Petersson products of singular and almost singular theta series
Let \(F\), \(G\) be Siegel modular forms of degree \(n\) and weight \(k= {m\over 2}\) with \(n> m\) and \(m\equiv 0\pmod 8\). Then \(F\) and \(G\) are square integrable (according to a theorem of Weissauer), and the authors show that the Petersson product \(\langle F,G\rangle\) is equal to the residue of a Dirichlet series of Rankin-Selberg type at \(s= {n\over 2}\) times some constant depending only on \(m\) and \(n\). The same result holds in the case \(k= {n\over 2}\) whenever \(F\) and \(G\) are linear combinations of theta series. Moreover, if \(n\geq m\) and \(F\) is a Siegel modular form of weight \(k= {m\over 2}\) which is a linear combination of theta series and which is a Hecke eigenform, then the scalar product of \(F\) against a theta series is equal to a known constant times the residue of the standard \(L\)-function of \(F\) at \(s= n+1-k\). Application of these beautiful theorems include a remarkable asymptotic orthogonality relation for theta series and several results on theta liftings. In the final section the authors briefly state how to extend the previous results when lattices of level greater than 1 are dealt with, and they give an inner product formula for Siegel modular forms on the Hecke subgroup of level \(N\). -- The proofs of the main results are based on an inner product formula in the adelic representation theoretic setting due to \textit{J.-S. Li} [Am. J. Math. 119, No. 3, 523--578 (1997; Zbl 0871.11036)].
scientific article; zbMATH DE number 2132054
scientific article