Vélu's x-map for μₚ on the split node

Let $F$ be a field of characteristic zero, let $p$ be a prime with $p \neq 2$, and let $\zeta \in F$ be a primitive $p$-th root of unity. Write $x_k = \zeta^k/(1-\zeta^k)^2$ for $k$ in the integer interval $[1, \lfloor p/2 \rfloor]$ (natural-number division, so $\lfloor p/2\rfloor = (p-1)/2$ as $p$ is odd). Let $X \in F$ satisfy $X \neq x_k$ for every such $k$. Then

where $p^2-1$ and $12$ are read in $F$ (legitimate since $\operatorname{char} F = 0$). The hypothesis $X \neq x_k$ is exactly what makes both sides defined; all terms are elements of $F$, so the assertion is a pointwise identity at every admissible $X$, equivalent to the corresponding identity of rational functions in $X$.

This is the explicit form of Vélu's $x$-coordinate map for the quotient of the standard split nodal cubic $y^2 + xy = x^3$, whose smooth points carry the multiplicative group structure with the point of parameter $\zeta^k$ having abscissa $\zeta^k/(1-\zeta^k)^2$, by the subgroup $\mu_p$: the resulting rational function is the $p$-th power map, up to the additive normalising constant $(p^2-1)/12$. It is used in WeierstrassCurve.inZeroComponentAt_veluCoord_iff_of_multiplicative to evaluate Vélu's $x$-map at a multiplicative place, giving the toric branch of the transport law for the zero component under a Vélu quotient.