Holomorphic Eisenstein Series Research Articles

On zeros of Eisenstein series for genus zero Fuchsian groups

Let r < SL 2 (R) be a genus zero Fuchsian group of the first kind with ∞ as a cusp, and let E Γ 2k be the holomorphic Eisenstein series of weight 2k on r that is nonvanishing at ∞ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on Γ, and on a choice of a fundamental domain F, we prove that all but possibly c(Γ, F) of the nontrivial zeros of E Γ 2k lie on a certain subset of {z ∈ h: j Γ (z) ∈ R}. Here c(Γ, F) is a constant that does not depend on the weight, S) is the upper half-plane, and j Γ is the canonical hauptmodul for Γ.

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

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 ∗ ( 4 n ) . 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: 〈 θ f 1 ⋅ E s , θ f 2 〉 G = 〈 f 1 , f 2 〉 H ⋅ L std ( f 1 , s ) . 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 ∗ ( 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. As a consequence we prove that the space of holomorphic cuspforms for O ∗ ( 4 n ) 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.

Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series

AbstractThe present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of n-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

Special values of triple product L-Functions and nearly holomorphic eisenstein series

Special values of triple product L-Functions and nearly holomorphic eisenstein series

On Eisenstein series on quaternion unitary groups of degree 2

In this paper we give an explicit Fourier expansion of the Eisenstein series on certain quaternion unitary groups of degree 2 by means of Shimura's method. Moreover using an explicit formula of the Fourier coefficients of holomorphic Eisenstein series and Oda's lifted cusp forms, we give some numerical examples.

Fourier coefficients of Eisenstein series on non-congruence subgroups

Let Γ be a subgroup of PSL2 (ℤ) of finite index. In this note we are concerned with the arithmetic nature of the Fourier coefficients of holomorphic Eisenstein series on Γ.

The rationality of holomorphic Eisenstein series

The rationality of holomorphic Eisenstein series