Eichler cohomology theorem for automorphic forms of small weights

Abstract:

Let $\Gamma$ 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 $\infty$), and with a weight $k,$multiplier system $v$ linear fractional action of $\Gamma$, is isomorphic to the space of cusp forms on $\Gamma$ of weight $2-k$ and multiplier system $\overline {v}$, 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).