Function: hyperellpadicfrobenius
Section: elliptic_curves
C-Name: hyperellpadicfrobenius
Prototype: GUL
Help: hyperellpadicfrobenius(Q,p,n): Q being a rational polynomial of degree
d and X being the curve defined by y^2=Q(x), return the matrix of the
Frobenius at p>=d in the standard basis of H^1_dR(X) to absolute p-adic
precision p^n.
Doc:
Let $X$ be the curve defined by $y^2=Q(x)$, where $Q$ is a polynomial of
degree $d$ over $\Q$ and $p\ge d$ a prime such that $X$ has good reduction
at $p$ return the matrix of the Frobenius endomorphism $\varphi$ on the
crystalline module $D_p(X) = \Q_p \otimes H^1_{dR}(X/\Q)$ with respect to the
basis of the given model $(\omega, x\*\omega,\ldots,x^{g-1}\*\omega)$, where
$\omega = dx/(2\*y)$ is the invariant differential, where $g$ is the genus of
$X$ (either $d=2\*g+1$ or $d=2\*g+2$). The characteristic polynomial of
$\varphi$ is the numerator of the zeta-function of the reduction of the curve
$X$ modulo $p$. The matrix is computed to absolute $p$-adic precision $p^n$.