Pfaffians, the G-Signature Theorem and Galois de Rham discriminants
  We study equivariant de Rham discriminants associated to arithmetic varieties which support a tame action by a finite group; we form these discriminants by endowing the de Rham cohomology with pairings arising from duality theory. Such equivariant discriminants are shown to break up naturally into a metric part and a signature part. In a previous paper we described the equivariant Arakelov discriminants, obtained by endowing the equivariant determinant of de Rham cohomology with various metrics. In this paper we study the associated equivariant signature information; in particular, we show that the symplectic signature invariants both determine and are determined by the symplectic Archimedean epsilon constants of the arithmetic variety. 
T. Chinburg, G. Pappas, M. J. Taylor