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On the existence of cusp forms over function fields. (English) Zbl 0667.10014
In this paper it is proved that there exist arithmetic groups of arbitrarily large co-volume which admit no cusp forms. They thus stand in contrast to congruence groups, for which it is known that the dimension of the space of cusp forms grows asymptotically to the volume. The groups we consider are discontinuous subgroups of GL(2) of a function field of genus 0 over the finite field \({\mathbb{F}}_ q\), and thus act on the associated building, which is the tree of regularity \(q+1\). We also find that (non-cuspidal) residual spectrum is abundant for these groups.
Reviewer: I.Efrat

11F06 Structure of modular groups and generalizations; arithmetic groups
11R58 Arithmetic theory of algebraic function fields
11F11 Holomorphic modular forms of integral weight
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