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This is exercise 3.7 from Silvermans AEC (2nd edition).

Let $E$ be a nonsingular elliptic curve over $\mathbb{C}$ given by

$$ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$

The $n^{th}$ division polynomls $\psi_{n}$ are defined using

$ \psi_{1} = 1, \\ \psi_{2} = 2y+a_{1}x+a_{3}, \\ \psi_{3} = 3x^{4} +b_{2}x^{3} +3b_{4}x^{2} + 3b_{6}x + b_{8}. \\ \psi_{4} = \psi_{2}(2x^{6} +b_{2}x^{5} + 5b_{4}x^{4} +10b_{6}x^{3} + 10b_{8}x^{2} +(b_{2}b_{8} - b_{4}b_{6})x + (b_{4}b_{8}-b_{6}^2)),$

then recursively by the formulas

$\psi_{2n+1} = \psi_{n+2}\psi_{n}^{3} - \psi_{n-1}\psi_{n+1}^{3} \\ \psi_{2n}\psi_{2} = \psi_{n-1}^{2}\psi_{n}\psi_{n+2}- \psi_{n-2}\psi_{n}\psi_{n+1}^{2}. $

Show that

$$ \psi_{m+n}\psi_{m-n}\psi_{r}^{2} = \psi_{m+r}\psi_{m-r}\psi_{n}^{2} - \psi_{n+r}\psi_{n-r}\psi_{m}^{2}.$$

Now, it seems that this should be done by considering div($\psi_{n}$), and by doing so (and considering $\psi_{n}$ as a function on $\mathbb{C} / \Lambda$) I can show

$$ \frac{\psi_{m+n}(z)\psi_{m-n}(z)}{\psi_{m}^{2}(z)\psi_{n}^{2}(z)} = \wp(nz) - \wp(mz). $$

which gives the result. However elliptic functions aren't covered until chapter 6.

So, my question is: How can this be done without using elliptic functions?

Somos
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Jeff Bleaney
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  • Also posted to, and commented on (but closed) at MO, http://mathoverflow.net/questions/137211/division-polynomials-of-elliptic-curves – Gerry Myerson Jul 20 '13 at 12:32
  • Note: there is a typo, it should say $\psi_2 = 2y + a_1 x + a_3$. Came across this searching for division polynomials, so it should probably be fixed. – dwbarkley Mar 02 '20 at 20:33
  • @dwbarkley It says that we are considering $y^2=x^3+Ax+B$ (short Weierstrass form). Thus, the $a_1=a_3=0$. – daruma Jun 05 '20 at 14:32

1 Answers1

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You can use the addition/duplication formulas and a bunch of algebra to solve the problem without using any complex analysis. The advantage of the algebraic proof is that it's valid in any characteristic

Joe Silverman
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  • I am curious to see if anyone has actually done the purely algebraic approach. I tried it myself and had to give up because the algebra was too messy. Other references I have checked with (e.g. Serge Lang's book or Lawrence Washington's book) rely on proving it using Weierstrass p function over char(K)= 0 case via the way that the OP mentioned. – daruma Jun 05 '20 at 14:39
  • @daruma You're actually asking three questions. (1) Has anyone actually done it purely algebraically? I expect that the answer is yes. (2) Can I tell you the name of someone who has done it? Offhand, the answer is no. (3) Has anyone ever published (even in the sense of posting online) a description of the computation? As far as I know, the answer is no. --- Having said that, the divisor calculation you mention can probably be verified algebraically by first proving div$(\psi_n)$ is the sum of the $n$-torsion points minus an appropriate multiple of $O$. ... – Joe Silverman Jun 05 '20 at 15:22
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    Well, at least that will probably work as long as the characteristic of your field doesn't divide $n$. Note that your proof using elliptic functions is strictly characteristic $0$. – Joe Silverman Jun 05 '20 at 15:23
  • Sorry to bother you with this, but do you happen to know of an on-line resource where the characteristic two versions of this recurrence relations have been published? I'm used to rederiving the first few every time I need them, but, as the years tick by, it is becoming more of a chore, and I would appreciate a reference. – Jyrki Lahtonen Nov 03 '22 at 07:32
  • The statement of that exercise in the second edition seems to contain errors. In the definition of $\omega_m$ and in the statement of part (a), one both divides by $y$ (or $2y$) but that works only when $a_1 = a_3 = 0$. I guess the statements are copied from the first edition where the same exercise only deals with curves of the form $y^2 = x^3 + Ax + B$ and many expressions don't get properly updated. @JoeSilverman Could you confirm these errors? – WhatsUp May 24 '23 at 15:18
  • Upon further reading, I see that the formula for $b_2$ on Page 42 is also wrong: it should be $b_2 = a_1^2 + 4a_2$. Where can I find a list of errata for the book? – WhatsUp May 24 '23 at 15:32
  • @WhatsUp If you google "arithmetic of elliptic curves errata", you'll find that which you seek. But here is the direct link: https://www.math.brown.edu/johsilve/AEC/AECErrata.pdf – Joe Silverman May 24 '23 at 17:09
  • Thanks for the link - actually I googled "GTM 106 errata" but didn't find it (: It seems that the typo on Page 42 is not mentioned in the errata though. It could be that the book has been updated. – WhatsUp May 24 '23 at 18:45