Zeta function of monodromy and counting points over C((t)) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:35:25Z http://mathoverflow.net/feeds/question/46523 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46523/zeta-function-of-monodromy-and-counting-points-over-ct Zeta function of monodromy and counting points over C((t)) Vivek Shende 2010-11-18T18:41:58Z 2010-11-21T00:02:43Z <p>If $X$ is a smooth, projective variety over $\mathbb{F}_q$, the <a href="http://en.wikipedia.org/wiki/Weil_conjectures" rel="nofollow">Weil conjectures</a> tell us:</p> <p>$$\prod \mathrm{det} (I - TF|_{H^i_c(X)})^{(-1)^{i+1}} = \mathrm{exp}\left(\sum_{m=1}^{\infty} \frac{N_m}{m} T^m \right)$$</p> <p>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 $X$. </p> <p>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. </p> <blockquote> <blockquote> <p>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 $\mathbb{C}((t^{1/m}))$ ?</p> </blockquote> </blockquote> <p>Note that if one naively takes the cardinality of the set of these points, one would often find $N_m = \mathrm{\infty}$.</p> <blockquote> <blockquote> <p>Is there any structure on the set of $\mathbb{C}((t))$ points which would allow me to take an Euler number? </p> </blockquote> </blockquote> <p>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. </p> <blockquote> <blockquote> <p>Can I "count" $\mathbb{C}((t^{1/m}))$ points of $X$ in terms of "counting" $\frac{1}{m} \mathbb{Z}$ points of $\mathrm{Trop}(X)$?</p> </blockquote> </blockquote> http://mathoverflow.net/questions/46523/zeta-function-of-monodromy-and-counting-points-over-ct/46789#46789 Answer by ACL for Zeta function of monodromy and counting points over C((t)) ACL 2010-11-21T00:02:43Z 2010-11-21T00:02:43Z <p>Hi Vivek. You should have a look to the following papers :</p> <blockquote> <p>J. Nicaise and J. Sebag (2007). <em>Motivic Serre invariants, ramification, and the analytic Milnor fiber</em>. Inventiones mathematicae, <strong>168</strong> (1), p. 133-173.</p> <p>J. Nicaise (2009). <em>A trace formula for rigid varieties, and motivic Weil generating series for formal schemes</em>. Math. Ann. <strong>343</strong>, p. 285–349.</p> </blockquote> <p>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 $\bar X$.</p>