In Strasbourg Part 2, I gave a bit of motivation for rigid cohomology, but I skirted defining anything substantial, except for the zeta function. Recall that we have an smooth algebraic variety $ X$ of pure dimension $ d$ defined over the finite field $ \mathbb{F}_q$, and initially we were interested in the rational points $ X(\mathbb{F}_q)$ of $ X$. This led us to define the zeta function

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

of $ X$. Furthermore, I asked the reader to have faith that there are rigid cohomology spaces $ H_\mathrm{rig}^i(X)$ of $ X$, that the Frobenius extends to an operator $ F$ on each of them, and that we have a product expansion

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

Obviously many details have been left out, but this will suffice for continuing. I should state the disclaimer that from now on things will be a bit more sketchy since I’m not familiar with the more specialized material, and eventually I will just give references. As I have mentioned previously, I am not an expert in these areas, and I ask the patience of the reader since these summaries may lack some of the polish of my usual posts.

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