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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?

Somos
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mainman
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    I swear this is duplicate – mercio May 22 '17 at 13:13
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    https://math.stackexchange.com/questions/447672/division-polynomials-of-elliptic-curves and https://math.stackexchange.com/questions/1492620/how-to-prove-that-multiplication-of-points-of-an-elliptic-curve-can-be-done-with (but no answer still) – mercio May 22 '17 at 13:16

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