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



Avis importants/Special Announcements:

  • Teaching evaluations!!! The teaching evaluations are available on-line at the Omnivox system until December 8. 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.


  • December 6 décembre (préliminaire/tentative) : minimal models, types of reduction, conductor, L-functions, Birch and Swinnertin-Dyer, modularity, the congruent number problem, Fermat Last's Theorem(!?).
  • December 4 décembre : Weierstrass forms, elliptic curves, Mordell-Weil.
  • November 29 novembre : X(Γ) is Hausdorff and compact, definition of Riemann surface, X(Γ) is a Riemann surface, genus, ramification and Riemann-Hurwitz formula. Projective plane and points at infinite (idea, in preparation for elliptic curves)
  • November 27 novembre : the functional equation, definition of modular curves and construction of its topology.
  • November 22 novembre : the eigenvector basis for the newforms also works for T_m when (m,N)>1, we get an orthogonal basis for the newforms, L-functions, the Mellin transform, the L-function of a modular form, half of the proof of the functional equation.
  • November 20 novembre : we will finish the proof that all the relevant spaces are preserved by the Hecke operators and their adjoints, Atkin-Lehner operator, primitive forms, an eigenvector for T_m when (m,N)=1 and the diamond operators is also an eigenvector for T_m when (m,N)>1.
  • November 15 novembre : the adjoints of the Hecke operators T_m and < d >, oldforms and newforms, we are in the middle of proving that all the relevant spaces are preserved by the Hecke operators and their adjoints.
  • November 13 novembre : the adjoints of the Hecke operators T_\alpha
  • November 8 novembre : Hecke eigenforms and their eigenvalues for < d >, M_k(Γ_1(N),χ), Ramanujan's identity, the Petersson inner product
  • November 6 novembre : The effect of T_n on q-expansions, Hecke eifenforms and their eigenvalues for T_n, normalized eigenforms
  • November 1st/er novembre : Hecke operators T_p formula, lattice interpretation, the Hecke algebra, the Hecke operators T_n
  • October 30 octobre : Hecke operators for Γ_1(N), < d > and T_p
  • October 18 octobre : more on Dirichlet characters (conductor, examples), sums of squares, Hecke operators, the operator T_α
  • October 16 octobre : the valence formula (end of the proof), bounds on dimension of the vector spaces of modular forms, Dirichlet characters (modulo, number of characters modulo N, principal character)
  • October 11 octobre : the Jacobi Θ function, the Eisenstein series of weight 2 strikes back, the valence formula (almost the whole proof)
  • October 4 octobre : the cusps of Γ_0(p), modular forms for congruence subgroups, holomorphicity at a cusp, an equivalent condition in term of growth of Fourier coefficients, the Jacobi Θ function (introduction).
  • October 2 octobre : width of a cusp, regular and irregular cusps, the sums of widths
  • September 27 septembre : Γ_1(N) as a subgroup of Γ_0(N), the lattice problem for Γ_0(N), some technical properties, an example of a fundamental domain, cusps
  • September 25 septembre : 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)