The eta invariant of the Atiyah-Patodi-Singer operator on compact flat manifolds

MIATELLO, ROBERTO J.; PODESTÁ RICARDO A.

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY

SPRINGER

Lugar: Berlin; Año: 2012 vol. 42 p. 171 - 194

0232-704X

Let D be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of L χ(s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h −p of the number field Q(−p−−−√) . In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h −p .