Zeta function of monodromy and counting points over C((t))

2010-11-18T18:41:58Z

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 $\mathrm{Trop}(X)$?

Answer by ACL for Zeta function of monodromy and counting points over C((t))

2010-11-21T00:02:43Z

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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Zeta function of monodromy and counting points over C((t))

Answer by ACL for Zeta function of monodromy and counting points over C((t))