Frobenius and pairings
If µ` ⊂ Fq, we have the Frey-Rück pairing
J(Fq)[`] × J(Fq)/`J(Fq) −→ F×
q /F×`
q
∼
−→ Z/`Z.
P = pictorspairinginit(J,13);
X = picrand(J);
pictorspairing(J,P,W,X)
pictorspairing(J,P,picmul(J,W,2),X)
We can analyse the action of Frobenius on J(Fq)[13]:
FW = picfrob(J,W);
pictorspairing(J,P,FW,X)
piceq(J,picmul(J,W,9),picfrob(J,W))
Nicolas Mascot p-adic computation of Galois representations