Abstract :
Let ℓ be any prime number greater-or-equal, slanted2, and let N be any integer greater-or-equal, slanted1 with (N, ℓ)=1. We generalize J.-P. Serreʹs Théorème 5.4 in (1976, Enseign Math. (2)22, 227–260) and Théorème 12 in (1973, Lecture Notes in Mathematics, Vol. 350, pp. 191–268, Springer-Verlag, Berlin/Heidelberg) for classical modular forms on Γ0(ℓm) and (Γ1(ℓ) for ℓgreater-or-equal, slanted3) and ℓ-adic modular forms on SL2(image) to arbitrary classical modular forms of nebentypus of level Nℓm and ℓ-adic modular forms of level N. We express any modular form of nebentypus of level Nℓm as an ℓ-adic modular form of level N. We extend Ribetʹs Theorem (2.1) in [8] on mod ℓ Galois representations for odd characteristics to the case of characteristic 2. Finally, we give another method using rigid analysis.
