Elliptic point groups under a variable change
Abstract
Elliptic point groups under a variable change.
Definition 1.1 (Elliptic point groups under a variable change).
Lean statement: D5/S3/Factorization/Mordell/PointVariableChange.equivVariableChange
Formalization. D5/S3/Factorization/Mordell/PointVariableChange.equivVariableChange (✓ std3).
Citation. Michael Stoll; David Kurniadi Angdinata; The Tau Ceti contributors; Kevin Buzzard (2026). Canonical heights, parallelogram constructions and arithmetic point transport. URL: https://github.com/TauCetiProject/TauCeti/tree/934db6ae0034643ffe7b5180242f9ec4c00a56ae.
Commentary.
For an elliptic Weierstrass curve W over a field and an admissible variable change C with nonzero scale, the coordinate transformation induces an additive equivalence from the nonsingular point group of C acting on W to the point group of W. The equivalence includes infinity.
The coordinate and derivative identities preserve the curve equation and nonsingularity. The inverse change supplies the inverse map. The finite-point addition, doubling and negation identities prove preservation of the point group law.