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If $X$ is a smooth, projective variety over $\mathbb{F}_q$, the 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.

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 $\mathbb{C}((t^{1/m}))$ ?

Note that if one naively takes the cardinality of the set of these points, one would often find $N_m = \mathrm{\infty}$.

Is there any structure on the set of $\mathbb{C}((t))$ points which would allow me to take an Euler number?

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.

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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Motivic integration gives suitable invariants from $\mathbb{C}((t))$-points. I am not sure if they answer all your questions. –  Felipe Voloch Nov 18 '10 at 19:12
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On the off chance that you don't know this, over $\mathbf{C}((t))$ the eigenvalues of the Galois action on cohomology are all roots of unity. So the characteristic polynomial is a fundamentally combinatorial object, which is very different to the situation over finite fields. (So the spectrum of a finite field is in some ways more like an annulus than a punctured disk.) Maybe the combinatorics of the characteristic polynomial is related to the combinatorics of the special fiber of models of $X$ over $\mathbf{C}[[t]]$. –  JBorger Nov 19 '10 at 11:24
    
@James: certainly if you put the whole family in some $\mathbb{P}^n \times \mathrm{Spec}\,\mathbb{C}((t))$ and then use embedded resolution, you'll get some description of the monodromy in terms of combinatorics of how the various exceptional divisors intersect. But what points are being counted and how? –  Vivek Shende Nov 19 '10 at 17:33
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up vote 8 down vote accepted

Hi Vivek. You should have a look to the following papers :

J. Nicaise and J. Sebag (2007). Motivic Serre invariants, ramification, and the analytic Milnor fiber. Inventiones mathematicae, 168 (1), p. 133-173.

J. Nicaise (2009). A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann. 343, p. 285–349.

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 $\bar X$.

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lovely, thank you! –  Vivek Shende Nov 21 '10 at 9:54
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