## Etale and crystalline companions

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