A conjecture of Serre (now a theorem of Gross, Edixhoven, and Coleman-Voloch) classifies pairs of weights where one finds modular forms congruent modulo a prime p in terms of local behavior at p. We discuss a generalization of this conjecture in higher rank. A key step in our work is the study of a certain subscheme of Gaitsgory's $A^1$ affine Grassmannian which shares properties with some affine Springer fibers. This is joint work with B. Le Hung, B. Levin, and S. Morra.