On the size of Satake parameters for unitary cuspidal automorphic representations for GL(4)

TL;DR

This paper establishes lower bounds on the number of places where Satake parameters of cuspidal automorphic representations for GL(4) are not excessively large, with implications for the distribution of tempered places.

Contribution

It provides unconditional lower bounds on the distribution of Satake parameters for GL(4) automorphic representations, extending Ramakrishnan's methods and analyzing Langlands conjugacy classes.

Findings

Lower bounds on non-large Satake parameters for GL(4)

Positive lower Dirichlet density of tempered places in self-dual cases

Extension of Ramakrishnan's analysis to GL(4) and discussion of GL(3)

Abstract

Let \Pi\ be a cuspidal automorphic representation for GL(4) over a number field F. We obtain unconditional lower bounds on the number of places at which the Satake parameters are not "too large". In the case of self-dual \Pi\ with non-trivial central character, our results imply that the set of places at which \Pi\ is tempered has an explicit positive lower Dirichlet density. Our methods extend those of Ramakrishnan by careful analysis of the hypothetical possibilities for the structure of the Langlands conjugacy classes, as well as their behaviour under functorial lifts. We then discuss the analogous problem in GL(3).