Arithmetic Chern–Simons Theory I

In this paper, we apply ideas of Dijkgraaf and Witten [24, 6] on 2+1 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons functionals on spaces of Galois representations. In the highly speculative section 6, we consider the far-fetched possibility of using Chern-Simons theory to construct L-functions. 1. The arithmetic Chern-Simons action: basic case We wish to move rather quickly to a concrete definition in this first section. The reader is directed to section 5 for a motivational discussion of L-functions. Let X = Spec(OF ), the spectrum of the ring of integers in a number field F . We assume that F is totally imaginary, for simplicity of exposition. Denote by Gm the étale sheaf that associates to a scheme the units in the global sections of its coordinate ring. We have the following canonical isomorphism ([19], p. 538): inv : H(X,Gm) ≃ Q/Z. (∗) This map is deduced from the ‘invariant’ map of local class field theory. We will use the same name for a range of isomorphisms having the same essential nature, for example, inv : H(X,Zp(1)) ≃ Zp, (∗∗) where Zp(1) = lim ←−i μpi , and μn ⊂ Gm is the sheaf of n-th roots of 1. This follows from the exact sequence 0→ μn → Gm (·) → Gm → Gm/(Gm) n → 0. That is, according to loc. cit., H(X,Gm) = 0, while by op. cit., p. 551, we have H(X,Gm/(Gm) ) = 0 for i ≥ 1. If we break up the above into two short exact sequences, 0→ μn → Gm (·) → Kn → 0, and 0→ Kn → Gm → Gm/(Gm) n → 0, 1991 Mathematics Subject Classification. 14G10, 11G40, 81T45 . M.K. Supported by grant EP/M024830/1 from the EPSRC.