Hilbert Cusp Forms Research Articles

On Hilbert and quaternionic cusp forms

The aim of this paper is to determine in a natural manner the subspace of the space of Hilbert modular newforms of level n which correspond to eigenforms of an appropriate quaternion algebra, in the sense of having the same eigenvalues with respect to the corresponding Hecke operators. This study may be seen as a particular case of the Jacquet–Langlands correspondence.

Unramified Hilbert modular forms, with examples relating to elliptic curves

We give a method to explicitly determine the space of unramified Hilbert cusp forms of weight two, together with the action of Hecke, over a totally real number field of even degree and narrow class number one. In particular, one can determine the eigenforms in this space and compute their Hecke eigenvalues to any reasonable degree. As an application we compute this space of cusp forms for Q(root 509), and determine each eigenform in this space which has rational Hecke eigenvalues. We find that not all of these forms arise via base change from classical forms. To each such eigenform f we attach an elliptic curve with good reduction everywhere whose L-function agrees with that of f at every place.

Petersson norms and liftings of Hilbert modular forms

We prove the algebraicity of the ratio of the Petersson norm of a holomorphic Hilbert modular form over a totally real number field and the norm of its Saito–Kurokawa lift. We prove a similar result for the Ikeda lift of an elliptic modular form. In order to obtain these we combine some results on local symplectic groups to generalize a special value of the standard L-function attached to a Siegel–Hilbert cuspform.

Poincaré Series and Hilbert Modular Forms

In this paper, we bound the square moment of the linear form in the Fourier coefficients of Hilbert modular forms by means of Poincare series, and obtain sharp estimate on the critical line for the fourth moment of L-functions associated with Hilbert cusp forms which are primitive Hecke eigenforms.

Hilbert cusp forms and special values of Dirichlet series of Rankin type

Let K be a totally real number field of degree nover ℚ and let c be an integral ideal of a maximal order of K. Given a nonnegative integer j and a Hecke character on the group of ideles of K, let denote the space of Hilbert cusp forms of holomorphic type on ℋn of weight j, level c and character ψ where ℋn is the n-th power of the Poincaré upper half plane ℋ.Let g be an element of , where 1 is the trivial character. If u ∈ Sk(c, ψ), then the product gu is an element of Sk+l (c, ψ), and therefore we can consider the linear map sending u to gu. Let be the adjoint of the linear map Φg with respect to the Petersson inner product.

Non-abelian base change for totally real fields

Let F be a number field. A cohomological Hecke eigen cusp form f on GL2(FA) for the adele ring FA of F is called a base change of f if L(s, f) = L(s, ρF ) for ρF = ρ ∣∣ Gal(Q/F ). When f exists, we call f “liftable to F” or “F -liftable”. When F is totally real, a Hilbert cusp form is cohomological if and only if it is holomorphic of weight k ≥ 2 at every archimedean place of F . Since the definition of the base change f is given in purely algebraic terms, it is a non-trivial problem to find the base-change form f in the space of cohomological forms characterized by analytic means. This problem was first studied by Doi and Naganuma [DN] and [N] for quadratic extensions F/Q, and the existence of f was later proved by Langlands [L] for relative prime cyclic cases (so eventually covering all solvable cases).

Diagonalizing Hilbert cusp forms

We develop an operator C^Q) on the space Sk{λί, Φ) of Hubert cuspforms as an alternative to the Hecke operator Tq for primes q dividing λί. For f G Sk{λί, Φ) a newform, we have f | Cq(Φg) = f | Γq. We are able to decompose the space Sk(Λf, Φ) into a direct sum of common eigenspaces of {Γ p, Cq(Ψg) : p λί, q I ΛΓ}, each of dimension one. Each common eigenspace is spanned by an element with the property that its eigenvalue with respect to Tp (resp. Cq(ΦQ)) is its p th (resp qth) Fourier coefficient. We finish by deriving bounds for the eigenvalues of

On the trace formula of the Hecke operators and the special values of the second L-functions attached to the Hilbert modular forms

In this paper, we consider 1) the explicit formula of the trace of the Hecke operators acting on the space of the Hilbert cusp forms, and 2) the special values of the second L-functions attached to the Hilbert cusp forms. The method is that of Zagier[12], and the results of this paper are the generalization of his results to the case of the Hilbert modular forms for the congruence subgroup Γ0 (M).

On an operator $U_\chi$ acting on the space of Hilbert cusp forms

On an operator $U_\chi$ acting on the space of Hilbert cusp forms