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

Abstract

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 $\mathrm{U}(n,n)\left({\mathbf{A}}_F\right)$ 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 $\ell$-adic properties of the Fourier coefficients of an Ikeda lift $I_{\varphi }$ (of an elliptic modular form $\varphi$) on $\mathrm{U}(n,n)\left({\mathbf{A}}_{\mathbf{Q}}\right)$, proving that they are $\ell$-adic integers which do not all vanish modulo $\ell$. Finally we combine these results to show that the condition of $\ell$ being a congruence prime for $I_{\varphi }$ is controlled by the $\ell$-divisibility of a product of special values of the symmetric square $L$-function of $\varphi$. We close the paper by computing an example when our main theorem applies.