F7+ and F4+: Generating Parameter Pairs with Equal tf

1. The generation problem

For a nondegenerate rational pair, put

f(a,b)=ab(a^2-b^2),\qquad \operatorname{tf}(a,b)=t(f(a,b)).

Simultaneous scaling does not change tf, so the natural objects are projective pairs [a:b]. The object to be described and generated is

\mathcal R= \left\{ ([a:b],[c:d]): \operatorname{tf}(a,b)=\operatorname{tf}(c,d) \right\}.

Here ab(a²−b²)cd(c²−d²)≠0. The two pairs remain ordered for now; interchanging them is a separate symmetry of 𝓡.

The main text uses a positive common class T. The same formulas apply to negative oriented tf: the equation of E_T depends on T², while the inverse pair [x:T] retains the sign of the class.

2. What F7+ provides

Fix a positive squarefree value T. The F7+ theorem gives a bijection between projective pairs with tf=T and nontrivial rational points on

E_T:\quad y^2=x^3-T^2x

up to the sign of y:

\begin{aligned} f(m,n)=Tq^2 &\quad\Longrightarrow\quad [m:n]\longmapsto \left[ \left(\frac{Tm}{n},\frac{T^2q}{n^2}\right) \right]_{\pm y},\\ [(x,y)]_{\pm y} &\quad\longmapsto\quad [x:T]. \end{aligned}

Thus every known point P∈E_T(ℚ)∖E_T[2] gives one parameter pair, and every nontrivial multiple nP gives another pair with the same tf. If two independent points P and Q are known, one may use

Before adding points, one must choose one of the two signs of y for every pair, that is, orient the corresponding point. Changing a sign may change the sum even though the original projective pair is unchanged.

R_{r,s}=rP+sQ,\qquad \Psi_T(R_{r,s})=[x(R_{r,s}):T].

This is already a two-dimensional lattice of solutions. Opposite points give the same projective pair, while the 2-torsion points correspond to degenerate pairs and are removed.

3. Why F7+ alone is not a search engine

F7+ is a complete coordinate theorem, but it does not supply an initial point. If T is given and no nontrivial point of E_T(ℚ) is known, the inverse formula [x:T] has no input: there is nothing to add in the group. Even one known point usually gives only the cyclic sequence of its multiples and does not by itself reveal a second independent direction.

Consequently, F7+ as a generator requires a seed set of points. This is precisely the gap filled by F4+.

4. F4+ as a complete source of pairs

Normalized-pair theorem

Every element of 𝓡 has a unique normalization to an ordered pair

([x:1],[x:y]),

for which there is a rational square τ∈(ℚ×)² satisfying

x^2-1=\tau y(x^2-y^2).

Conversely, every nondegenerate rational point on such an F4+ fiber gives an element of 𝓡.

Forward passage

Suppose tf(u,v)=tf(r,s). Put

x=\frac uv,\qquad y=\frac{us}{rv}.

Then [u:v]=[x:1] and [r:s]=[x:y]. Equality of tf means that the quotient of the two f-values is a rational square:

\tau= \frac{f(x,1)}{f(x,y)} = \frac{x^2-1}{y(x^2-y^2)} \in(\mathbb Q^\times)^2.

Reverse passage

If (x,y,τ) satisfies the F4+ equation and τ is a rational square, then

f(x,1)=\tau f(x,y),

and hence tf(x,1)=tf(x,y). Independent scalings recover arbitrary representatives of the two projective classes:

(a,b)=(\lambda x,\lambda),\qquad (c,d)=(\mu x,\mu y), \qquad \lambda,\mu\in\mathbb Q^\times.

5. The F4+ elliptic generator

Let τ=ρ² and τ≠1. The F4+ cubic is birational to the auxiliary curve

\mathcal E_\tau:\quad Y^2=X^3-3\tau^2X+\tau^2(\tau^2+1).

A rational point on this curve recovers

x=\frac{Y}{\tau(X-1)},\qquad y=\frac{X-\tau^2}{\tau(X-1)}.

After removing the boundary xy(x²−1)(x²−y²)=0, one obtains two nondegenerate pairs [x:1] and [x:y] with a common tf. To generate two distinct points, one also removes the diagonal y=1. The surface has rational sections; although the simplest sections themselves lie on the degenerate boundary, combinations such as P−Q, 2P, and P+2Q already give explicit nondegenerate families. Thus F4+ can choose T and simultaneously supply two seed points on E_T.

At τ=1 the elliptic model degenerates, but the original problem remains. Its nontrivial component is the conic

x^2=y^2+y+1,

which is the old F4 family. It is handled by its own complete rational parametrization.

6. The combined F4+ → F7+ algorithm

Choose a rational ρ and a nondegenerate rational point on the corresponding F4+ fiber, directly or from a combination of the universal sections.
Recover the two projective pairs [x:1] and [x:y].
Compute their common squarefree class T=tf(x,1)=tf(x,y).
Map both pairs through F7+ to points P,Q∈E_T(ℚ), choosing the signs of their y-coordinates.
For integers r,s, compute R=rP+sQ and return every nontrivial point to the pair [x(R):T].

Over the function field of the full F4+ surface, the two resulting points are independent, so a lattice of rank at least 2 appears universally. Independence must still be checked in an individual rational specialization, where it may degenerate.

7. Exact example: T=210

On the τ=1 fiber, take

x=\frac73,\qquad y=\frac53, \qquad 7^2=3^2+3\cdot5+5^2.

It gives the two projective pairs

[7:3],\qquad [7:5],\qquad f(7,3)=f(7,5)=840=210\cdot2^2.

Under F7+, they correspond to

P=(490,9800),\qquad Q=(294,3528) \quad\text{на}\quad E_{210}:v^2=u^3-210^2u.

They are independent. Their first sum already gives

P+Q=(240,-1800),\qquad \Psi_{210}(P+Q)=[240:210]=[8:7].

Indeed, f(8,7)=840 and tf(8,7)=210. Further combinations rP+sQ generate new projective pairs in the same square class.

8. Completeness and exact scope

F7+ completely describes pairs of fixed tf through the rational points E_T(ℚ); this is not a ready finite list.
F4+ completely describes ordered tf coincidences after normalization to a common first parameter and also supplies explicit families of seed pairs.
The F4+ → F7+ combination generates the subgroup ⟨P,Q⟩ and every pair corresponding to it. For an individual T, proving that this subgroup equals all of E_T(ℚ) requires determining the Mordell–Weil group of that particular curve.

Full proof of F7+ →Geometry and rank of F4+Theory contents