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

AbstractWe 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∗(4n). For f1 and f2 two cuspforms on H, consider their theta liftings θf1 and θf2 on G. Then we compute a Rankin–Selberg type integral and obtain an integral representation of the standard L-function:〈θf1⋅Es,θf2〉G=〈f1,f2〉H⋅Lstd(f1,s). Also a short proof the Siegel–Weil–Kudla–Rallis formula is given. This implies that at the critical point s=s0=m−n+12 Eisenstein series Es have rational Fourier coefficients. Via the natural embedding G×G↪G˜=O∗(8n) 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∗(4n) 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