Adjoint $L$-functions, congruence ideals, and Selmer groups over ${\mathrm{G}\mathrm{L}}_n$

In this paper, we relate $L(1,\pi ,{\mathrm{A}\mathrm{d}}^{\circ })$ to the congruence ideals for cohomological cuspidal automorphic representations $\pi$ of ${\mathrm{G}\mathrm{L}}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM and totally real fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,\pi ,{\mathrm{A}\mathrm{d}}^{\circ })$.