Exposition on point counting using rigid cohomology

Given an algebraic variety $X$ over the finite field ${\bf F}_{q}$, it is known that the zeta function of $X$,

$$ Z(X,T):=\mbox{exp}\left( \sum_{k=1}^{\infty} \frac{#X({\bf F}_{q^{k}})T^{k}}{k} \right) $$

is a rational function of $T$. It is an ongoing topic of research to efficiently compute $Z(X,T)$ given the defining equation of $X$.

I will summarize how we can use Berthelot's rigid cohomology (sparing you the actual construction) to compute $Z(X,T)$, first done for hyperelliptic curves by Kedlaya. I will go on to describe Lauder's deformation algorithm, and the promising fibration algorithm, outlining the present drawbacks.