Eta invariants and class numbers

MIATELLO R.,J., PODESTA' R.,A.

PURE AND APPLIED MATHEMATICS QUARTERLY

INT PRESS BOSTON, INC

Lugar: Beijing; Año: 2009 vol. 5 p. 729 - 754

1558-8599

Abstract: 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 and the eta invariant eta(0). We prove that eta(s) = e(s)L(s, chi ) where e(s) is a linear combination of exponentials and L(s, chi) is the Dirichlet L-function attached to  the Legendre symbol. Furthermore, we show that eta is an explicit integral multiple of the class number h_p of the imaginary quadratic field Q(sqrt −p) and we provide alternative expressions as finite cotangent or cosecant sums.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 and the eta invariant eta(0). We prove that eta(s) = e(s)L(s, chi ) where e(s) is a linear combination of exponentials and L(s, chi) is the Dirichlet L-function attached to  the Legendre symbol. Furthermore, we show that eta is an explicit integral multiple of the class number h_p of the imaginary quadratic field Q(sqrt −p) and we provide alternative expressions as finite cotangent or cosecant sums.