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

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. As a continuation of our previous work, we deal with the problem of computing explicitly the eta invariant  for any orientable compact flat manifold M. After giving an explicit expression for the eta function eta(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 eta(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. D be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. As a continuation of our previous work, we deal with the problem of computing explicitly the eta invariant  for any orientable compact flat manifold M. After giving an explicit expression for the eta function eta(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 eta(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. M. As a continuation of our previous work, we deal with the problem of computing explicitly the eta invariant  for any orientable compact flat manifold M. After giving an explicit expression for the eta function eta(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 eta(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. for any orientable compact flat manifold M. After giving an explicit expression for the eta function eta(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 eta(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. (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 eta(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. (0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. p ge 7, eta(s) can be expressed as a multiple of L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. L(s), an L-function associated to a quadratic character mod p, while eta(0) is a (nonzero) integral multiple of the class number h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp. h_p of the number field Q(sqrt -p). In the case of metacyclic groups  of odd order pq, with p; q primes, we show that eta(0) is a rational multiple of hp. multiple of hp.