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

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*(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:left angle bracketθf1dot operatorEs,θf2right-pointing angle bracketG=left angle bracketf1,f2right-pointing angle bracketHdot operatorLstd(f1,s). Also a short proof the Siegel–Weil–Kudla–Rallis formula is given. This implies that at the critical point image Eisenstein series Es have rational Fourier coefficients. Via the natural embedding image we restrict the holomorphic Siegel-type Eisenstein series image 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.