Arithmetic Chern-Simons Theory I

Abstract

In this paper, we apply ideas of Dijkgraaf and Witten 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 5, we consider the far-fetched possibility of using Chern-Simons theory to construct L-functions.

with Appendix B by Behrang Noohi. A section added with brief sketch of a glueing formula. Section 3 corrected and simplified

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