A note on exponential sums (Q5917683)

This paper along with \textit{J. Angel} [Finite Fields Appl. 2, 62-86 (1996; see the preceding review)] and \textit{R. Evans} [Finite Fields Appl. 1, 376-394 (1995; Zbl 0844.11078)] proves that finite upper half plane graphs are Ramanujan in characteristic 2. Ramanujan graphs were first defined by \textit{A. Lubotzky}, \textit{R. Phillips} and \textit{P. Sarnak} in [Combinatorica 8, 261-277 (1988; Zbl 0661.05035)]. The character sums that arise are slightly different from those that appear for odd characteristic fields. The method the author uses to estimate these sums is similar to that which he used in odd characteristic. It involves algebraic geometry; in particular, \(\ell\)-adic étale cohomology, the Grothendieck-Lefschetz trace formula, Weil's proof of the Riemann hypothesis for zeta functions of curves over finite fields, and Artin-Schreier theory. For odd characteristic, finite upper half plane graphs were proved Ramanujan using the result of \textit{J. Angel}, \textit{S. Poulos}, the reviewer, \textit{C. Trimble} and \textit{E. Velasquez} [Contemp. Math. 173, 15-70 (1994; Zbl 0813.11034)], the author [J. Reine Angew. Math. 438, 143-161 (1993; Zbl 0798.11053)], and \textit{J. Soto}-\textit{Andrade} [Proc. Symp. Pure Math. 47, 305-316 (1987; Zbl 0652.20047)]. See \textit{Winnie W. C. Li} [Number theory with applications, World Scientific, Singapore (1995; Zbl 0849.11006)] for a different method of estimating the character sums in odd characteristic -- using only Weil's proof of the Riemann hypothesis for zeta functions of curves over finite fields and class field theory.