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

Abstract

Let $B$ be a quaternion division algebra over a number field $k$ that gives rise to arithmetic Fuchsian or Kleinian groups, and let $\mathcal{O}$ be a maximal order of $B$. We show that every closed orientable Riemannian manifold isospectral to a principal congruence hyperbolic surface $X_\mathcal{O}(\mathfrak{a})$ is isometric to $X_{\mathcal{O}'}(\mathfrak{a})$ for some maximal order $\mathcal{O}'$ of $B$, provided that no prime factor of $\mathfrak{a}$ ramifies in $B$ and that $\mathfrak{a}$ 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 $X_\mathcal{O}(\mathfrak{a})$ has at most $t(B)$ elements, and $X_\mathcal{O}(\mathfrak{a})$ 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 $\mathrm{SL}(2)$ 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.