## Geometric Ramanujan conjecture and Drinfeld reciprocity law.(English)Zbl 0674.10025

Number theory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway 1987, 201-218 (1989).
[For the entire collection see Zbl 0661.00005.]
This paper contains a discussion of the authors’ recent work concerning cuspidal automorphic forms. Let F be a global field of positive characteristic p, $${\mathbb{A}}$$ the ring of F-adèles, $$G=GL(r)$$, and $$\pi$$ an irreducible admissible G($${\mathbb{A}})$$-module. The authors prove (the “Purity theorem”) that if $$\pi$$ is cuspidal with supercuspidal component $$\pi_{\infty}$$ then each conjugate of each Hecke eigenvalue $$z_ i(\pi_ V)$$, for almost all unramified components $$\pi_ v$$, lies on the unit circle in $${\mathbb{C}}$$. Their proof relies on a new form of the Selberg trace formula for a test function with supercuspidal component.
In addition, they give a higher rank analogue of the classical theory of congruence relations that arises from the geometry of certain correspondences.
Using methods similar to those in the proof of the Purity theorem they also show how the Drinfel’d explicit reciprocity law follows from a conjecture of Deligne.
Reviewer: S.Kamienny

### MSC:

 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F80 Galois representations 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Zbl 0661.00005