Hecke eigenspaces for the projective line

TL;DR

This paper studies the action of Hecke operators on automorphic forms over function fields of the projective line, explicitly computing eigenspaces dimensions for various ramification cases in the context of GL_2.

Contribution

It provides explicit calculations of Hecke eigenspaces dimensions for both unramified and ramified operators, extending to higher degree ramifications.

Findings

Computed dimensions of unramified Hecke eigenspaces.

Explicit description of ramified Hecke operator actions.

Extended analysis to higher degree ramifications.

Abstract

In this article we investigate the action of (ramified and unramified) Hecke operators on automorphic forms for the function field of the projective line defined over a finite field and for the group GL_2. We first compute the dimension of the Hecke eigenspaces for every generator of the unramified Hecke algebra. Thus, we consider the ramification in a point of degree one and describe explicitly the action of certain ramified Hecke operators on automorphic forms. Moreover, for those ramified Hecke operators, we also compute the dimensions of its eigenspaces. We finish the article considering more general ramifications, namely, those one attached to a closed point of higher degree.