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Let $ E $ be an elliptic curve over a field $ K $ given by a long Weierstrass equation in Jacobian coordinates $ (x : y : z) $ $$ y^2 + a_1xyz + a_3yz^3 = x^3 + a_2x^2z^2 + a_4xz^4 + a_6z^6. $$ What should the definition of division polynomials $ \psi_m, \phi_m, \omega_m \in \mathbb{Z}[a_1, a_2, a_3, a_4, a_6, x, y] $ for $ m \in \mathbb{N}^+ $ be, so that $$ [m](x : y : 1) = \left(\phi_m : \omega_m : \psi_m\right), $$ for any point $ (x : y : 1) $ on $ E $?

In AEC Exercise 3.7, Silverman defines these recursively by \begin{align*} \psi_1 & = 1, \\\\ \psi_2 & = 2y + a_1x + a_3, \\\\ \psi_3 & = 3x^4 + b_2x^3 + 3b_4x^2 + 3b_6x + b_8, \\\\ \psi_4 & = \psi_2 \cdot (2x^6 + b_2x^5 + 5b_4x^4 + 10b_6x^3 + 10b_8x^2 + (b_2b_8 - b_4b_6)x + (b_4b_8 - b_6^2), \\\\ \psi_{2m + 1} & = \psi_{m + 2}\psi_m^3 - \psi_{m - 1}\psi_{m + 1}^3, & (m \ge 2) \\\\ \psi_{2m} & = \dfrac{1}{\psi_2}(\psi_{m - 1}^2\psi_m\psi_{m + 2} - \psi_{m - 2}\psi_m\psi_{m + 1}^2), & (m \ge 3) \\\\ \phi_m & = x\psi_m^2 - \psi_{m + 1}\psi_{m - 1}, & (m \ge 1) \\\\ \omega_m & = \dfrac{1}{2\psi_2}(\psi_{m - 1}^2\psi_{m + 2} + \psi_{m - 2}\psi_{m + 1}^2). & (m \ge 2) \end{align*}

First of all I do think the $ + $ in the definition of $ \omega_m $ is a typo and should probably be a $ - $, but more importantly $ \phi_m $ and $ \omega_m $ are not defined for small $ m $. Silverman attempts to fix this in the errata, by claiming either that $ \psi_{-m} $ should be defined for small $ m $, or that $ \phi_m $ and $ \omega_m $ should be defined for small $ m $. How should these be defined?

Clearly $ [0](x : y : 1) = (1 : 1 : 0) $ and $ [1](x : y : 1) = (x : y : 1) $, so the values $ \psi_0 = 0 $ and $ \psi_{-1} = -1 $ are essentially determined by the recursive relations for $ \psi_m $ and $ \phi_m $, assuming they are defined correctly. On the other hand, this forces $ \omega_1 = \psi_2 / 2 $, which is not a polynomial unless $ a_1 = a_3 = 0 $, in which case $ \omega_1 = y $ as expected. What went wrong?

I am inclined to believe that there is an uncorrected error in the definition of $ \omega_m $. Most references I could find avoided defining these for $ \mathrm{char}(K) = 2 $ where $ a_1 $ and $ a_3 $ may be non-zero, or proved the identity for $ [m](x : y : 1) $ using the Weierstrass $ \wp $-function in $ \mathrm{char}(K) = 0 $, which I am also trying to avoid.

Context: I was attempting to prove the EDS recurrence in characteristic two purely symbolically.

Multramate
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  • Your guess about the sign error is wrong. The $\omega_m = \psi_{2m}/(2\psi_m).$ – Somos Apr 10 '23 at 01:55
  • Then $ \omega_1 = -\psi_2/2 $, which is $ -y $ and not $ y $ when $ a_1 = a_3 = 0 $? – Multramate Apr 10 '23 at 12:09
  • Oops! I meant that you are correct about the sign error. I checked other sources Wikipedia Division polynomials and arXiv:1108.3051, pp.6-7, and the sign is minus. However the denominator should be $4y$ instead of $2\psi_2$. – Somos Apr 10 '23 at 15:27
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    The denominator is $ 4y $ when $ a_1 = a_3 = 0 $, but I still think it should be $ 2\psi_2 $ otherwise: for instance assuming that $ \psi_{-2} = -\psi_2 $, it can be shown that $ \omega_0 = \tfrac{1}{2\psi_2}(\psi_2 - \psi_{-2}) = 1 $ as expected. As far as I am aware, there are no definitive references when $ a_1 $ and $ a_3 $ are non-zero other than Silverman's AEC Exercise 3.7, which is why I am asking this question. – Multramate Apr 10 '23 at 17:54
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    I think you are correct. Great question! – Somos Apr 10 '23 at 19:26
  • I found an old article containing a slightly different definition: $$ \omega_m = \dfrac{\psi_{2m}}{2\psi_m} - \dfrac{1}{2}\psi_m(a_1\phi_m + a_3\psi_m^2), $$ which I think is correct, based on numerical experiments. It would be cool to see an amended proof somewhere though! – Multramate Apr 10 '23 at 21:40
  • It might help that I was chasing the original source for these a while ago, see: https://mathoverflow.net/questions/438656/reference-request-for-recurrence-relation-of-division-polynomials/ – Krijn Apr 12 '23 at 15:12

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