  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 which
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.
