Cubic Map Nonimage on Actual Golden Blocks
Abstract
The two canonical points of every positive cubic-block layer have no rational preimage under the explicit cubic map.
Definition 1.1 (The explicit affine image relation).
Formalization. D5/S3/Factorization/Mordell/GoldenCubicBlockIsogenyNonimage.CubicIsogenyAffineImage (✓ std3).
Source. Repository-derived.
Commentary.
The source coordinates s,t satisfy t squared equals s cubed minus 27b, with s nonzero. The target coordinates are X=(s cubed minus 108b)/(9s squared) and Y=t(s cubed plus 216b)/(27s cubed). This relation is defined only for finite rational source coordinates with s nonzero.
Theorem 1.2 (Neither canonical point has a rational affine preimage).
Proof. Machine-checked in Lean as D5/S3/Factorization/Mordell/GoldenCubicBlockIsogenyNonimage.actual_block_cubic_isogeny_nonimage (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every j at least one, d_j is the cubefree factor and c_j is the cube-part root of B_j=L_(3^j)^2+3. The negative-twist point has coordinates (d_j c_j,d_j L_(3^j)); the positive-twist point has coordinates (5d_j c_j,25d_j F_(3^j)). Neither belongs to the rational affine image relation at its displayed curve constant.
A prime factor of B_j with cubefree exponent one or two exists because B_j is not a cube. The block congruences exclude the primes two, three and five. At that prime, the target abscissa has valuation e+k and the curve constant has valuation 2e, with e equal to one or two and k nonnegative. The equation s^3=9Xs^2+108b has a unique least valuation among its terms for every integer valuation of s, so it has no nonzero rational solution.
The formal image relation uses the stated affine formulas. A group-homomorphism construction and degree calculation for the global isogeny are not part of this statement.