Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form f on the unitary group U(n,n)(AF) for a large class of totally real fields F via a divisibility of a special value of the standard L-function associated to f. We also study l-adic properties of the Fourier coefficients of an Ikeda lift Iϕ (of an elliptic modular form ϕ) on U(n,n)(AQ), proving that they are l-adic integers which do not all vanish modulo l. Finally we combine these results to show that the condition of l being a congruence prime for Iϕ is controlled by the l-divisibility of a product of special values of the symmetric square L-function of ϕ. We close the paper by computing an example when our main theorem applies.