30 octobre
  • due/date limite October 11 octobre (modified date!)


Avis importants/Special Announcements:

  • Homework 2's due date has been moved to October 11, since October 9 is a holiday.
  • 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:

  • September 25 septembre (préliminaire/tentative) : dimensions of M_k and S_k, and consequences of this, congruence subgroups of SL_2(Z)
  • September 20 septembre : The modular form Δ, the Dedekind η function, the modular function j, the valence formula, definition of M_k and S_k, zeros for E_4, E_6, Δ
  • September 18 septembre : The Eisenstein series of weight 2 and its transformation respect to the modular group
  • September 13 septembre : Meromorphic/holomorphic at infinity, the q-series of a weakly modular function that is meromorphic at infinity, definition of modular forms and cusp forms for the full modular group, Eisenstein series, convergence, q-expansions of Einsenstein series
  • September 11 septembre : Properties of the fundamental domain of the full modular group, definition of weakly modular functions for the full modular group and the slash operator
  • September 6 septembre : Bienvenue/Welcome! Pourquoi on est intéressé aux formes modulaires? / Why do we care about modular forms? Le groupe modulaire et le demi-plan de Poincaré/The modular group and the complex upper half-plane, homogeneous functions on lattices, a description of the fundamental domain of H by the action of SL_2(Z)


Ouvrages complémentaires:

  • D. Zagier, "Introduction to modular forms". From number theory to physics (Les Houches, 1989), 238-291, Springer, Berlin, 1992.




Dernière mise à jour: le 11 août 2017 (ou plus tard)