Shintani function and its application to automorphic \(L\)-functions for classical groups. I: The case of orthogonal groups
Let \(H\) be the orthogonal group of a quadratic form over \(\mathbb{Q}\) which is positive definite over \(\mathbb{R}\) and \(G\) the isotropy subgroup of some nonzero vector. If \(F\) (resp. \(f)\) is an automorphic form on \(H(\mathbb{Q}) \backslash H (\mathbb{A})\) (resp. \(G(\mathbb{Q}) \backslash G (\mathbb{A}))\), the inner product of \(f\) with the restriction of \(\rho (h)F\) to \(G(\mathbb{Q}) \backslash G (\mathbb{A})\) is called Shintani function by the authors. It appears in the ``basic identity'' for a certain Rankin-Selberg type zeta- integral. Assume \(F\) and \(f\) are invariant under \(H(\mathbb{R})\) resp. \(G(\mathbb{R})\) and are eigenfunctions of the Hecke algebras. Then the zeta- integral in question is shown to be (essentially) the quotient of two standard \(L\)-functions for \(H\) and \(G\). This is dense by explicitly computing the local zeta-integrals. The result is proved under cerrtain assumptions on the quadratic form. Conjectures concerning the functional equation and the poles of the \(L\)-functions considered here are formulated and partially proved.
scientific article; zbMATH DE number 590001
scientific article