A dimension formula for a certain space of automorphic forms of SU(p,1). II: The case of \(\Gamma\) (N) with N\(\geq 3\)
The author gives more explicit expressions for the dimensions of the spaces of cusp forms of \(SU(p,1)\) in the case of the congruence subgroups \(\Gamma(N)\) with \(N\geq 3\) in terms of arithmetic quantities, using his previous paper [Math. Ann. 266, 457-477 (1984; Zbl 0517.10023)]. We denote the space of such cusp forms by \(S_ m(\Gamma (N))\). Suppose \(m>2p\) and \(N\geq 3\). Then one obtains the following: \[ \begin{multlined} \dim S_ m (\Gamma (N)) = [\Gamma (1) : \Gamma (N)] (m-1)! ((m-p-1)!)^{-1} \pi^{-p} \text{vol}(\Gamma (1)\setminus G_\mathbb{R}) + \\ + [\Gamma (1) : \Gamma (N)] 2^ t \delta^ p N^{-p} \zeta (1-p) e(p,d), \end{multlined} \] where \[ e(p,d) = 2^{-t-p(p-1)/2} \pi^{-p(p-1)/2} \prod^{p-2}_{r=0}r! | D|^{p(p-1)/4} L(1,\chi) \prod^{(p-2)/2}_{r=1} L(2r+1,\chi) \zeta(2r), \] \(D\) is the discriminant of \(k\), \(\delta\) is 1 or 2 according as \(-d\equiv 1 (4)\) or \(-d\equiv 2,3 (4)\), \(t\) is the number of primes which divide the discriminant of \(k\), \(\zeta(s)\) is the Riemann zeta-function and \(\chi(p)\) is the Dirichlet character associated with the field extension \(k/\mathbb{Q}\), \(G_\mathbb{R}\) the group of real points. Notice that \(\text{Vol}(\Gamma(1) \setminus G_\mathbb{R})\) was calculated by \textit{H. Zeltinger} [Bonner Math. Schriften 136 (1981; Zbl 0501.10027)]. The author also proves a certain result concerning the number of \(\Gamma(1)\)-inequivalent cusps conjectured by Zeltinger.
number of \(\Gamma \) (1)-inequivalent cusps
scientific article; zbMATH DE number 3931111
scientific article