In Strasbourg Part 1, I promised to deliver a few summaries of the minicourses given at the special week hosted at the Institut de Recherche Mathématique Avancée. In the next few posts, I will highlight a few things that occurred in Bernard le Stum‘s lectures on rigid cohomology. My posts are not meant to be a complete summary, and indeed I would like to be as succint as possible. Furthermore, I intend to add additional material explaining some of of the prerequisites. This first post covering rigid cohomology will try to explain the motivation for considering rigid cohomology in the first place.

Introduction

The lectures on rigid cohomology were given by Bernard le Stum who wrote the only existing textbook on rigid cohomology developed primarily by Pierre Berthelot, so the reader should definitely consult le Stum’s book for more details, as well as consult the slides when they appear, which contain many wonderful examples that I am omitting.

But what is rigid cohomology, and why is it useful?

Suppose $ X$ is an algebraic variety over $ \mathbb{F}_q$. Then how many rational points does $ X$ have? This is a very natural question because finding rational solutions to systems of polynomial equations boils down to finding rational points. Now, there are many ways to approach this problem. For finite fields, we can be barbarous and just check every possbility: after all, our $ X$ will be given by some finite set of polynomials in $ \mathbb{F}_q[x_1,\dots,x_n]$ and there are only finitely many possibilites for a morphism to $ \mathbb{F}_q$.

However, this solution is unsatisfying from a theoretical point of view because we want general results about general varieties, and this brute-force method really only says something about a given specific variety that is given explicitly in terms of equations. Of course for more practical things like cryptography, it is still useful to have good algorithms over finite fields, although I shall not discuss this here.

Zeta Functions

Fine. As number theorists, we know that we can attach an $ L$-function to everything, from a Dirichlet character to the cat, so let’s introduce a zeta function to keep track of rational points, but instead of just rational points, we will consider also the $ \mathbb{F}_{q^r}$ points for $ r=1,2,3,\dots$. Let us denote $ X(\mathbb{F}_{q^r})$ by $ N_r(X)$.

We define the zeta function associated to $ X$ by

$ Z(X,t) = \mathrm{exp}( \sum_{r=1}^\infty N_r(X)t^r/r)$.

Notice that $ t$ now can be interpreted as a complex variable, so we have shifted the domain of the problem, which naively was $ \mathbb{F}_q$, to the complex numbers. By taking the logarithimic derivative of this series, and evaluating at $ 0$, we get back $ N_1(X) = X(\mathbb{F}_q)$! Of course, if we start to use analysis, we should be careful then about bringing the derivative inside of this series.

Of course, just because we have written down this little analytic gadget doesn’t mean we’ve a clue about it. If we could get enough information about $ Z(X,t)$, perhaps from $ X$ in some devious way, we might actually be able to compute the number of rational points without having to use the old-fashioned way. Keep in mind that we can always compute as many coefficients as we want for a specific variety, because finite fields are…uh…finite, but it would be nice if we never had to do this.

Cohomology

So, given that the title of this post and the topic of the lectures was rigid cohomology, it’s no surprise that we can use rigid cohomology to compute the zeta function defined above. In fact, given that we have enough information about the rigid cohomology groups of $ X$, we can write down a formula for $ Z(X,t)$ using this information. In fact, one can do the same using etale cohomology, but for certain computational purposes, it’s easier to use rigid cohomology.

So our next task is to take a closer look at rigid cohomology, what this “information” is, and how it is related to the zeta function we want. Now, like most cohomology theories, there are many ways to define rigid cohomology. Rigid cohomology, like etale cohomology, is defined in terms of sheaf cohomology: that is, the rigid cohomology vector spaces are just the right-derived functors of the global sections functor with respect to a sheaf on a certain site. This is one definition of rigid cohomology we will see in this post series.

While as theoretically satisfying as any definition involving universal delta functors, this is not always the most computationally-friendly. However, there are indeed other definitions that can be used in various cases that allow us to be pretty explicit in determining rigid cohomology. The purpose of le Stum’s lectures, was to give a brief glimpse of these different definitions, and state the relevant theorems that show the equivalence of these definitions.

Bring on the Details!

Now we will start to get into details. Recall $ X$ is a variety over $ \mathbb{F}_q$, and we will denote its dimension by $ d$. To simplify things, we will also assume $ X$ is smooth of pure dimension $ d$.

Before we define the rigid cohomology spaces, first suppose that we already had them. We will denote the $ i$th rigid cohomology space of $ X$ by $ H_\mathrm{rig}^i(X)$. The Frobenius map extends to a map $ F$ (omitting indices) acting on each of these spaces and a Lefschetz-fixed point formula in this setting implies that

$ Z(X,t) = \prod_{i=0}^{2d}\mathrm{det}(1 – tq^dF^{-1}|_{H_\mathrm{rig}^i(X)})^{{-1}^{i+1}}$.

Note that the assumption that $ X$ is smooth of pure dimension $ d$ implies that $ F$ is invertible. So now we need to know these spaces $ H_\mathrm{rig}^i(X)$ and the action of the Frobenius on them well enough. At first glance it may seem that we’ve just made the problem much harder, especially since we now will have to concern ourselves with sites, derived categories, and other exotic creatures. However, remember that our goal is not the number of rational points for some specific variety, but rather hard theoretical statements.

In the next two posts, we will cover some of the easier definitions of rigid cohomology, and point the reader to references for the complete theory.

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