The Arithmetic of Automorphic Forms With Respect to a Unitary Group

Abstract

utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms. One expresses a given modular form F as a function of complex variables u,, * * *, u. with an expansion (0.1) F(u,, *. Us) = Ex c exp(2wi. * 1U'), where the coefficients c. are complex numbers and x runs over a lattice. Especially important are those F for which all c, are algebraic, or more restrictedly, cyclotomic. They form a distinguishable class which is stable under the transformation by the elements of the algebraic group in question. The algebraicity of c., is also indispensable if the value of a modular function at a special point is the problem, as in the theory of complex multiplication. In general, an expansion of type (0.1) exists for an automorphic form if it is defined on a tube domain and the group contains sufficiently many translations. There are, however, cases in which no such expansion is available. A typical example is provided by the symmetric domain