Character sums, automorphic forms, equidistribution, and Ramanujan graphs Part II. Eigenvalues of Terras graphs

We study two types of character sums related to Terras graphs using the method ℓ-adic cohomology. These character sums also arise as traces of Frobenii of some two dimensional linear representations of a global function field. Detailed information about these Galois representations at ramified places are obtained from analysis of vanishing cycles. Consequently we give a complete description of the automorphic forms of which these character sums appear as Fourier coefficients. These character sums are shown to be equidistributed with respect to the Sato-Tate measure.