Theta functions and a presentation of [formula omitted

Abstract

We investigate a Curtis–Tits style group presentation based on the Dynkin diagram Cg, in which the short root generators have order two while the long root generators have order four. We prove that this describes a finite group with an almost extraspecial normal subgroup of order 22g+2 and quotient isomorphic to the symplectic group Sp(2g,2). Such an extension has to be non-split if g⩾3, and was proved to exist in papers of Bolt, Room and Wall from 1961/2 and Griess from 1973. Our presentation proves that it is a double cover of a finite quotient of Sp(2g,Z). We investigate a 2g dimensional complex representation on a suitable space of theta functions, and produce some consequences for the signatures of 4-manifolds described as surface bundles over surfaces. In particular, we prove that if the monodromy is contained in the theta subgroup Spq(2g,Z)⩽Sp(2g,Z) then the signature of the 4-manifold is divisible by eight.