Quantum chaos is a branch of mathematical physics which studies how quantum dynamical systems relate to their classical version. We argue that it combines well with number theory and the study of arithmetic periods in particular. We illustrate this with the classical problem of bounding the sup-norm of eigenfunctions on locally symmetric spaces where as a new result we show that for SL(n) eigenfunctions achieve their maximum in the cusp. The argument relies on uniform estimates for special functions which are of independent interest. In some cases we establish a formula that involves 2F1 hypergeometric sums. We expect that large values of local periods occur at certain Lagrangian singularities and we give evidence using methods in representation theory.