I'm working on ex. 3.7 (d) in Silvermans "Arithmetic on Elliptic Curves", has anyone done that proof using an elementary way and not complex analysis?
The Exercise is the following:
Let $E : y^2=x^3+Ax+B$ be an Elliptic Curve
For integers $n\ge 0$, define $\psi_n\in Z[x,y,A,B]$ by
$\psi_0 = 0$,
$\psi_1 = 1$,
$\psi_2 = 2y$,
$\psi_3 = 3x^4+6Ax^2+12Bx-A^2$,
$\psi_4 = 4y(x^6+5Ax^4+20Bx^3-5A^2x^2-4ABx-8B^2-A^3)$,
$\psi_{2m} = \frac{1}{2y}\psi_m(\psi_{m+2}\psi_{m-1}^2-\psi_{m-2}\psi_{m+1}^2)\qquad(m\ge 3),\\$
$\psi_{2m+1} = \psi_{m+2}\psi_m^3-\psi_{m-1}\psi_{m+1}^3$ ($m\ge2$)
Prove that for any Point $(x,y)$ on $E$ we have:
$[m]P = \Big( x - \frac{\psi_{m-1}\psi_{m+1}}{\psi_{m}^2}$,$\frac{\psi_{2m}}{2\psi_{m}^4}\Big)$ evaluated in $(x,y)$.
I've tried an induction, proved it for $P+P$ (what is a long calculation!) and want to use the goup law on $E$ and $mP = (m-1)P+P$ to do the induction step, but i don't get anything than lousy long equations. Is there a better way using just elementary mathmatics?