Members’ Seminar

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 l distinct from p, l-adic representations of the etale fundamental group do not occur in isolation: they always exist in compatible families that vary across l, including a somewhat more mysterious counterpart for l=p (the "petit camarade cristallin"). We explain in more detail what this all means, indicate some key ingredients in the proof (particularly the role of the Langlands correspondence for function fields), and describe some concrete applications.