Quotient and arithmetic descent identities for X₁(18) #
The order-three automorphism of the order-eighteen sextic has invariant functions
u = (x³ - 3x + 1)/(x(x-1)),
v = y/(x(x-1)),
which satisfy
v² = u² - 4u + 12.
This file parameterizes that rational conic, identifies the degree-three fiber as the cyclic cubic
z³ - uz² + (u-3)z + 1,
and records its orbit factorization and square discriminant. After the conic parameter is introduced, the cleared cubic has discriminant
(t²+3)²(7t²+6t+3)².
The two quadratic factors are explicit norms in Eisenstein coordinates.
Writing ω²+ω+1=0, the second is
N((3+ω)m + (1-ω)n) = 7m²+6mn+3n²,
and N(3+ω)=7. We prove these norm identities, multiplicativity, two
primitive-form Bézout identities, and the fact that the two norm forms
cannot both be divisible by 7 for coprime integer parameters.
For a rational root, comparison of all three coefficients of the cyclic
cubic gives two binary quadratic--cubic identities. Their resultant is
16; a mod-2 and mod-16 calculation then proves that the remaining
integer quotient is one of -8, -4, 4, or 8. The final boundary is
therefore a four-case primitive integral obstruction.
These are algebraic and local prerequisites for the classical
π = 3+ω descent. They do not construct the induced endomorphism of the
Jacobian, prove π-surjectivity on its Mordell--Weil group, determine its
torsion, or solve the four remaining integral cases.
The rational conic quotient #
The slope used to recover the conic parameter away from u=1.
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- MazurTorsion.XOneEighteenDescent.conicSlope u v = (v - 3) / (u - 1)
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The cyclic cubic fiber #
The exceptional quotient fiber u=1 has no rational abscissa.
The invariant abscissa of a noncuspidal point satisfies its cubic fiber equation.
A noncuspidal rational abscissa never lies over the exceptional
quotient value u=1.
Product over the three-element Möbius orbit, viewed as a degree-three norm polynomial.
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The orbit norm is exactly the cyclic cubic fiber polynomial.
The standard discriminant formula for a cubic
a z³ + b z² + c z + d.
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Clearing the conic denominator gives clearedFiber.
A root of the parameterized fiber gives a root of the cleared cubic.
The homogeneous cubic obtained by writing the conic parameter as
t=m/n.
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Substitution t=m/n turns the cleared cubic into its homogeneous
form after multiplication by n².
A root of the cleared rational-parameter cubic is a root of the primitive homogeneous cubic attached to its canonical fraction.
The homogeneous cubic discriminant is the product of the squares of the two quadratic norm forms.
Every rational noncuspidal point on the sextic admits the explicit conic parameter and is a root of the resulting denominator-free cubic.
Canonical numerator-denominator coordinates sharpen the preceding statement to primitive integer parameters.
The coefficient discriminant of the cleared cubic factors into two squared quadratic forms.
Eisenstein-coordinate norm identities #
The coordinate norm is multiplicative.
Multiplication by the conjugate gives the rational integer norm.
Conjugation preserves the Eisenstein norm.
The Eisenstein prime above 7, represented by π=3+ω.
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The norm of π=3+ω is 7.
The conjugate prime is conj(π)=2-ω.
The coordinate factorization of the rational prime 7 as
π·conj(π).
Divisibility by π in the coordinate ring.
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A factor of π forces a factor of 7 in the norm.
Up to multiplication by ω, the two norm arguments multiply to the
pair formed by the trace coefficient and three times the leading
coefficient of the split cubic.
The two split-coefficient identities package as a scalar-times-cube factorization in integral Eisenstein coordinates. This is the direct input for the remaining unique-factorization argument.
The two primitive quadratic norm forms.
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For integer homogeneous parameters, the two factors in the cubic discriminant are exactly the two named norm forms.
A Bézout-style identity eliminating the denominator parameter.
For primitive integer parameters, the two Eisenstein norm forms
cannot both be divisible by the rational prime below π.
Rational-root divisibility #
If a primitive rational number a/b is a root of the homogeneous
cyclic cubic, then the three pairwise-coprime cusp factors of its
Möbius orbit divide the leading coefficient together.
Equating the two presentations of the cyclic-cubic discriminant gives an integral norm equation for primitive quotient and root coordinates.
After dividing by the nonzero product of the three cusp factors, the discriminant identity becomes a square times the cube of the Eisenstein orbit norm.
Once the quotient parameter and a primitive rational root are both cleared, equality of the two invariant presentations determines the other two coefficients of the split cyclic cubic.
A primitive split fiber can only occur in the odd-odd parameter
class. Opposite parity would give the rootless cubic
z³ + z² + 1 modulo two.
The two independent split-fiber coefficients have resultant sixteen as binary quadratic forms in the primitive conic parameters.
Primitivity and the split coefficient identities make the apparent
quotient k a divisor of sixteen with exact two-adic order two or three.
In particular, the remaining arithmetic boundary has only four possible
signed values for k.
The quotient in a primitive split cyclic cubic is one of the four signed integers of two-adic order two or three.
Modulo four, the primitive norm identity separates the four possible
quotients into two pairs: the parameters agree for quotient ±8 and differ
for quotient ±4.
The complete elementary descent package obtained here from a noncuspidal rational point. Besides primitive quotient and root coordinates, it supplies the exact quotient of the cubic's leading coefficient by the three cusp factors and the resulting square-times-cube Eisenstein norm equation.
The remaining arithmetic statement after the checked quotient and rational-root descent. It rules out primitive integral parameters for which the cyclic cubic has a rational root and satisfies the resulting square-times-cube norm equation.
This predicate is deliberately narrower than the original rational-point classification: all coordinate changes, denominator conditions, and norm identities needed to reach it are proved above.
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The finite integral boundary left after comparing all three
coefficients of the split cyclic cubic. The quotient is restricted to
-8, -4, 4, or 8; no rational functions or denominator conditions
remain in the statement.
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The refined finite boundary also exposes the scalar-times-cube
Eisenstein product and the modulo-four distinction between quotient ±4
and quotient ±8.
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The refined finite boundary implies the original four-case boundary. Both extra certificates are derived from the old hypotheses, so existing consumers retain their public types.
The finite split-coefficient boundary implies the earlier primitive rational-root obstruction.
The refined finite split boundary implies the primitive rational-root obstruction through the compatibility-preserving four-case interface.
The primitive cyclic-cubic obstruction consumes the complete descent
package and rules out a noncuspidal rational point on the X₁(18) model.
A proof of the explicit primitive cyclic-cubic obstruction excludes exact rational order eighteen through the already checked Tate-normal-form and genus-two reduction.
The four-case integral split obstruction excludes exact rational order eighteen through the checked primitive descent.
The scalar-cube and modulo-four refined finite boundary excludes exact rational order eighteen through the compatibility-preserving conversion.