Motives (1/3)
A hypergeometric motive (HGM) is a pair (H, t), where H is a hypergeometric template and
t ∈ Q∗. For t 6= 1, this data is (conjecturally) attached to a pure motive M of weight w,
essentially the middle cohomology group of some algebraic variety. Traces of Frobenius on Mℓ,
are given by an explicit formula (Katz) involving Jacobi sums, equivalently by a finite
hypergeometric sum evaluated at t: for each finite field Fq, we can compute an integer
Nq(H, t) = Tr(Frq | (H, t)).
This formula only makes sense for good primes p; there are two kinds of bad primes:
p is wild if it divides a denominator of the αj or βi (equivalently, one of the cyclotomic
parameters)
else it is tame if vp(t) 6= 0 or vp(t − 1) 6= 0.
Atelier 2022 (13/01/2022) – p. 7/12