Integral tfmn Forms: Containers, Kummer Coordinates, and the Group Law

1. From the function to the form

For rational m,n put

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

Definition

The ordered tuple of the four linear factors

\Phi(m,n)=(n;\,m-n,m,m+n)

will be called the tfmn form of the pair (m,n), and each entry will be called a container. The semicolon separates the first container from the three consecutive containers; algebraically all four are on equal footing.

\prod\Phi(m,n)=n(m-n)m(m+n)=f(m,n).

Simultaneously scaling the pair multiplies every container by one factor and their product by a fourth power:

\Phi(\lambda m,\lambda n)=\lambda\Phi(m,n), \qquad f(\lambda m,\lambda n)=\lambda^4f(m,n).

Hence the square class of f depends only on the projective pair [m:n]. On the chart n≠0 put r=m/n. Dividing the form by its first container gives the rational normalization

\Phi_{\mathrm{rat}}(r)=(1;\,r-1,r,r+1).

In the integral setting one chooses a primitive representative with gcd(m,n)=1. In the positive chamber one takes m>n>0; primitive progressions of squares additionally use the usual opposite-parity condition on m and n. These conventions make the notation canonical in the chosen chamber, but they are not part of the rational definition of the form.

2. A harmonic rank-two frame

Let V=ℚ² and v=(m,n). The containers are the values of four fixed linear functionals:

\ell_0(v)=n,\qquad \ell_-(v)=m-n,\qquad \ell_c(v)=m,\qquad \ell_+(v)=m+n.

This is not a basis of V*: there are four functionals in dimension two. The correct linear object is a redundant rank-two frame. Any two distinct functionals form a basis, while all four satisfy the relations

\ell_c=\ell_-+\ell_0,\qquad \ell_+=\ell_c+\ell_0,\qquad \ell_-+\ell_+=2\ell_c.

Harmonicity

The directions [0:1], [1:−1], [1:0], [1:1] in the projective line of V* have cross-ratio −1 in the ordering (ℓ−,ℓ+;ℓc,ℓ0). Thus the form is obtained by evaluating a fixed harmonic quadruple of functionals on one vector v.

The form genuinely resembles a coordinate system, but it is not a vector basis: its four coordinates are redundant even before square classes are taken.

3. The elliptic point and the square constituent

Let T≠0 be squarefree and suppose

f(m,n)=Tq^2.

Then the form corresponds to a point on the curve

E_T:\quad Y^2=X^3-T^2X=X(X-T)(X+T)

through the formulas

[m:n]\longmapsto P=\left(\frac{Tm}{n},\,\pm\frac{T^2q}{n^2}\right), \qquad (X,\pm Y)\longmapsto[X:T].

X^3-T^2X =\frac{T^3}{n^4}f(m,n) =\frac{T^4q^2}{n^4} =Y^2.

In normalized coordinates

r=\frac XT=\frac mn, \qquad w=\frac{Y}{T^2}=\pm\frac q{n^2}, \qquad T w^2=r(r-1)(r+1).

Why q should not be discarded

The integer q depends on scale: under (m,n)↦(λm,λn), it becomes ±λ²q. But w=q/n² is scale invariant and is the vertical coordinate of the point. It distinguishes P from −P and enters the exact addition formulas.

A rational form without the sign of q determines the pair ±P and naturally lives on the quotient E_T/{±1}. A form together with w lifts to the elliptic curve itself.

4. Square classes and the Kummer shadow

Let K be the group of nonzero rational square classes, written additively as a vector space over 𝔽₂:

K=\mathbb Q^\times/(\mathbb Q^\times)^2, \qquad \overline u+\overline v=\overline{uv}.

For a point P=(X,Y) away from rational 2-torsion, the full 2-descent map is

\kappa(P)= (\overline{X-T},\overline X,\overline{X+T})\in K^3.

The product of the three components is the class of Y² and is therefore trivial. For e∈{−T,0,T}, the functions X−e have divisors 2(e,0)−2(𝒪), so κ is the standard Kummer map and factors through E_T(ℚ)/2E_T(ℚ).

The square-class signature of the normalized form is

\sigma(P)= (\overline{r-1},\overline r,\overline{r+1}).

Because X=Tr, all three coordinates differ from the Kummer coordinates by the same class T. If

\tau=(\overline T,\overline T,\overline T),

then

\sigma(P)=\tau+\kappa(P).

The affine law

\sigma(P+Q)=\sigma(P)+\sigma(Q)+\tau.

After translating the origin by σ̂(P)=σ(P)+τ=κ(P), one obtains an ordinary homomorphism. Thus the square-class form is not a linear vector with a natural zero, but a point of an affine space over the image of 2-descent.

For every nondegenerate multiple [k]P the signature depends only on the parity of k:

\sigma([k]P)= \begin{cases} \sigma(P),&k\ \text{odd},\\ \tau,&k\ \text{even}. \end{cases}

5. The exact square correction under addition

The Kummer law records only square classes. The exact form also recovers the square factors. Let Pᵢ=(rᵢ,wᵢ) lie on Tw²=r(r²−1), assume r₁≠r₂, and let P₃=P₁+P₂. Put

\mu=\frac{w_2-w_1}{r_2-r_1}, \qquad r_3=T\mu^2-r_1-r_2, \qquad w_3=\mu(r_1-r_3)-w_1.

For each root a∈{−1,0,1}, define the value of the secant above that root:

h_a= \frac{w_1(r_2-a)-w_2(r_1-a)}{r_2-r_1}.

Containerwise identity

(r_1-a)(r_2-a)(r_3-a)=T h_a^2, \qquad a\in\{-1,0,1\}.

Derivation

Let L(r) be the line through P₁ and P₂. Its third common x-parameter with the cubic is r₃: reflecting the third intersection across the horizontal axis gives P₁+P₂. Comparing leading coefficients gives

T L(r)^2-r(r^2-1) =-(r-r_1)(r-r_2)(r-r_3).

At r=a the cubic term vanishes, and L(a)=hₐ. Substitution immediately proves the identity. Passing to square classes removes the factors hₐ² and leaves the affine law of the previous section.

This reveals the origin of the square constituent of the result: it is assembled from the values of the secant above the three rational 2-torsion points. The signature sees only their classes; the exact form sees the rational squares themselves.

6. Doubling and tripling

Doubling

For the point represented by [m:n], put

M_2=(m^2+n^2)^2, \qquad N_2=4f(m,n).

Then [M₂:N₂] represents [2]P, and the three consecutive containers factor as exact squares:

\begin{aligned} M_2-N_2&=(m^2-2mn-n^2)^2,\\ M_2&=(m^2+n^2)^2,\\ M_2+N_2&=(m^2+2mn-n^2)^2. \end{aligned}

Consequently

\Phi(M_2,N_2)= \bigl(4f(m,n);\,D_-^2,D_0^2,D_+^2\bigr),

D_-=m^2-2mn-n^2, \quad D_0=m^2+n^2, \quad D_+=m^2+2mn-n^2.

If f(m,n)=Tq², then the square constituent of this homogeneous representative is, up to sign,

q_{2P}^{\mathrm{hom}}=2qD_-D_0D_+.

Tripling

Define four quartics

\begin{aligned} A&=m^4+6m^2n^2-3n^4,\\ B&=3m^4-6m^2n^2-n^4,\\ C_-&=m^4-4m^3n-6m^2n^2-4mn^3+n^4,\\ C_+&=m^4+4m^3n-6m^2n^2+4mn^3+n^4. \end{aligned}

The point [3]P corresponds to the pair [mA²:nB²], and the entire form factors containerwise:

\Phi(mA^2,nB^2)= \bigl(nB^2;\,(m-n)C_-^2,\,mA^2,\,(m+n)C_+^2\bigr).

Tripling preserves the square class of each source container separately, and the square constituent is

q_{3P}^{\mathrm{hom}}=qABC_-C_+.

Primitive reduction

The formulas refer to the displayed homogeneous representatives. If both output parameters have a common factor d and the pair is reduced, then f is divided by d⁴ and the exact rational square constituent by d². Thus the sequence of primitive q-values contains additional gcd arithmetic and need not literally be a standard elliptic divisibility sequence.

7. What is already contained in 2-descent and the Monsky matrix

The triple of normalized square classes is not a new replacement for 2-descent. It is the same Kummer data carried by X−T, X, and X+T, written in the coordinate r=X/T and translated by the common class T. For congruent-number curves, local conditions on these classes have long been converted into linear algebra over 𝔽₂.

The Monsky matrix encodes local solubility of 2-Selmer candidates at the primes dividing 2T. Its kernel describes locally admissible square-class shadows, not necessarily rational points: the gap between the Selmer group and the image of E_T(ℚ)/2E_T(ℚ) is measured by the 2-torsion of the Tate–Shafarevich group. Odd and even T use different block formulas for the matrix.

Level	What is retained	Best use
Integral form	Four representatives, linear relations, parity, divisibility, and exact q.	Explicit formulas, primitive reduction, and the connection with progressions of squares.
Kummer signature	Three square classes and the point class modulo 2E_T(ℚ).	The group law, parity of multiples, and full 2-descent.
Monsky matrix	Local restrictions on two independent square-class parameters.	Computation of the pure 2-Selmer group and an upper bound for the rank.

The contribution of the language of forms is not a reinvention of Kummer classes, but their unnormalized four-container lift together with exact square-correction formulas. For the present problem this lift is the more useful primary object; 2-descent and the Monsky matrix remain its natural linear X-ray.

Literature context

D. R. Heath-Brown, with an appendix by P. Monsky, The Size of Selmer Groups for the Congruent Number Problem, II — the matrix description of 2-Selmer groups for congruent-number curves.
Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, and Nicholas F. Rogers, Ranks of Quadratic Twists of Elliptic Curves — the split form uv(u−v)(u+v), primitive conventions, and practical use of Monsky matrices.
Tom Fisher, Edward F. Schaefer, and Michael Stoll, Higher Descents on an Elliptic Curve with a Rational 2-Torsion Point — higher descent layers beyond square-class data.
Katherine E. Stange, Elliptic Nets and Elliptic Curves — the general setting of division polynomials and exact recurrence factors.

8. The T=210 example and exact scope

Two primitive pairs give the same value of f without an additional square factor:

\Phi(5,2)=(2;3,5,7), \qquad \Phi(6,1)=(1;5,6,7), \qquad f=210.

Choosing the positive signs q=1 gives the points

P=(525,11025), \qquad Q=(1260,44100) \quad\text{on}\quad E_{210}:Y^2=X^3-210^2X.

Addition and reversal of the second point give two different outputs:

\begin{aligned} P+Q&=(240,1800) &&\longleftrightarrow [8:7], &f(8,7)&=210\cdot2^2,\\ P-Q&=(3840,237600) &&\longleftrightarrow [128:7], &f(128,7)&=210\cdot264^2. \end{aligned}

Their integral forms are different:

\Phi(8,7)=(7;1,8,15), \qquad \Phi(128,7)=(7;121,128,135).

Yet taking containerwise square classes gives in both cases

(\overline{7};\overline{1},\overline{2},\overline{15}), \qquad \sigma=(\overline{7},\overline{14},\overline{105}).

The Kummer shadow is the same, while the exact square constituents 2 and 264 distinguish the outputs. The two-valued behavior is unavoidable: the projective pair [m:n] remembers the x-coordinate but forgets whether the original point was P or −P.

Scope of the statements

All square-class formulas are stated on the nondegenerate locus r(r−1)(r+1)≠0.
The identity and the points (0,0), (±T,0) require the special conventions of 2-descent and have status not_assessed here.
The addition formulas with hₐ refer to a secant with r₁≠r₂; the tangent case is handled separately by the exact doubling formula.
Equality of signatures does not mean equality of forms or points; it means equality of the corresponding classes modulo 2E_T(ℚ).

In summary, the integral tfmn form is the primary arithmetic object, its normalized signature is an affine Kummer shadow, and the square constituent is the vertical coordinate that restores the exact data of the group law.

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