Abstract :
Let GF(q) be a field of q elements. Let G denote the group of matrices M(x, y) = [formula] over GF(q) with y ≠ 0. Fix an irreducible polynomial u2 + tu + n GF(q)[u]. For each a GF(q), let Xa be the graph whose vertices are the q2 − q elements of G, with two vertices M(x, y), M(v, w) joined by an edge iff (x − v)2 + t(x − v) (y − w) + n(y − w)2 = wya. The graphs Xa with a {0, t2 − 4n} are (q + 1)-regular connected graphs. These Xa are known to be Ramanujan graphs for odd q, largely by the work of Katz, Soto-Andrade, and Terras. Using representation theory for GL(2, q) and recent character sum estimates of Katz, we show that these Xa are Ramanujan graphs for even q as well. This settles the beautiful conjecture of Terras that all (q + 1)-regular finite upper half plane graphs Xa are Ramanujan graphs. Further, let L(G) denote the Hilbert space of functions mapping G into the complex numbers, with inner product (f, h) = ∑g Gf(g) . We give q2 − q character sums which comprise a common basis in L(G) of mutually orthogonal eigenfunctions for the ajacency operators of all the connected graphs Xa.
