Eta invariants and class numbers

ROBERTO J. MIATELLO; RICARDO A. PODESTÁ

PURE AND APPLIED MATHEMATICS QUARTERLY

INT PRESS BOSTON, INC

Año: 2009 vol. 5 p. 729 - 753

1558-8599

Let M be a compact flat spin Riemannian manifold, having cyclic holonomy group of odd prime order p. If D is the Dirac operator acting on spinor fields of M, we give explicit expressions for the eta series eta(s) and the eta invariant eta=eta(0). We prove that eta(s) = e(s)L(s; X) where e(s) is a linear combination of exponentials and L(s; X) is the Dirichlet L-function attached to X(k) = (k/p), the Legendre symbol. Furthermore, for p >=5, we show that eta is an explicit integral multiple of the class number h(-p) of the imaginary quadratic field Q(sqrt(-p)). We also provide alternative expressions for eta as finite cotangent or cosecant sums.