On the dimensions of spaces of Siegel modular forms of weight one
Let \(G=Sp_n(\mathbb{R})\) be the real symplectic group of degree \(n\), and, given a positive integer \(m\), let \(\Gamma(m)\) be the principal congruence subgroup of \(Sp_n(\mathbb{Q})\) of level \(m\). For any integer \(k\geq 0\) let \(\pi_k\) be the irreducible holomorphic unitary representation of \(G\) having lowest weight \((k+1,1,\dots,1)\). Thus automorphic forms of type \(\pi_k\) correspond to classical Siegel modular forms with values in the \(k\)th symmetric power twisted by the determinant; in particular \(\pi_0\) corresponds to scalar valued Siegel modular forms of weight one. For \(k\geq n\), \(\pi_k\) is a unitary representation with non-vanishing relative Lie algebra cohomology, with twisted coefficients if \(k>n\). If \(\pi\) is an irreducible unitary representation of \(G\) let \(m(\pi,\Gamma(m))\) denote the multiplicity with which \(\pi\) occurs in the discrete spectrum of \(L^2(\Gamma(m) \setminus G)\). Suppose \(n\geq 3\). Then as shown \(m(\pi_k,\Gamma(m))=0\) unless either \(4\mid m\), or \(m\) has a prime factor \(p\) with \(p\equiv-1\bmod 4\). For an odd prime \(p>3\) with \(p\equiv-1\bmod 4\) and \(r\geq 1\) the following formula for the multiplicity \(m(\pi_k, \Gamma(p^r))\), \(k>0\), holds: \[ m(\pi_k, \Gamma(p^r))= (h/2)(p^{n(2r-1)}+ (-1)^k), \] where \(h\) denotes the class number of the field \(\mathbb{Q}(\sqrt{-p})\). A similar formula holds for \(k=0\). As a consequence, there is an analogous formula for the dimension of the \(L^2\)-cohomology \(H^n_{(2)} (\Gamma(p^r) \setminus H_n)\) of the associated locally symmetric space \(\Gamma(p^r) \setminus H_n\) [\(H_n\) the Siegel upper half space of degree \(n\)]. The method behind the proof is the following: The representations \(\pi_k\) come up in a list of all irreducible unitary representations of \(G\) which are of rank 2 in the sense of \textit{R. Howe} [On a notion of rank for unitary representations of classical groups, in: C.I.M.E. Summer School in Harmonic Analysis, 223-331 (1980)]. Globally (and locally) the irreducible automorphic representations \(\pi\) of \(Sp_n(\mathbb{A})\) are well understood. There is a binary quadratic form \(q\) over \(\mathbb{Q}\) and an automorphic representation \(\tau\) of the corresponding isometry group \(O(q)(\mathbb{A})\) such that \(\pi=\Theta(\tau)\), i.e. \(\pi\) is the theta lift of \(\tau\). The binary form \(q\) must be anisotropic over \(\mathbb{Q}\). Then the multiplicity in question can be expressed as \[ m(\pi_h,\Gamma(m))= \sum_q \sum_\tau \prod_{p\mid m}\dim \Theta(\tau_p)^{Kp(m)}, \] where the outer sum is over all isometry classes of anisotropic binary forms over \(\mathbb{Q}\) which are unramified at all primes \(\neq\infty\) and not dividing \(m\). The inner sum ranges over all automorphic representations \(\tau=\otimes \tau_p\) of \(O(q)(\mathbb{A})\) with \(\Theta(\tau_\infty)= \pi_k\) and \(\tau_p\) unramified for all primes as above.
scientific article; zbMATH DE number 912462