p-adic level raising on the eigenvariety for U(3)

We prove level raising results for $p$-adic automorphic forms on definite unitary groups $U(3)\mathrm{/}\mathbb{Q}$ and deduce some intersection points on the eigenvariety. Let $l$ be an inert prime where the level subgroups varies, if there is a non-very-Eisenstein point $\phi$ on the old component (generically parametrizing forms old at $l$) satisfying $T_l(\phi )=l(l^3+1)$, then this point also lies in the new component (generically parametrizing forms new at $l$). This provides a $p$-adic analogue of Bellaïche and Graftieaux’s mod $p$ level raising for classical automorphic forms on $U(3)$, and also generalizes James Newton’s $p$-adic level raising results for definite quaternion algebras. Key ingredients include abelian Ihara lemma (proved for any definite unitary group $U(n)$) and some duality arguments about certain Hecke modules. Finally we also discuss some methods to construct such points explicitly and further development.