In this article we give an explicit formula for the number of k-normal elements, hence answering Problem 6.1. of Huczynska et al. (Existence and properties of k-normal elements over finite fields, Finite Fields Appl 2013; 24: 170-183). Furthermore, for some cases we provide formulas that require the solutions of some linear Diophantine equations. Our results depend on the explicit factorization of cyclotomic polynomials.

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.

We prove Galois equivariance of ratios of Petersson inner products of holomorphic cuspforms on symplectic, unitary, or Hermitian orthogonal groups. As a consequence, we show that the ratios of Petersson norms of such cuspforms with the same Hecke eigenvalues are algebraic. We also show that spaces of such cuspforms of sufficiently high fixed weight and level are spanned by theta series.

No abstract

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.