Hypergeometric templates (2/4): cyclotomic and γ formats
The defined over Q assumption allows to abbreviate each occurence of [a1/D, . . . , aϕ(D)/D]
(where the ai range in (Z/DZ)∗) to [D]. We have three possible ways of giving a
hypergeometric template:
by the two GP vectors [α1, . . . , αd] and [β1, . . . , βd] (α, β parameters),
or by their denominators [D1, . . . , Dm] and [E1, . . . , En] (cyclotomic parameters); note
that
P
j ϕ(Dj) =
P
k ϕ(Ek) = d.
a third and final way is to give the gamma vector (γn) defined by
A(X)/B(X) =
Q
n(Xn − 1)γn , which satisfies
P
n nγn = 0.
To any such data we associate a hypergeometric template using the function hgminit; then the
αj and βk are obtained using hgmalpha, cyclotomic parameters using hgmcyclo and the
gamma vectors using hgmgamma.
N.B. β = (0, . . . , 0) or E = (1, . . . , 1) can be omitted in (α, β) and cyclotomic formats,
respectively.
Atelier 2022 (13/01/2022) – p. 4/12