Integral representations of L-functions and Siegel-Weil-Kudla-Rallis formulas

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternionic groups Sp*(m, 0). We consider the reductive dual pair G = O*(4n), H = Sp*(m, 0) (skew-hermitian and hermitian quaternionic groups). For f1 and f 2 two cuspforms on H, consider their theta-liftings qf1 and qf2 on G. We construct a family of Siegel Eisenstein series Es on G. Then we compute a Rankin-Selberg type integral over Gk GA involving two theta-liftings and Eisenstein series and obtain &angl0;qf1˙Es ,qf2&angr0;=&angl0;f1 ,f2&angr0;˙Lstd f1,s. The L-function is the standard Langlands L-function which can also be given an integral representation by the doubling method. Both the above integral representation and that from doubling method yield the meromorphic continuation and functional equation of the L-function.;In the other part of the thesis, we prove a Siegel-Weil-Kudla-Rallis formula for the pair G, H by invoking irreducibility of degenerate principal series and uniqueness of certain distributions. This implies that at the critical point s = s0 = m - n + ½ Eisenstein series Es have rational Fourier coefficients.;Via the natural embedding G x 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. We obtain a new proof of Shimura's fundamental result in the theory of arithmetic automorphic forms, namely 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. Finally we obtain special value results for the L-functions.