On type numbers of split orders of definite quaternion algebras (Q1911192)

Let \(q\) be a product of \(r\) distinct primes, where \(r\) is odd, and let \(B\) be the definite quaternion algebra over \(\mathbb{Q}\) with discriminant \(q\). Let \(N\) be a positive integer with \((q,N) = 1\) and let \(O\) be an order with level \(qN\), a so-called split order of type \((q,N)\) (for squarefree \(N\) this is an Eichler order). Let \(T_{q,N}\) denote the type number of this split order \(O\) of type \((q,N)\). Explicit formulas for \(T_{q,N}\) are known [see \textit{A. Pizer}, Acta Arith. 31, 61-89 (1976; Zbl 0348.12018)]. The authors associate \(T_{q,N}\) with the dimension of some subspace of \(S_2 (qN)\), where \(S_2 (M)\) is the space of cusp forms of weight 2 for \(\Gamma_0 (M)\): more precisely, let \(S_2 (q,N)\) be the subspace of \(S_2 (qN)\) defined by the sum \[ S_2 (q,N) = \bigoplus_{m |N} \bigoplus_{d |{N \over m}} S^0_2 (qm)^{[d]}, \] where \(S^0_2 (M)\) is the subspace of \(S_2 (M)\) spanned by newforms for \(\Gamma_0 (M)\) and \(S^0_2 (M)^{[d]} = \{f(d \tau) \mid f (\tau) \in S^0_2 (M)\}\). Then the main result is \[ T_{q,N} = 1 + \dim S_2 (q,N)^{(-, +)}, \] where \(S_2 (q,N)^{(-, +)}\) is the common \((-1, +1)\)-eigenspace for the Atkin-Lehner involutions corresponding to the prime divisors of \(q,N\) respectively.