# Purity Theorem

• Yuval Z. Flicker
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## Abstract

The purpose of this section is to prove Ramanujan’s conjecture for cuspidal representations π of $$GL(r, \mathbb{A})$$ over a function field, which have a cuspidal component, namely that all unramified components of such a π are tempered, namely that all of their Hecke eigenvalues have absolute value one. This is deduced from a form of the trace formula of Arthur, as well as the theory of elliptic modules developed above, Deligne’s purity of the action of the Frobenius on the cohomology, standard unitarity estimates for admissible representations, and Grothendieck’s fixed point formula. Once we assume and use Deligne’s (proven) conjecture on the fixed point formula, in the following sections, we no longer need the complicated full trace formula, but the simple trace formula suffices. Thus this section is for pedagogical purposes only, to show what can be done without Deligne’s conjecture.

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