Point counting and random torsion points
The zeta function of C/Fp is
Z(C/Fp, x)
def
= exp
X
n≥1
#C(Fpn )
xn
n
!
=
L(x)rev
(1 − x)(1 − px)
where L(x) = det(x − Frobp |J) ∈ Z[x].
Theorem
We have #J(Fpn ) = Res(L(x), xn
− 1) ∈ N for all n ∈ N.
factor(piccard(J))
W = picrandtors(J,13);
picmember(J,W)
piciszero(J,picmul(J,W,13))
piciszero(J,W)
picistorsion(J,W,13)
Nicolas Mascot p-adic computation of Galois representations