Wissam Raji

EICHLER COHOMOLOGY FOR GENERALIZED MODULAR FORMS

By using Stokess theorem, we prove an Eichler cohomology theorem for generalized modular forms with some restrictions on the relevant multiplier systems.

FOURIER COEFFICIENTS OF GENERALIZED MODULAR FORMS OF NEGATIVE WEIGHT

The Fourier coefficients of classical modular forms of negative weights have been determined for the case for which F(τ) belongs to a subgroup of the full modular group [9]. In this paper, we determine the Fourier coefficients of generalized modular forms of negative weights using the circle method.

EICHLER COHOMOLOGY THEOREM FOR GENERALIZED MODULAR FORMS

We show starting with relations between Fourier coefficients of weakly parabolic generalized modular forms of negative weight that we can construct automorphic integrals for large integer weights. We finally prove an Eichler isomorphism theorem for weakly parabolic generalized modular forms using the classical approach as in [3].

Eichler cohomology for generalized modular forms ii

In the present work we derive further results on the Eichler cohomology of generalized modular forms by means of the methods of [4]. In a departure from [4], we here shift the focus from generalized modular forms of integral weight to those of arbitrary real weight. As it turns out, the all-important application of Stokes’s theorem suggested by J. Lehner [4, sections 4-5] could work equally well in this broader context, including in particular, for the difficult range of ”small” weights k ∈ (0, 2). However the use of Stokes’s theorem is rendered unnecessary here by our use instead of the appropriate known cohomology theorem with unitary multiplier systems (see [3] and [8]), together with Theorem 4.2 of [7]. We note that Stokes’s theorem is essential in [8], which deals with weights in (0, 2) and unitary multiplier systems. Since the weight here is not necessarily in Z, polynomials of fixed degree cannot serve (as they do in [4]) as the underlying space of functions in the definition of the Eichler cohomology groups we study. Instead, we employ as the underlying space the collection P of all functions holomorphic in H, the upper half-plane, which are also of polynomial growth upon approach to the real line R and to i∞. This space was introduced in [3, p.612] in the context of the Eichler cohomology theory for unitary (i.e. the usual) modular forms of arbitrary real weight.

Generalized Maass wave forms

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Eichler cohomology of generalized modular forms of real weights

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A new proof of the transformation law of Jacobi’s theta function ₃(,)

. We present a new proof, using Residue Calculus, of the transformation law of the Jacobi theta function θ 3 (ω,τ) defined in the upper half plane. Our proof is inspired by Siegels proof of the transformation law of the Dedekind eta function.