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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?

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