## Arithmetic Geometry and Number Theory RTG Seminar: Etale and crystalline companions

Seminar | May 8 | 3:10-5:10 p.m. | 939 Evans Hall | Note change in date, time, and location

Kiran Kedlaya, UCSD

Department of Mathematics

Deligne's "Weil II" paper includes a far-reaching conjecture to the effect that for a smooth variety on a finite field of characteristic $p$, for any prime $\ell$ distinct from $p$, $\ell$-adic representations of the etale fundamental group do not occur in isolation: they always exist in compatible families that vary across $\ell$, including a somewhat more mysterious counterpart for $\ell =p$ (the "petit camarade cristallin"). We explain what such an object is; indicate the role of the Langlands correspondence for function fields in the approach to Deligne's conjecture; and report on prior and ongoing work towards the conjecture (including results of Deligne, Drinfeld, Abe-Esnault, and the speaker).

events@math.berkeley.edu