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.

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

algebraic number field.

Let p be an odd prime and k an

algebraic number field of finite degree such that the p-rank of Cl(k) is equal to 2.

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.