Hecke Eigenfunctions of Quantized Cat Maps Modulo Prime Powers

This paper continues the work done in Olofsson Commun Math Phys 286(3):1051–1072, 2009] 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 properties of N play an important role. We prove the sharp estimate ||ψ||_∞ = O(N ^1/4) 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 = p ⁿ , with n > 2, 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/\sqrt{1\pm 1/p}}This paper continues the work done in Olofsson Commun Math Phys 286(3):1051–1072, 2009] 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 properties of N play an important role. We prove the sharp estimate ||ψ||_∞ = O(N ^1/4) 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 = p ⁿ , with n > 2, 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/?{1±1/p}{2/\sqrt{1\pm 1/p}} and the supremum norm of the newforms in the second half have at most three different values, all of the order N ^1/6. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.