# Reference request for recurrence relation of division polynomials

The recurrence relations for division polynomials of elliptic curves are well known: $$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$ $$\Psi_{2n+1} = \Psi_{n+2} \Psi_n^3 - \Psi_{n+1}^3 \Psi_{n-1}$$

Who originally discovered these relations?

For some reason, a number of works do not cite the origin of the relations, e.g. Silverman's 2nd edition of The Arithmetic of Elliptic Curves, exercise 3.7 just gives them plainly. See edit:

SageMath cites Appendix A from Mazur-Tate from 1991 and they take a few pages to describe the discovery in full. However, they also show up in this paper (p.485, pdf p.4) from Schoof from 1985, citing Lang's Elliptic Curves; Diophantine Analysis from 1978. Indeed, on page 37 of Lang we find the above result as Theorem 1.3, but again without a reference.

EDIT: I was wrong about Silverman not giving citations, they are just in the back of the book (p. 461) and he cites Lang, and also the same work by Cassels as given in the answer by Jeremy Rouse.

## 2 Answers

These recurrences are stated explicitly in Weber's Lehrbuch der Algebra (published in 1908). See volume 3, section 58, page 200. Weber doesn't give a citation to them, so it's hard to know if they were worked out by him or not.

I found the passage in Weber via J.W.S. Cassels 1949 article ''A note on the division values of $$\wp(u)$$'' which I believe is the source for the exercise in Silverman's Arithmetic of Elliptic Curves. Cassels gives references for these recurrences to Fricke (1922), Weber (1908) and Jordan (1913).

• Before accepting, I'm trying to chase Jordan (1913) further. The 3rd edition is from 1913, but the 2nd is 1896, making it older than Weber. The reference to Fricke (1922) can be found here for those interested. Commented Jan 17, 2023 at 22:50

An early reference is Morgan Ward, Memoir on elliptic divisibility sequences (1948). (See equations 4.5 and 4.6.)