A conjecture on the Eichler cohomology of automorphic forms
Let \(\Gamma\) be a Fuchsian group of the first kind. That is: (i) \(\Gamma\) is a discrete group of real linear fractional transformations (fixing \(H\), the upper half-plane); (ii) \(\Gamma\) contains translations; (iii) \(\Gamma\) has a fundamental region of finite hyperbolic volume. In [Bull. Am. Math. Soc. 80, 607-632 (1974; Zbl 0292.10022)] the reviewer introduced Eichler cohomology groups connected with automorphic forms on such \(\Gamma\), of arbitrary real weight and multiplier system (MS), and with an underlying space \({\mathcal P}\) of functions holomorphic in \(H\) and of polynomial growth at points of \(\mathbb{R}^*\), the real line with \(\infty\) adjoined. With the action defined by \[ (f|_{-k}^v M)(z)= \overline{v}(M) (cz+d)^{-k} f(Mz) \qquad \left(M= \left( \begin{smallmatrix} * &*\\ c &d \end{smallmatrix} \right)\in \Gamma \text{ and }z\in H \right), \] the cohomology group \(H_{k,v}' (\Gamma,{\mathcal P})\) is defined to be the space of cocycles modulo coboundaries (all in \({\mathcal P})\). Let \(C^0 (\Gamma,k+2, \overline{v})\) denote the space of cusp forms on \(\Gamma\), of weight \(k+2\) and MS \(\overline{v}\) (invariant under the action \(|_{k+2}^{\overline{v}}M\), \(M\in \Gamma\)). Theorem 1 of the reviewer's cited article proves that \(H_{k,v}' (\Gamma,{\mathcal P})\) and \(C^0 (\Gamma,k+2,\overline{v})\) are isomorphic if \(k\geq 0\) or \(k\leq-2\), and the article closes with the conjecture: This isomorphism is valid in the range \(-2< k< 0\), as well. The article under review establishes this conjecture in part: Theorem: If \(-2< k< -1\) and \(C^0 (\Gamma,k+2, \overline{v})=0\), then \(H_{k,v}' (\Gamma,{\mathcal P})=0\). The proof employs the Hecke convergence factor [\textit{E. Hecke}, Abh. Math. Sem. Hamb. Univ. 5, 199-224 (1927; JFM 53.0345.02)], the analytic continuation of the Selberg zeta-function [\textit{A. Selberg}, Proc. Symp. Pure Math. 8, 1-15 (1965; Zbl 0142.33903)] and ``some estimations''. The estimations to which the author refers undoubtedly require application of the nontrivial upper bound for the Selberg zeta-function given by \textit{D. Goldfeld} and \textit{P. Sarnak} [Invent. Math. 71, 243-250 (1983; Zbl 0507.10029)]. The author's methods very likely can be extended to give a proof of the full conjecture.
scientific article; zbMATH DE number 1542546
scientific article