On the Ramanujan conjecture over number fields

Abstract

We extend to an arbitrary number field the best known bounds towards Ramanujan for the group $\mathrm{GL}_n$, $n=2,3,4$. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite group of units.

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Authors

Valentin Blomer

Mathematisches Institut
Universität Göttingen
37073 Göttingen
Germany

Farrell Brumley

Institut Élie Cartan
Université Henri Poincaré Nancy 1
54506 Vandœuvre-les-Nancy
France