Given an algebraic curve X defined over a finite field, let Bun( X ) be the set of isomorphism classes of rank- r vector bundles over X . The Langlands conjecture states that there should be a one-to-one correspondence between Hecke eigenfunctions on Bun( X ) and r -dimensional representations of the fundamental group of X . In particular, eigenfunctions in the space of cusp forms should correspond to irreducible representations.
Cusp forms make up the discrete spectrum of the Hecke operators, while Eisenstein series give the continuous spectrum. With enough information about the structure of Bun( X ), it should be possible to explicitly calculate the action of the Hecke operators, giving information both about cusp forms and about Eisenstein series.
In this paper, I have done these calculations for vector bundles of arbitrary rank over the projective line and for vector bundles of rank two over any elliptic curve. I have derived a formula for Eisenstein series over the projective line based on Macdonald's formula for spherical functions and have derived formulas describing cusp forms on rank-two vector bundles over elliptic curves.