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A topic related to hypergeometric motives is that of Jacobi sum motives.
These are indeed simpler, and in fact the tame prime information for
hypergeometric motives can be determined from Jacobi motives, possibly
twisted by Kummer and Tate characters.
The classical Jacobi sums were indicated by Weil to come from
Grössencharacters [Wei52], and this functionality
is also included, with it indeed being the preferred method to
compute Euler factors and the L-series, once the reciprocity
correspondence has been established and the Grössencharacter identified.
Let nj∈(Z) and xj∈(Q)/(Z)
with θ=∑j nj< xj > an element
of the free group on (Q)/(Z) with ∑njxj∈(Z).
Letting m be the least common multiple of the denominators of the xj,
the field of definition Kθ is a subfield of (Q)(ζm),
corresponding by class field theory to quotienting out by
((Z)/m(Z))star by elements which leave θ fixed
when scaling by them. When scaling by -1 fixes θ this field Kθ
is totally real, and otherwise it is a CM field.
For primes p with gcd(p, m)=1, we consider Gauss sums corresponding
for prime ideals p in (Q)(ζm), defined by
Ga/mψ(p)= - ∑_(x∈(F)ppstar)
((x/p))am
ψ((Tr)_((F)p)^((F)pp) x),
where ψ is a nontrivial additive character on (F)p
and the power residue symbol takes values in the roots of unity
of (Q)(ζm) with
((x/p))am ≡ x(q - 1)a/m
((mod) p).
The associated Jacobi sum evaluation for θ at p
is then given by ∏j Gxj(p)nj with the result
being independent of the choice of additive character ψ.
This defines the Jacobi sum for good primes p up to a
choice of ζm into (C).
If one is just interested in Euler factors over (Q) and not Kθ,
then a p-adic method using the Gross-Koblitz formula can also be used.
There are known bounds on the conductor of the resulting L-function,
the first being that of Weil [Wei52].
A Jacobi motive can also be Kummer twisted by tρ for some
rational ρ and nonzero rational t. This can increase the field
of definition so that m includes the denominator of ρ.
This corresponds to multiplying the various Jacobi sum evaluations
by suitable roots of unity.
Often one wants to Tate twist the Jacobi sum to gets its effective weight,
and for this reason the full unit is sometimes called a Jacket motive
(for Jacobi, Kummer, and Tate).
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