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Elliptic surfaces for 6/9 patterns · 5.2

The ABCDEJ Pattern: Two Progressions with a Common Endpoint

The AEJ and BDJ progressions meet at J. Parametrizing them simultaneously leaves one genus-one quartic; its split Jacobian is an elliptic K3 surface with a non-torsion section.

ABCDEFGHJ
ABCDEJ: the AEJ and BDJ progressions meet at J

1. The three independent conditions

Let a,b,c,d,e,j be the rational square roots of the six selected entries. The initial ABCDEJ system is

{a2+j2=2e2,b2+d2=2j2,a2+d2=c2+e2. \begin{cases} a^2+j^2=2e^2,\\ b^2+d^2=2j^2,\\ a^2+d^2=c^2+e^2. \end{cases}

The first two equations describe two progressions of squares. The last equation is the yellow ACDE relation that determines when both progressions belong to one magic square.

2. Gluing the progressions at J

Use the standard polynomials

L(z)=z2−2z−1,C(z)=z2+1,R(z)=z2+2z−1, L(z)=z^2-2z-1,\qquad C(z)=z^2+1,\qquad R(z)=z^2+2z-1,L(z)2+R(z)2=2C(z)2.L(z)^2+R(z)^2=2C(z)^2.

Two parameters p,q make the root j common to both progressions:

a=L(p)C(q),e=C(p)C(q),j=R(p)C(q),b=L(q)R(p),d=R(q)R(p). \begin{aligned} a&=L(p)C(q),& e&=C(p)C(q),& j&=R(p)C(q),\\ b&=L(q)R(p),& d&=R(q)R(p). \end{aligned}

After this substitution both red equations hold identically. It remains to require c²=a²+d²−e² to be a square.

3. The residual quartic

Writing c=V and simplifying gives

V2=R(p)2R(q)2−4p(p2−1)C(q)2. V^2=R(p)^2R(q)^2-4p(p^2-1)C(q)^2.

This is a quartic in q. At q=0 its right-hand side is C(p)², so the generic fiber has the rational base point (0,C(p)). At q=1 there is another rational point (1,2C(p)); this point will produce the non-torsion section.

Exact criterion

Rational p,q,V give an ABCDEJ solution through the formulas of Section 2 and c=V exactly when they lie on this quartic. The magic-square coordinates are recovered as E=e², x=a²−e², and y=e²−c².

4. The split Jacobian

The classical binary-quartic invariants give the short model

Y2=X3−1728P(p)X+27648Q(p), Y^2=X^3-1728P(p)X+27648Q(p),P(p)=p8+6p7+16p6+6p5−18p4−6p3+16p2−6p+1,Q(p)=(p2+1)2⋅(p4+3p3+2p2−3p+1)⋅(p4+6p3+2p2−6p+1). \begin{aligned} P(p)={}&p^8+6p^7+16p^6+6p^5-18p^4\\ &-6p^3+16p^2-6p+1,\\ Q(p)={}&(p^2+1)^2\cdot (p^4+3p^3+2p^2-3p+1)\\ &\cdot(p^4+6p^3+2p^2-6p+1). \end{aligned}

The cubic polynomial splits completely:

X0=24(p2+1)2,X1=24(p4+6p3+2p2−6p+1),X2=−48(p4+3p3+2p2−3p+1). \begin{aligned} X_0&=24(p^2+1)^2,\\ X_1&=24(p^4+6p^3+2p^2-6p+1),\\ X_2&=-48(p^4+3p^3+2p^2-3p+1). \end{aligned}

Thus full rational 2-torsion is visible over ℚ(p). In Legendre form the surface has parameter

λ(p)=−p4+2p3+2p2−2p+12p(p2−1). \lambda(p)=- \frac{p^4+2p^3+2p^2-2p+1} {2p(p^2-1)}.

5. A non-torsion section and an infinite subgroup

The quartic point q=1, V=2C(p) maps to a rational section P of the Jacobian. Exact specialization at p=2 and the Kummer map show that P is non-torsion. Therefore

⟨2P⟩={ 2nP:n∈Z }≅Z. \langle2P\rangle =\{\,2nP:n\in\mathbb Z\,\} \cong\mathbb Z.

The inverse birational transformation sends each 2nP to rational functions qₙ(p),Vₙ(p), and then to six ABCDEJ roots. This is an infinite mechanism for generating parametrizations.

6. An explicit family from the section 2P

For the first even multiple, p is the only free parameter:

q(p)=−2C(p)2R(p)2,V(p)=C(p)V0(p)R(p)4, \begin{aligned} q(p)&=-\frac{2C(p)^2}{R(p)^2},\\ V(p)&=\frac{C(p)V_0(p)}{R(p)^4}, \end{aligned}

where

U(p)=(p4+6p2−8p+5)⋅(5p4+8p3+6p2+1),B0(p)=7p8+8p7+12p6+8p5+74p4−8p3+12p2−8p+7,V0(p)=p8−24p7−44p6−24p5+102p4+24p3−44p2+24p+1,D0(p)=p8+24p7+20p6+24p5−26p4−24p3+20p2−24p+1. \begin{aligned} U(p)={}&(p^4+6p^2-8p+5) \cdot(5p^4+8p^3+6p^2+1),\\ B_0(p)={}&7p^8+8p^7+12p^6+8p^5+74p^4\\ &-8p^3+12p^2-8p+7,\\ V_0(p)={}&p^8-24p^7-44p^6-24p^5+102p^4\\ &+24p^3-44p^2+24p+1,\\ D_0(p)={}&p^8+24p^7+20p^6+24p^5-26p^4\\ &-24p^3+20p^2-24p+1. \end{aligned}

Multiplying the roots by the common denominator R(p)⁴ gives the polynomial family

a=L(p)U(p),b=R(p)B0(p),c=C(p)V0(p),d=−R(p)D0(p),e=C(p)U(p),j=R(p)U(p). \begin{aligned} a&=L(p)U(p),& b&=R(p)B_0(p),\\ c&=C(p)V_0(p),& d&=-R(p)D_0(p),\\ e&=C(p)U(p),& j&=R(p)U(p). \end{aligned}

Identity-level correctness

The six polynomials identically satisfy all three ABCDEJ equations. Values of p at which the required nondegeneracy conditions fail must be excluded. The multiples 4P,6P,… produce further families in the same way.

7. Surface passport and rank bound

Up to a nonzero constant, the discriminant is

Δ(p)∼p2(p−1)2(p+1)2R(p)4⋅(p4+2p3+2p2−2p+1)2. \Delta(p)\sim p^2(p-1)^2(p+1)^2R(p)^4\cdot (p^4+2p^3+2p^2-2p+1)^2.

The two roots of R(p) give I₄ fibers. The points p=0,±1, the four roots of the last factor, and infinity give eight I₂ fibers. Their Euler numbers sum to 24, so the minimal surface is a K3 surface.

Proved rank bound

1≤rank⁡E(Q‾(p))≤4. 1\le\operatorname{rank}E(\overline{\mathbb Q}(p))\le4.

The non-torsion section gives the lower bound 1. The root rank of the 2I₄+8I₂ configuration is 14; Shioda–Tate and ρ≤20 for a complex K3 surface give the upper bound 4. The exact rank is not yet determined.

8. Scope of the result

An explicit generating K3 surface and an infinite cyclic subgroup of its sections have been constructed. This gives infinitely many one-parameter ABCDEJ families, beginning with the displayed 2P family.

It is not claimed that this subgroup exhausts the full Mordell–Weil group or that the chosen quartic chart enumerates every rational solution of the original pattern.

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