An analogy between number theory and spectral geometry

Abstract: Sunada’s construction of non-isometric, isospectral manifolds proceeds in the same way as Gassmann’s construction of non-isomorphic number fields with the same zeta function, using a group G with two non-conjugate subgroups H and K such that the permutation representations given by G acting on their cosets are isomorphic. In Gassmann’s example, G was the permutation group on 6 letters and H and K the groups generated by (12)(34) and (13)(24), and (12)(34) and (12)(56), respectively. These can be realized as covering groups of a compact Riemann surface of genus 2. Recently, the speaker and collaborators showed that isomophism of number fields can be detected by equality of suitable L-series. This talk is about the finding the analogous result for manifolds. The result says that if two manifolds are finite Riemannian covers of a developable orbifold, and such that a certain homological condition is satisfied, then the manifolds are isometric if and only if the spectra of finitely many Laplacians twisted by suitable unitary representations of the fundamental group are equal. The result is explicit: in the above example, one needs 56 spectral equalities corresponding to 180-dimensional representations. (Joint work with Norbert Peyerimhoff.)

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar