On the Ramanujan conjecture over number fields

TL;DR

This paper extends the best known bounds towards the Ramanujan conjecture for GL(n) over arbitrary number fields, introducing a technique to handle the analytic challenges posed by infinite units.

Contribution

It provides a novel method to overcome analytic obstacles caused by infinite units, extending bounds for the Ramanujan conjecture to all number fields.

Findings

Extended bounds for GL(2), GL(3), GL(4) over arbitrary number fields.

Developed a new technique to manage infinite units in number fields.

Achieved progress towards the Ramanujan conjecture in a more general setting.

Abstract

We extend to an arbitrary number field the best known bounds towards the Ramanujan conjecture for the groups 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.