Hecke eigenfunctions of quantized cat maps modulo prime powers

This paper continues the work done in [16] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic prop-erties of N play an important role. We prove the sharp estimate}ψ} 8 OpN1{4q for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N pn, with n ¡ 1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about 2{ a 1 1{p and the supremum norm of the newforms in the second half have at most three different values, all of the order N1{6. The only dependence of A is that the normalization factor is different if A has eigen-vectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.