# Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin L-functions $L_\sigma$ looks like a product over characteristic polynomials of the monodromies/holonomies of that flat connections. I suppose.

If one considers an actual (differential geometric) flat connection $A$ on a suitable hyperbolic manifold with twisted Dirac operator $D_A$, then its Selberg zeta function as in def. 4.1 arXiv:dg-ga/9407012 has essentially this form, too: a product over characteristic polynomials of, essentially, the monodromies/holonomies of this flat connection. I suppose.

Is there a systematic way -- or at least discussion of more analogy -- of Artin L-functions induced by Galois representations being like zeta functions of (Laplace operators of) Dirac operators twisted by flat connections?

• John Baez points out that discussion of more analogy along these lines is in D. Brown "Lifting properties of prime geodesics" ncatlab.org/nlab/show/Selberg+zeta+function#Brown09 . On p. 9 there is a comprehensive table of pertinent analogies. Dec 11 '14 at 0:52