Differential addition on E/Aut(E) for elliptic curves with extra automorphisms

Let E be an elliptic curve over a field k with char k != 2,3. The Kummer line E/{+-1} carries a doubling map and a differential addition law, and these underlie the Montgomery ladder. When j(E) in {0, 1728} the automorphism group is strictly larger than {+-1} — it is mu_6 or mu_4 — and the quotient E/Aut(E) is a strictly smaller line, coordinatized by x^{|Aut(E)|/2}. We show that this larger quotient still carries both operations. Explicitly, u(P+Q) is a rational function of u(P), u(Q) and u(P-Q), where u = x^{|Aut(E)|/2}. The proof identifies the obstruction: the addition and subtraction maps are equivariant only for the diagonal Aut(E)-action, so the quadratic supplied by the theorem of the cube lives not on (E/Aut E)^2 but on a degree-(|Aut E|/2) cover of it; the third input u(P-Q) carries exactly the information needed to descend. The classical Kummer line is the case where that cover is trivial, which is why the classical construction stops there. We give the laws explicitly, uniformly in the curve parameter, and record a formula for the length of doubling orbits on E/Aut(E). In characteristics 2 and 3, where Aut(E) is non-commutative, both operations still descend — commutativity turns out to be inessential — and the same argument gives the result for abelian varieties of any dimension, quotiented by any finite group of origin-fixing automorphisms containing -1 (for instance the automorphisms of a polarized abelian variety). One principle organizes the explicit laws: differential addition on A/Aut(A) is the classical Kummer-variety law lifted through the cover with group Aut(A)/{+-1}, and that lift is a single radical, a solvable tower of radicals, or irreducible to the Kummer law according as the group is cyclic, solvable, or neither — the last case first arising for the Klein quartic in genus three. We emphasize that these differential-addition laws confer no cryptographic advantage over the x-line: the sqrt(|Aut(E)|) speedup of automorphism-assisted Pollard rho is information-theoretic, and on a general-purpose processor the higher-degree laws obtained here are less efficient than the x-line ladder.