Function: ellpadicheightmatrix
Section: elliptic_curves
C-Name: ellpadicheightmatrix
Prototype: GGLG
Help: ellpadicheightmatrix(E,p,n,Q): gives the height-pairing matrix for vector
of points Q on elliptic curve E.
Doc: $Q$ being a vector of points, this function returns the ``Gram matrix''
$[F,G]$ of the cyclotomic $p$-adic height $h_E$ with respect to
the basis $(\omega, \eta)$ of $D=H^1_{dR}(E) \otimes_\Q \Q_p$
given to $n$ $p$-adic digits. In other words, if
\kbd{ellpadicheight}$(E,p,n, Q[i],Q[j]) = [f,g]$, corresponding to
$f \omega + g \eta$ in $D$, then $F[i,j] = f$ and $G[i,j] = g$.
\bprog
? E = ellinit([0,0,1,-7,6]); Q = [[-2,3],[-1,3]]; p = 5; n = 5;
? [F,G] = ellpadicheightmatrix(E,p,n,Q);
? lift(F) \\ p-adic entries, integral approximation for readability
%3 =
[2364 3100]
[3100 3119]
? G
%4 =
[25225 46975]
[46975 61850]
? [F,G] * [1,-ellpadics2(E,p,n)]~
%5 =
[4 + 2*5 + 4*5^2 + 3*5^3 + O(5^5) 4*5^2 + 4*5^3 + 5^4 + O(5^5)]
[ 4*5^2 + 4*5^3 + 5^4 + O(5^5) 4 + 3*5 + 4*5^2 + 4*5^3 + 5^4 + O(5^5)]
@eprog