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Elliptic geometry of tfmn · 4.1

F7+: Parameter Pairs and the Congruent-Number Surface

A fixed tf value determines a congruent-number elliptic curve. Nondegenerate rational parameter pairs are in exact bijection with its nontrivial rational points, up to the sign of y.

1. Square class and projective pair

Put

f(m,n)=mn(m−n)(m+n). f(m,n)=mn(m-n)(m+n).

A pair (m,n) is nondegenerate when m, n, m−n, and m+n are all nonzero.

Definition of a projective pair

Two nonzero rational pairs are equivalent when one is obtained from the other by simultaneously multiplying both parameters by the same nonzero rational number:

(m,n)∼(m′,n′)  ⟺  ∃ λ∈Q×:(m′,n′)=(λm,λn). (m,n)\sim(m',n') \iff \exists\,\lambda\in\mathbb Q^\times: \quad (m',n')=(\lambda m,\lambda n).

The equivalence class of (m,n) is denoted by [m:n] and is called a projective pair. The set of these classes is the projective line ℙ¹(ℚ).

Under this simultaneous substitution, f(λm,λn)=λ⁴f(m,n). Since λ⁴ is a rational square, the square class of f and the value of tf are the same for every representative of [m:n]. Scaling only one parameter is not part of this equivalence.

Fix a positive squarefree integer T. Consider the set

PT={[m:n]∈P1(Q):mn(m−n)(m+n)≠0,f(m,n)∈T(Q×)2}. \mathcal P_T= \left\{ [m:n]\in\mathbb P^1(\mathbb Q): mn(m-n)(m+n)\ne0,\quad f(m,n)\in T(\mathbb Q^\times)^2 \right\}.

Thus [m:n] belongs to 𝒫_T exactly when f(m,n)=Tq² for some q∈ℚ×, equivalently tf(m,n)=T.

About the name

The label F7+ arose historically. In this article it denotes the entire surface of fixed-square-class curves and does not depend on any finite numbering of formulas.

2. The curve for fixed T

Associated with T is the congruent-number elliptic curve

ET:y2=x3−T2x=x(x−T)(x+T). E_T:\qquad y^2=x^3-T^2x=x(x-T)(x+T).

Its rational 2-torsion points and identity are

ET[2](Q)={O, (0,0), (T,0), (−T,0)}. E_T[2](\mathbb Q)= \{\mathcal O,\ (0,0),\ (T,0),\ (-T,0)\}.

Let E_T°(ℚ) be the complement of these four points. There y≠0. The involution (x,y)↦(x,−y) negates a point in the group E_T(ℚ) while preserving its x-coordinate.

3. The bijection theorem

F7+ theorem

For every positive squarefree T there is a bijection

ΦT:PT⟶∼ET∘(Q)/{(x,y)∼(x,−y)}. \Phi_T:\quad \mathcal P_T \overset{\sim}{\longrightarrow} E_T^\circ(\mathbb Q)\big/\{(x,y)\sim(x,-y)\}.

If f(m,n)=Tq², the forward map is

ΦT([m:n])=[(Tmn,T2qn2)]±y. \Phi_T([m:n]) = \left[ \left( \frac{Tm}{n}, \frac{T^2q}{n^2} \right) \right]_{\pm y}.

The inverse map has the particularly simple form

ΨT([(x,y)]±y)=[x:T]. \Psi_T([(x,y)]_{\pm y})=[x:T].

4. Derivation of the forward map

Let [m:n]∈𝒫_T and f(m,n)=Tq². Set

x=Tmn,y=T2qn2. x=\frac{Tm}{n},\qquad y=\frac{T^2q}{n^2}.

Then the curve equation follows directly from the definition of f:

x3−T2x=T3m(m2−n2)n3=T3n4 mn(m2−n2)=T3n4 f(m,n)=T4q2n4=y2. \begin{aligned} x^3-T^2x &=\frac{T^3m(m^2-n^2)}{n^3}\\ &=\frac{T^3}{n^4}\,mn(m^2-n^2)\\ &=\frac{T^3}{n^4}\,f(m,n)\\ &=\frac{T^4q^2}{n^4} =y^2. \end{aligned}

Nondegeneracy gives x≠0,±T, while q≠0 gives y≠0. Hence the constructed point lies in E_T°(ℚ).

Why the map is well defined

The equation f=Tq² determines q only up to sign; replacing q by −q replaces y by −y. This is exactly why the target is quotiented by the sign of y.

Replacing the representative by (λm,λn) replaces q by ±λ²q. Both fractions Tm/n and T²q/n² remain unchanged, apart from the already-quotiented sign of y. Thus Φ_T depends only on [m:n].

5. Derivation of the inverse map

Let (x,y)∈E_T°(ℚ). The curve equation gives

f(x,T)=xT(x−T)(x+T)=T(x3−T2x)=Ty2. \begin{aligned} f(x,T) &=xT(x-T)(x+T)\\ &=T(x^3-T^2x)\\ &=Ty^2. \end{aligned}

Therefore [x:T] belongs to 𝒫_T. Replacing y by −y does not change this pair, so Ψ_T is well defined on the quotient.

It remains to check the two compositions. For the original pair,

ΨT(ΦT([m:n]))=[Tmn:T]=[m:n]. \Psi_T(\Phi_T([m:n])) = \left[\frac{Tm}{n}:T\right] =[m:n].

For the original point, apply the forward formula to [x:T]. In f(x,T)=Ty² one may take q=y, and then

ΦT([x:T])=[(TxT,T2yT2)]±y=[(x,y)]±y. \Phi_T([x:T]) = \left[ \left( \frac{Tx}{T}, \frac{T^2y}{T^2} \right) \right]_{\pm y} =[(x,y)]_{\pm y}.

Both compositions are identities. This proves existence, injectivity, and surjectivity at once.

6. The exact degenerate boundary

The four removed points correspond exactly to the four ways in which a factor of f can vanish:

точка на ETпроективная паравырождениеO[1:0]n=0(0,0)[0:1]m=0(T,0)[1:1]m=n(−T,0)[−1:1]m=−n \begin{array}{c|c|c} \text{точка на }E_T & \text{проективная пара} & \text{вырождение}\\ \hline \mathcal O &[1:0]&n=0\\ (0,0)&[0:1]&m=0\\ (T,0)&[1:1]&m=n\\ (-T,0)&[-1:1]&m=-n \end{array}

There are no further exceptions: every rational point with y≠0 gives a nondegenerate pair, and every nondegenerate pair gives such a point.

7. The right triangle in both coordinate systems

The equality f(m,n)=Tq² immediately gives the rational right triangle

A=m2−n2q,B=2mnq,H=m2+n2q. A=\frac{m^2-n^2}{q},\qquad B=\frac{2mn}{q},\qquad H=\frac{m^2+n^2}{q}.

Indeed, A²+B²=H², and the area is

AB2=mn(m2−n2)q2=f(m,n)q2=T. \frac{AB}{2} =\frac{mn(m^2-n^2)}{q^2} =\frac{f(m,n)}{q^2} =T.

In the point coordinates (x,y), the same triangle is

A=x2−T2y,B=2Txy,H=x2+T2y. A=\frac{x^2-T^2}{y},\qquad B=\frac{2Tx}{y},\qquad H=\frac{x^2+T^2}{y}.

Substituting x=Tm/n and y=T²q/n² turns these expressions exactly into the preceding ones. Thus a parameter pair, an elliptic-curve point, and a triangle of area T are three coordinate representations of the same object.

8. The group law as an operation on pairs

To add points, one chooses one of the two signs of q and hence one sign of y. After the addition, the inverse map forgets the sign of the result and returns a new projective pair. In particular, for point doubling,

x(2P)=(x2+T2)24y2. x(2P) = \frac{(x^2+T^2)^2}{4y^2}.

For the point corresponding to [m:n], put

V=(m2+n2)2,D=4f(m,n). V=(m^2+n^2)^2,\qquad D=4f(m,n).

Substituting x=Tm/n, y=T²q/n², and f(m,n)=Tq² gives

x(2P)=T4(m2+n2)2/n44T4q2/n4=(m2+n2)24q2=T VD. \begin{aligned} x(2P) &=\frac{T^4(m^2+n^2)^2/n^4}{4T^4q^2/n^4}\\ &=\frac{(m^2+n^2)^2}{4q^2}\\ &=T\,\frac{V}{D}. \end{aligned}

By the inverse formula, the point 2P corresponds to the pair

[x(2P):T]=[TVD:T]=[V:D]. [x(2P):T] = \left[T\frac VD:T\right] =[V:D].

Thus the familiar self-recurrence of the parameters is not a separate family of solutions but ordinary point doubling on each fiber E_T. More general sums of points similarly define operations on pairs, although their explicit formulas may be considerably more complicated.

9. The full surface and the scope of the result

Allowing T to vary gives the congruent-number surface

E:y2=x3−T2x,T≠0. \mathcal E:\qquad y^2=x^3-T^2x,\qquad T\ne0.

Every nondegenerate rational pair with f(m,n)>0 has a unique positive squarefree representative T of its square class and therefore lies on exactly one such fiber. This is precisely the domain of the usual congruent-number problem.

If the oriented sign of f is retained, the same theorem works verbatim for every nonzero squarefree integer T. Here E_T=E_{−T}, because the equation depends on T², while the inverse formula [x:T] distinguishes the two square-class signs. Thus the positive version loses no curve geometry, and the signed extension covers every oriented tf value.

Completeness of the coordinate description is not a ready enumeration of points. For fixed T, determining E_T(ℚ), its rank, and its generators remains a separate arithmetic problem. The group law turns known points into new pairs through the inverse formula [x:T], but does not guarantee that the known points generate the entire group.

Thus F7+ covers every solution not by a list of formulas but by a bijection with rational points on an elliptic surface. This is the exact boundary of the claim.

Generating pairs with F7+ and F4+ →The F4+ elliptic surfaceTheory contents