Value distribution for eigenfunctions of desymmetrized quantum maps

We study the value distribution and extreme values of eigenfunctions for the “quantized cat map.” This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map—a commutative group of unitary operators that commute with the map,which we called “Hecke operators.” The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call “Hecke eigenfunctions.”In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the “split primes” for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover, in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil's theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.