On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

Some computational results on Hecke eigenvalues of modular forms on a unitary group

In this paper we present some computational results on Hecke eigenforms and eigenvalues for a unitary group in three variables. Our results are based on the work of Shiga [SHig], Holzapfel [Holz1,Holz2] and Feustel ]Feustel] which gives in a special case a generating system for the ring of (holomorphic) modular forms consisting of powers of theta constants. We compute all Hecke eigenforms in this ring for weights up to 12 and for each eigenform the first Hecke eigenvalues. Received: 25 July 1997 / Revised version: 7 January 1998     

Linear dependence among Siegel modular forms

Theorems are given which describe when high enough vanishing at the cusps implies that a Siegel modular cusp form is zero. Formerly impractical computations become practical and examples are given in degree four. Vanishing order is described by kernels, a type of polyhedral convex hull. Received: November 19, 1998 / Revised: July 5, 1999 / Published online: September 5, 2000     

Jacobi–Hecke algebras and a rationality theorem for a formal Hecke series

It is well known that the L-function associated to a Siegel eigenform f is equal to a Rankin-Selberg type zeta-integral involving f and a restricted Eisenstein series ([3], [14]). At some point in the proof one has to show the equality of a certain Dirichlet series and the L-function, which follows from a rationality theorem for a certain formal power series over the Hecke algebra. The main purpose of this paper is to develop a Hecke theory for the Jacobi group and to prove such a rationality theorem. Received: 17 August 1998 / Revised version: 17 February 1999     

Siegel cusp forms of degree 12k and weight 12k

In this paper we prove the existence of cusp forms relative to the full modular group whose genus is equal to the weight. These cusp forms are linear combination of theta series. Received: 26 July 1999 / Revised version: 16 September 1999     

Probabilistic number theory in additive arithmetic semigroups II

We prove two quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups. On the basis of the two theorems, a central limit theorem of additive functions on additive arithmetic semigroups is proved with a best possible error estimate. This generalizes the vital results of Halász and Elliott in classical probabilistic number theory to function fields. Received October 26, 1998; in final form April 5, 2000 / Published online October 11, 2000     

A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

Summary. We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems. Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements. Received March 17, 1998 / Revised version received June 7, 1999 / Published online January 27, 2000     

L-functions for Jacobi forms and the basis problem

This paper deals with Jacobi forms Φ on ?×ℂ. The Rankin–Selberg doubling method is employed to study properties of the standard L-function of Hecke–Jacobi eigenforms. It is shown that every analytic Klingen–Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke–Jacobi cusp eigenforms of weight k > 4 and k≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved. Received: 12 March 2000 / Revised version: 17 September 2001     

Multilevel ILU decomposition

Summary. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method. Received June 13, 1997 / Revised version received March 17, 1998     

On the use of rational iterations and domain decomposition methods for the Helmholtz problem

An iterative algorithm for the numerical solution of the Helmholtz problem is considered. It is difficult to solve the problem numerically, in particular, when the imaginary part of the wave number is zero or small. We develop a parallel iterative algorithm based on a rational iteration and a nonoverlapping domain decomposition method for such a non-Hermitian, non-coercive problem. Algorithm parameters (artificial damping and relaxation) are introduced to accelerate the convergence speed of the iteration. Convergence analysis and effective strategies for finding efficient algorithm parameters are presented. Numerical results carried out on an nCUBE2 are given to show the efficiency of the algorithm. To reduce the boundary reflection, we employ a hybrid absorbing boundary condition (ABC) which combines the first-order ABC and the physical $Q$ ABC. Computational results comparing the hybrid ABC with non-hybrid ones are presented. Received May 19, 1994 / Revised version received March 25, 1997     

Ordinal arithmetic and \Sigma_{1}-elementarity

We will introduce a partial ordering on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use to provide a new characterization of the ubiquitous ordinal . Received: 18 August 1997