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 * (4n).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: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 * (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.

Arithmeticity of holomorphic cuspforms on Hermitian symmetric domains

“…Shimura in [40] studied a similar decomposition very generally for the cases of symplectic and unitary groups. Ürtiş studied a case of quaternion groups in [42]. The following (now standard) lemma is proved in [9].…”

“…See [27] for a survey of recent work in this direction. For similar extensions in the unitary group case see Theorems 4.1 and 4.2 of [22] and Theorem 1.1 of [23], and a version for Hermitian orthogonal groups is Theorem 2 of [42].…”

Algebraicity of L-values attached to quaternionic modular forms

“…This domain arises when we select to be a definite quaternion algebra and the form skew-hermitian (see the next section for details). There are already some works for these modular forms (for example, [3, 16, 35, 37]), but it is fair to say that these modular forms are not as intensively studied as the Siegel or Hermitian ones. Even more importantly, most, if not all, of the works are restricted to the case when the dimension of V is even.…”

Algebraicity of L-values attached to Quaternionic Modular Forms

“…This domain arises when we select D to be a definite quaternion algebra and the form , skew hermitian (see next section for details). There are already some works for these modular forms, for example [2,14,33,35], but it is fair to say that these modular forms are not as intensively studied as the Siegel or Hermitian ones. Even more importantly most, if not all, of the works are restricted to the case when the dimension of V is even.…”