On the trace of the representation of \(\text{SL}(2,{\mathbb Z}/N{\mathbb Z})\) in the space of modular forms of half integral weight (Q2725504)

For positive integers \(N\) and \(k\), let \(\Gamma(N)\) be the principal congruence subgroup of level \(N\) in the modular group \(\Gamma= \text{SL}(2,\mathbb{Z})\), and let \(S= S_{(2k+1)/2} (\Gamma(4N))\) be the space of cusp forms of half integral weight \(k+\frac 12\) on \(\Gamma(4N)\). A representation \(\Pi= \Pi_{4N,(2k+1)/2}\) of the factor group \(\Gamma(4)/ \Gamma(4N)\) is defined by \((\Pi[\alpha]f)(z)= j(\alpha,z)^{-(2k+1)} f(z)\) for \([\alpha]= \Gamma(4N)\alpha\), \(\alpha\in \Gamma(4)\), and \(f\in S\), where \(j(\alpha,z)\) is the automorphic factor coming from Jacobi's theta function. Note that the groups \(\Gamma(4)/ \Gamma(4N)\) and \(\text{SL}(2,\mathbb{Z}/N\mathbb{Z})\) are isomorphic. The author uses \textit{G. Shimura}'s trace formula [Acta Math. 132, 245-281 (1974; Zbl 0285.10018)] to prove an explicit formula for the trace of \(\Pi[\alpha]\) on \(S\). For primes \(N=p\) he computes the multiplicities of all irreducible representations of \(\text{SL}(2,\mathbb{Z}/p\mathbb{Z})\) in the representation \(\Pi\). The multiplicities involve the class number \(h(-p)\) of the imaginary quadratic field \(\mathbb{Q}(\sqrt{-p})\). This corresponds to Hecke's classical result on the representations of \(\text{SL}(2,\mathbb{Z}/p\mathbb{Z})\) in the space of cusp forms of weight 2 on \(\Gamma\).