Character sums and abelian Ramanujan graphs

Let F be a finite field of q elements. In this paper we obtain several estimates on character sums derived from the Riemann hypothesis for curves over F. In particular, we establish an estimate on twisted generalized Kloosterman sums as conjectured by P. Deligne (1977, "Cohomologie étale (SGA 4 1 2)," Lecture Notes in Mathemmatics, Vol. 569, Springer-Verlag, Berlin/Heidelberg/New York) for the case n = 2: |Σ_{x ∈ N2} χ(x) ψ(x)| ≤ 2q ^{1 2} for all nontrivial characters (χ, ψ) of N₂ × F₂. Here F₂ is a quadratic extension of F and N₂ consists of norm 1 (to F) elements in F₂. We also present new constructions of Ramanujan graphs based on abelian groups. The character sum estimates are used to prove that these are indeed Ramanujan graphs.