Special values of L-functions by a Siegel–Weil–Kudla–Rallis formula
✍ Scribed by Çetin Ürtiş
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 292 KB
- Volume
- 125
- Category
- Article
- ISSN
- 0022-314X
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✦ Synopsis
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 = mn + 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.