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Elliptic surfaces for 6/9 patterns · 5.14

The ABCGHJ Pattern: A K3 Surface Without a Red Progression

Two Gaussian norms and one norm of the form x²+2y² define a smooth intersection of three quadrics. Gaussian factorization reveals an elliptic 4I₄+4I₂ K3 surface, a non-torsion section, and an explicit infinite family of squares.

ABCDEFGHJ
ABCGHJ: yellow ACGJ and ABHJ; blue ACHJ

1. The exact yellow-yellow-blue system

Let a,b,c,g,h,j be rational square roots of the selected entries. The ABCGHJ pattern is characterized by the three conditions

a2+j2=c2+g2,a2+j2=b2+h2,a2+2c2=2h2+j2. \begin{aligned} a^2+j^2&=c^2+g^2,\\ a^2+j^2&=b^2+h^2,\\ a^2+2c^2&=2h^2+j^2. \end{aligned}

The first two equations are equalities of Gaussian norms on ACGJ and ABHJ. The third is the blue norm on ACHJ. This pattern contains no red three-term progression.

The full magic square is recovered from

E0=a2+j22,x=a2−j22,y=c2−g22, E_0=\frac{a^2+j^2}{2},\qquad x=\frac{a^2-j^2}{2},\qquad y=\frac{c^2-g^2}{2},(E0+xE0−x−yE0+yE0−x+yE0E0+x−yE0−yE0+x+yE0−x). \begin{pmatrix} E_0+x&E_0-x-y&E_0+y\\ E_0-x+y&E_0&E_0+x-y\\ E_0-y&E_0+x+y&E_0-x \end{pmatrix}.

2. Why the original surface is already K3

After projectivization, the three quadrics form a complete intersection of degree (2,2,2) in ℙ⁵. To check smoothness, arrange the coefficients of a²,b²,c²,g²,h²,j² as columns:

M=(10−1−1011−100−111020−2−1). M= \begin{pmatrix} 1&0&-1&-1&0&1\\ 1&-1&0&0&-1&1\\ 1&0&2&0&-2&-1 \end{pmatrix}.

Smoothness certificate

All twenty 3×3 minors of M are nonzero. Hence no nontrivial solution of the three quadrics can have fewer than four nonzero coordinates, and the Jacobian columns on its support have rank 3. The projective surface is smooth.

By adjunction, the canonical bundle of a smooth intersection of three quadrics in P5 is trivial. Moreover, H1(O)=0 by the Lefschetz hyperplane theorem; the same vanishing follows from the Koszul complex of the complete intersection. Hence the original ABCGHJ surface is K3. The Jacobian of the fibration chosen below is an additional elliptic model of this surface.

3. The Gaussian chart

Introduce three Gaussian factors with rational parameters

α=1+ip,β=1+iq,γ=1+ir \alpha=1+ip,\qquad \beta=1+iq,\qquad \gamma=1+ir

and define the roots by three products

a+ij=αβγ,c+ig=αβγ‾,h+ib=αβ‾γ. a+ij=\alpha\beta\gamma,\qquad c+ig=\alpha\beta\overline\gamma,\qquad h+ib=\alpha\overline\beta\gamma.
a=1−pq−pr−qr,j=p+q+r−pqr,c=1−pq+pr+qr,g=p+q−r+pqr,h=1+pq−pr+qr,b=p−q+r+pqr. \begin{aligned} a&=1-pq-pr-qr,& j&=p+q+r-pqr,\\ c&=1-pq+pr+qr,& g&=p+q-r+pqr,\\ h&=1+pq-pr+qr,& b&=p-q+r+pqr. \end{aligned}

All three complex numbers have the same norm (1+p²)(1+q²)(1+r²), so both yellow quadrics hold identically. The blue quadric leaves one condition:

A(p,q)(r2−1)+4B(p,q)r=0, A(p,q)(r^2-1)+4B(p,q)r=0,
A(p,q)=p2+q2+12pq−p2q2−1,B(p,q)=p2q−pq2+p−q. \begin{aligned} A(p,q)&=p^2+q^2+12pq-p^2q^2-1,\\ B(p,q)&=p^2q-pq^2+p-q. \end{aligned}

This is a quadratic equation in r. Hence, on this chart, the K3 surface is a double cover of the (p,q)-plane:

W2=A(p,q)2+4B(p,q)2. W^2=A(p,q)^2+4B(p,q)^2.

The chart is birational on a dense open subset: the inverse parameters are recovered directly from the roots:

r=jc−aga2+j2+ac+jg,q=jh−aba2+j2+ah+jb,p=j(1−qr)−a(q+r)a(1−qr)+j(q+r). \begin{aligned} r&=\frac{jc-ag}{a^2+j^2+ac+jg},& q&=\frac{jh-ab}{a^2+j^2+ah+jb},\\ p&=\frac{j(1-qr)-a(q+r)} {a(1-qr)+j(q+r)}. \end{aligned}

4. A genus-one quartic over ℚ(q)

Taking q as the base parameter turns the double cover into a quartic in p:

W2=(q2+1)2p4+32q(1−q2)p3+2(q4+66q2+1)p2+32q(q2−1)p+(q2+1)2. \begin{aligned} W^2={}&(q^2+1)^2p^4+32q(1-q^2)p^3\\ &+2(q^4+66q^2+1)p^2\\ &+32q(q^2-1)p+(q^2+1)^2. \end{aligned}

The point (p,W)=(0,q²+1) provides a rational origin. Thus the generic smooth fiber is not merely a genus-one curve but a pointed elliptic curve over ℚ(q).

5. The split Jacobian

The classical binary-quartic invariants are

I=16(q8+228q6+710q4+228q2+1),J=128(q4−24q3+18q2−24q+1)⋅(q4+18q2+1)⋅(q4+24q3+18q2+24q+1). \begin{aligned} I={}&16(q^8+228q^6+710q^4+228q^2+1),\\ J={}&128(q^4-24q^3+18q^2-24q+1)\\ &\quad\cdot(q^4+18q^2+1)\\ &\quad\cdot(q^4+24q^3+18q^2+24q+1). \end{aligned}

In the short model

Y2=X3−27I(q)X−27J(q) \mathsf Y^2=\mathsf X^3-27I(q)\mathsf X-27J(q)

the cubic splits completely:

Y2=(X−24(q4+18q2+1))⋅(X+12(q4−24q3+18q2−24q+1))⋅(X+12(q4+24q3+18q2+24q+1)). \begin{aligned} \mathsf Y^2={}& \bigl(\mathsf X-24(q^4+18q^2+1)\bigr)\\ &\cdot\bigl(\mathsf X+12(q^4-24q^3+18q^2-24q+1)\bigr)\\ &\cdot\bigl(\mathsf X+12(q^4+24q^3+18q^2+24q+1)\bigr). \end{aligned}

Consequently, the elliptic curve has full rational 2-torsion.

6. The 4I₄+4I₂ passport and rank bound

Up to a nonzero constant, the discriminant is

Δ(q)∼q2(q2+1)2(q2−4q+1)4(q2+4q+1)4. \Delta(q)\sim q^2(q^2+1)^2(q^2-4q+1)^4(q^2+4q+1)^4.

The four roots of q²±4q+1 give four I₄ fibers. The points q=0 and q=±i give three I₂ fibers, and the minimal chart at q=∞ supplies the fourth I₂. On the relatively minimal model with a section, these fibers have Euler-number sum 4·4+4·2=24, consistent with the K3 property already proved.

The trivial lattice of the fibration has rank 2+4·3+4·1=18. Since the Picard number of a K3 surface is at most 20, the Shioda–Tate formula gives rank≤2.

7. A non-torsion section and an infinite family

A parabola tangent to the quartic at the marked point gives the rational section

p(q)=−2q(q2−2q−1)(q2+2q−1)(q−1)(q+1)(q2+1)2,W(q)=q12−26q10+207q8−300q6+207q4−26q2+1(q−1)2(q+1)2(q2+1)3. \begin{aligned} p(q)&=\frac{-2q(q^2-2q-1)(q^2+2q-1)} {(q-1)(q+1)(q^2+1)^2},\\ W(q)&=\frac{q^{12}-26q^{10}+207q^8-300q^6+207q^4-26q^2+1} {(q-1)^2(q+1)^2(q^2+1)^3}. \end{aligned}

It lifts through the quadratic equation in r:

r(q)=(q−1)(q6−13q4−16q3−13q2+1)(q+1)(q6−13q4+16q3−13q2+1). r(q)= \frac{(q-1)(q^6-13q^4-16q^3-13q^2+1)} {(q+1)(q^6-13q^4+16q^3-13q^2+1)}.

The section has infinite order

At q=2 it specializes to (34,240) on the minimal curve y²=x³+x²−9040x+324500. Good reductions modulo 7 and 17 have 8 and 24 points respectively, so rational torsion has order dividing 8. Yet the chosen point reduces to a point of order 3 modulo 17. Hence the point, and therefore the generic section, is non-torsion.

1≤rank⁡E(Q‾(q))≤2.1\le \operatorname{rank}E(\overline{\mathbb Q}(q))\le2.

Substituting p(q), q, and r(q) into the Gaussian formulas of Section 3 gives an explicit one-parameter family. A common denominator is

D(q)=(q−1)(q+1)2(q2+1)2⋅(q6−13q4+16q3−13q2+1). D(q)=(q-1)(q+1)^2(q^2+1)^2 \cdot(q^6-13q^4+16q^3-13q^2+1).

Multiplying the six roots by D and removing their common constant factor produces primitive degree-14 polynomials in ℤ[q]. All three original quadrics vanish identically, while a/j is nonconstant. Hence the family contains infinitely many projectively distinct rational ABCGHJ solutions.

8. An exact positive example

At q=−3, after removing the common factor of the roots, one obtains the following magic square. Root signs do not affect the entries.

A54 · 7332
B72 · 50092
C172092
D580800481
E620456425
F660112369
G72 · 43912
H33912
J52 · 112 · 5472
Square entries are written as prime factorizations.

The magic sum is 1,861,369,275. All nine entries are positive and pairwise distinct; exactly A,B,C,G,H,J are perfect squares.

9. Exact scope of the result

Smoothness is proved for the entire projective surface, the complete intersection of three quadrics. Gaussian factorization and the double cover describe a dense open subset; zeros of the denominators used above require adjacent projective charts.

The 4I₄+4I₂ configuration, full rational 2-torsion, existence of a non-torsion section, the bound 1≤rank≤2, and an explicit infinite family are proved. The exact rank, completeness of this family, and positivity of every rational specialization are not asserted.

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