Special values of L-functions by a Siegel–Weil–Kudla–Rallis formula

Abstract

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp ∗ ( m , 0 ) which form a reductive dual pair with G = O ∗ ( 4 n ) . For f 1 and f 2 two cuspforms on H, consider their theta liftings θ f 1 and θ f 2 on G. Then we compute a Rankin–Selberg type integral and obtain an integral representation of the standard L-function: 〈 θ f 1 ⋅ E s , θ f 2 〉 G = 〈 f 1 , f 2 〉 H ⋅ L std ( f 1 , s ) . Also a short proof the Siegel–Weil–Kudla–Rallis formula is given. This implies that at the critical point s = s 0 = m − n + 1 2 Eisenstein series E s have rational Fourier coefficients. Via the natural embedding G × G ↪ G ˜ = O ∗ ( 8 n ) we restrict the holomorphic Siegel-type Eisenstein series E ˜ on G and decompose as a sum over an orthogonal basis for holomorphic cusp forms of fixed type. As a consequence we prove that the space of holomorphic cuspforms for O ∗ ( 4 n ) of given type is spanned by cuspforms so that the finite-prime parts of Fourier coefficients are rational and obtain special value results for the L-functions.