Notes on automorphic functions: an entire automorphic form of positive dimension is zero

Abstract

L It is a result familiar in the theory of automorphic form s that an entire automorphic form of positive dimension on an H-group is identically zero (see sec. 2 for the definitions). This follows immediately , for exa mple, from the well-known exac t formula for the Fourier coefficie nts of automorphic forms of positive dimension ([1], p. 314).1 Another proof is by means of a formula for the numbe r of zeros minus the number of poles of an automorphic form in a fundamental domain. This formula (obtained by contour integration around the fundamental domain) shows that whe n the dime nsion of the form is positive, thi s difference is negative, and he nce such a form mu st have poles. In section 3 of thi s note we give what appears to be a new proof of this result by using the method Hecke e mployed to estimate the Fourier coeffi cients of cusp forms of negative dimension ([1] , p. 281). I This proof is simpler and more direc t than the proofs mentioned above. In sections 45 we give two variations of this method. The me thod of section 5 is applicable to a larger class of groups than the H-groups , and in particular applies to compact groups and groups conjugate to H-groups. 2. A group r of real linear fractional transformations acting on :J't', the upper half-plane 1m 7 > 0, is an H-group provided (i) r is discontinuous on :J't', but is not di scontinuous at any point of the real line, (ii) r is finitely generated, and (ii i) r contains translations. With each transformation v~r we associate a real