Hoffman and Wakatsuki study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group and the split symplectic group of rank two over any

algebraic number field. In particular, they say, they express the global coefficients of unipotent orbital integrals in terms of Dideking zeta functions, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.

In this paper, we study the remaining parts, namely, we give such bounds for modular forms with Fourier coefficients in an arbitrary algebraic number field K and for any prime ideal p in K.

Let K be an algebraic number field and O = [O.sub.K] the ring of integers in K.

(i) The usual definition is more general, the coefficients of S can be taken in an arbitrary

algebraic number field. For our purposes it is sufficient and convenient to restrict to rational coefficients.

Let k be an

algebraic number field. Let p be a prime number and G a p-group.

Faticoni (mathematics, Fordham U.) presents an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an

algebraic number field, point set topological spaces, and classical noncommutative localization.

of Tokyo) explore similarities between

algebraic number fields and algebraic function fields in one variable over finite fields, explain adele rings and idele groups, derive several prime number theorems, and prove the main theorem of class field theory.

Stan, Florin, University of Illinois, Urbana-Champaign, Trace problems in

algebraic number fields and applications to characters of finite groups.