Etale and crystalline companions

Etale and crystalline companions - Kiran Kedlaya

Kiran Kedlaya
University of California, San Diego; Visiting Professor, School of Mathematics
April 15, 2019

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.