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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.<\/p>\n
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.<\/p>\n
Further information is available on the event page<\/a> on the Indico platform.<\/p>\n","protected":false},"excerpt":{"rendered":" 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;…<\/p>\n