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For an elliptic curve $E:y^2=x^3+Ax+B$ in characteristic different from two and three, division polynomials are elements of its coordinate ring encoding multiplication and torsion points of an elliptic curve. The initial terms includeThe recurrences for higher $n$ yieldFor odd $n$, ${\psi }_n$ is a polynomial in $x$; for even $n$, it has a factor $2y$, and ${\psi }_n/(2y)$ is a polynomial in $x$. Thus the phrase “roots of the division polynomial” must specify whether it means points on the curve or roots in the $x$-line. When $n$ is prime to the characteristic, its zero points are exactly the nonidentity $n$-torsion points of an elliptic curve; for odd $n$, the two opposite points share an $x$-root.^[ref]

Jacques Vélu worked on elliptic curves and explicit isogenies. The original Bordeaux seminar archive verifies the name and his 1970–1971 contribution on elliptic curves.^[ref]

Vélu's formulas construct an isogeny from an elliptic curve and a finite kernel subgroup by summing coordinate corrections over the nonidentity kernel points. They turn the abstract quotient by a subgroup into explicit equations for both the map and the quotient curve. The simple short-Weierstrass formulas require characteristic restrictions and do not replace the distinction between geometric kernel points and inseparable degree.^[ref]

For an elliptic curve $E$ and a finite subgroup $G$ of its geometric points, Jacques Vélu's formulas construct the separable isogeny with kernel $G$. On a Short Weierstrass form away from characteristics two and three, its normalized coordinates areThese rational functions extend to a morphism of the smooth projective curves, sending $G$ to the new identity. Its degree is $\#G$. If $G$ is stable under the absolute Galois group, the summed expressions and quotient curve are defined over the base field, even when the individual points of $G$ are not rational.^[ref]

For $E:y^2=x^3-x$ and $G=\{O,(0,0)\}$, they give $x^{\prime }=x-1/x$ and $y^{\prime }=y(1+1/x^2)$, with quotient $y^{\prime 2}=x^{\prime 3}+4x^{\prime }$. The apparent pole at $(0,0)$ represents its image at infinity, not a missing value of the projective isogeny.