Eichler cohomology theorem for automorphic forms of small weights

Abstract

Abstract. Let Γ be an H-group. In 1974 Marvin Knopp conjectured that the Eichler cohomology group, with base space taken to be the set of all functions holomorphic in the upper half-plane, of polynomial growth at the real line (including ∞), and with a weight k,multiplier system v linear fractional action of Γ, is isomorphic to the space of cusp forms on Γ of weight 2 − k and multiplier system v, in the range 0 < k < 2. In this article the authors prove the conjecture by making essential use of Hans Petersson's "principal parts condition" for automorphic forms (1955).

Holomorphic modular forms and cocycles

“…Then the theorem gives the vanishing of the cohomology group. Knopp proved most of the statements in his 1974 paper and completed the proof with Mawi [7]. Knopp used a simpler notation for the module of functions with at most polynomial growth.…”

Modular forms of real weights and generalised Dedekind symbols

“…In this section, I fix notation and give a brief survey of relevant definitions and results from [11,12], and [4]. I adopt conventions of [4], where modular forms of real weights are holomorphic functions on the upper complex half-plane, whereas their period integrals, analogues of (1.3), are holomorphic functions on the lower half-plane.…”

“…Functions invariant with respect to this action and having exponential decay at cusps (in terms of geodesic distance, cf. [12] and [3]) are called cusp forms for the full modular group of weight k with multiplier system v. For such a form F (z), one can define its period function P F (t) by the formula similar to (1.3). Generally, it is defined only on H − and satisfies the polynomial growth condition near the boundary.…”

Period functions of half-integral weight modular forms

“…Knopp and Mawi also proved that there is an isomorphism between the space of cusp forms of a half-integral weight and the cohomology group. More precisely, in [9] Knopp and Mawi proved that…”

“…For half-integral weights Knopp [8] defined a bigger space P, which is the space of holomorphic functions g(z) on the upper half plane H satisfying for some positive constants K, ρ and σ, where z = x + iy ∈ H. In [8] and [9] Knopp and Mawi proved that the space of cusp forms of a half-integral weight is isomorphic to the cohomology group associated with P.…”