Jacobians and Galois representations
Let C be a curve of genus g over Q, let J be its Jacobian, and
let ` ∈ N.
Then J(Q)[`] ' (Z/`Z)2g
, and the points of J[`] are not
defined over Q in general
Galois representation
ρJ,` : Gal(Q/Q) −→ Aut(J[`]) ' GSp2g (Z/`Z).
If p - ` is a prime of good reduction of C, then ρJ,` is
unramified at p, and the characteristic polynomial of
ρJ,`(Frobp) is L(x) mod `, where Z(C/Fp) =
L(x)rev
(1 − x)(1 − px)
.
We wish to compute ρJ,`.
Nicolas Mascot p-adic computation of Galois representations