L ∞ Norms of Eigenfunctions of Arithmetic Surfaces

Abstract

Here and elsewhere A ,l1o0 as A -xc is a basic problem of Quantum Chaos [S]. In any case almost nothing beyond (0.2) is known about 110,joc, when the curvature is negative (one can push the standard techniques and replace A1/4 by A1/4/ log A in this case). In this paper we use arithmetic techniques, in particular modular correspondences to obtain the first improvement in the exponent (0.2). We also obtain lower bounds on the L' norms in these arithmetic cases. In more detail, let A = (,b) be a quaternion division algebra over Q. A is linearly generated by 1, w, Q, wQ over Q and w2 = a, Q2 = b, wQ + Qw = 0. Here a, b E Z are square free and we will assume that a > 0. As usual the norm and trace are defined by N(a) = a-d, tr(a) = a + oY where if a = Xo + x1W + x2Q + X3WQ,