If $X$ is a smooth, projective variety over $\mathbb{F}\_q$, the [Weil conjectures](http://en.wikipedia.org/wiki/Weil_conjectures) tell us:


$$\prod \mathrm{det} (I - TF|\_{H^i\_c(X)})^{(-1)^{i+1}} = \mathrm{exp}\left(\sum\_{m=1}^{\infty} \frac{N\_m}{m} T^m \right)$$


here, $T$ is a formal variable, $H^i\_c(X)$ is an appropriate cohomology theory, $F$ is the Frobenius automorphism, and $N\_m$ is the number of $\mathbb{F}\_{q^m}$ points of $X$. 


I would like to replace $\mathbb{F}\_q$ with $\mathbb{C}((z))$, on the pretext that both have absolute Galois group $\hat{\mathbb{Z}}$. I am thinking of $\mathbb{C}((z))$ as the ring of functions on a very small punctured disc, and of a variety over $\mathbb{C}((z))$ as a family over this punctured disc. I will also conflate the Frobenius automorphism with the monodromy action. 



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> > Let $X$ be a variety over $\mathbb{C}((t))$; interpret the LHS of the equation above by understanding $F$ as the monodromy action. In what, if any, sense does the number $N\_m$ count points over the field $\mathbb{C}((t^{1/m}))$ ?
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Note that if one naively takes the cardinality of the set of these points, one would often find $N\_m = \mathrm{\infty}$.



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Finally, let me view $\overline{\mathbb{C}((t))}$ as the field over which tropical geometry happens. Making the modifications appropriate to discuss the non-projective case, let me take $X$ to be an affine variety. 



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> > Can I "count" $\mathbb{C}((t^{1/m}))$ points of $X$ in terms of "counting" $\frac{1}{m} \mathbb{Z}$ points of $\mathrm{Trop}(X)$?
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Zeta function of monodromy and counting points over C((t))
Hi Vivek. You should have a look to the following papers :



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> J. Nicaise and J. Sebag (2007).
>  *Motivic Serre invariants, ramification, and the analytic Milnor
>  fiber*. Inventiones mathematicae,
>  **168** (1), p. 133-173.
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> J. Nicaise (2009). *A trace formula
>  for rigid varieties, and motivic Weil
>  generating series for formal schemes*.
>  Math. Ann. **343**, p. 285–349.
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They prove a trace formula along the lines you suspect: Let $X$ be a proper $\mathbf C((t))$-variety, let $S(X)$ be its motivic Serre invariant: this is an element of the Grothendieck group of complex varieties modulo the ideal generated by $\mathbf L-1$, where $\mathbf L$ is the class of the affine line; it is computed as the special fiber of any weak Néron model of~$X$ over $\mathbf C[[t]]$. The Euler characteristic of $S(X)$ is well-defined
and $\chi(S(X))$ equals the (alternate sum) of the traces of the action of the monodromy on the étale cohomology of $\bar X$.


