Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras

Let B be a quaternion division algebra over a number field k that gives rise to arithmetic Fuchsian or Kleinian groups, and let O be a maximal order of B. We show that every closed orientable Riemannian manifold isospectral to a principal congruence hyperbolic surface XO(a) is isometric to XO'(a) for some maximal order O' of B, provided that no prime factor of a ramifies in B and that a satisfies a mild condition at the primes over $2$. This is a converse to the construction of Vignéras, which produces isospectral manifolds from maximal orders of different types. In particular the spectral genus of XO(a) has at most $t(B)$ elements, and XO(a) is spectrally rigid when the type number $t(B)$ is one. In dimension three we prove the same statements for manifolds that, in addition, have the same complex lengths of closed geodesics. The proof rests on a local result that characterizes, by traces alone, the subgroups of SL(2) over a p-adic ring that are conjugate into a principal congruence subgroup by traces. As applications, every principal congruence Hurwitz surface is spectrally rigid, among them Klein's quartic, the Fricke-Macbeath surface and the surfaces of the first Hurwitz triplet, which partially answers a question of Reid.