When studying the zeta functions of algebraic varieties over finite fields of characteristic $p$, there are essentially two approaches to put these in the context of a Weil cohomology theory. One approach is the familiar one using etale cohomology; in this theory, the cohomology groups are vector spaces over an $\ell$-adic field where $\ell$ is distinct from $p$. The other is the one derived from Dwork’s proof of rationality of zeta functions and incorporating Berthelot’s work on crystalline cohomology.
We first describe the locally constant coefficient objects in these two approaches; in the etale approach these are representations of certain fundamental groups (lisse $\ell$-adic sheaves), while in the crystalline approach these are certain vector bundles with connection (overconvergent $F$-isocrystals). We then assert a theorem that says that these objects do not occur in isolation: on a smooth variety over a finite field, any object in one category admits corresponding objects in all of the other categories which carry the same arithmetic information. This builds on the Langlands correspondence for $GL_n$ (Drinfeld, L. Lafforgue, Abe), which shows that on a curve, coefficient objects always have “geometric origins”. It also incorporates results of Drinfeld, Deligne, Abe, Esnault, and the speaker, which together work around the fact that geometric origins on higher-dimensional varieties seem to be quite hard to establish.
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