Congruence of modular forms and arithmetic of Shimura varieties

The congruence of modular forms is an important phenomenon in the arithmetic study of modular forms, or more generally, automorphic forms. For classical modular forms, many results have been obtained by Serre, Ribet, et al, for more than thirty years. In particular, Ribet used the arithmetic geometry of modular curves to find such congruence relation, also known as level raising. We recall as follows: Fix a prime $l$; consider a weight-$2$ level-$N$ newform $f$ satisfying the $\mathrm{mod}~l$ level-raising condition at a prime $p$ coprime to $Nl$. Ribet proved that the first Galois cohomology of the $\mathrm{mod}~l$ Galois representation of $\mathbb{Q}_p$ associated with $f$ can be realized as the Abel-Jacobi image of the supersingular locus of the level-$N$ modular curve over $F_p$.
In ongoing joint work with Yichao Tian (MCM) and Liang Xiao (PKU), we generalize this phenomenon to higher-dimensional unitary Shimura varieties at inert places (which remains a conjecture in general), and its relation with a certain Ihara type lemma for such varieties. In the talk, I will explain cases for which we have confirmed such conjecture; and if time permits, we will mention its number-theoretical implications.