Hilbert 12: cubic divisor development

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1 \section{Canonical fifth-torsion normalization}\label{sec:theta}
2 \noindent\textit{Source anchor N: preserved mathematical development; exact source path and hash are in the claim/source inventory. Retained local equation numbers carry this prefix.}\par
3 \hypertarget{N-analytic-input-and-conventions}{%
4 \subsection{Analytic input and
5 conventions}\label{N-analytic-input-and-conventions}}
6 
7 Let \(t\) be any point of the upper half-plane and
8 \(E_t=\mathbb C/(\mathbb Z+\mathbb Zt)\). The classical product
9 convention is \cite{BCG}
10 
11 \[
12  \theta_0(z;t)=\prod_{n\ge0}(1-e^{2\pi i(z+nt)})(1-e^{2\pi i((n+1)t-z)}).
13 \]
14 
15 Retain from the preceding multiplication-norm proof:
16 
17 \[
18  \operatorname{div}(q_t)=4[0]-[P]-[-P]-[2P]-[-2P],\qquad P=1/5,
19 \]
20 
21 and, for \(n\) prime to five,
22 
23 \[
24  \prod_{\eta\in E_t[n]}q_t(z+\eta)=q_t(nz).                  \tag{N1.1}
25 \]
26 
27 The scalar in (N1.1) is exactly one. This is important: an unspecified
28 norm multiplier would not suffice below. The inherited proof derives
29 (N1.1) from the theta product by finite real root-of-unity averaging and
30 vertical residue splitting. This block also tests it at the required
31 nonzero two- and three-torsion points independently of the rational
32 formula.
33 
34 The normalization can also be checked directly here. Finite root
35 averaging gives
36 
37 \[
38  \prod_{a=0}^{n-1}\theta_0(w+a/n;t)=\theta_0(nw;nt).
39 \]
40 
41 Next, splitting the two defining products into vertical residues gives
42 
43 \[
44  \prod_{b=0}^{n-1}\theta_0(nz+bt;nt)=\theta_0(nz;t).
45 \]
46 
47 For the smoothing denominator, multiplication by five permutes the real
48 residues modulo n; the same two steps apply with period 5t. The
49 numerator becomes \(\theta_0(nz;t)^5\) and the denominator becomes
50 \(\theta_0(5nz;5t)\). This proves (N1.1) with no omitted multiplier and
51 without using the unknown scalar \(c_t\).
52 
53 Write
54 
55 \[
56  x(z)=\wp_t(z),\quad y(z)=\wp_t'(z),\quad
57  f(X)=4X^3-g_2X-g_3.
58 \]
59 
60 The classical equation is \(y(z)^2=f(x(z))\). The addition and
61 duplication laws used below are those of this convention
62 \cite{DLMFWeierstrass}. In particular the y-coordinate here is the full
63 Weierstrass derivative, not its half.
64 
65 Set
66 
67 \[
68  a=x(P),\quad b=x(2P),\quad d=a-b,\quad y_a=y(P),\quad y_b=y(2P),
69 \]
70 
71 \[
72  Q_5(X)=(X-a)(X-b).
73 \]
74 
75 All of \(d,y_a,y_b\) are nonzero: a point of exact order five is neither
76 two-torsion nor equal up to sign to its double. Comparing divisors
77 yields a nonzero scalar \(c_t\) with
78 
79 \[
80  q_t(z)=c_t/Q_5(x(z)).                                    \tag{N1.2}
81 \]
82 
83 Also \(q_t(z)\sim c_tz^4\) at zero. The problem in this development is
84 to determine this scalar exactly without a choice of cube root or eta
85 multiplier.
86 
87 \hypertarget{N-a-scalar-is-fixed-by-two-coprime-norm-exponents}{%
88 \subsection{A scalar is fixed by two coprime norm
89 exponents}\label{N-a-scalar-is-fixed-by-two-coprime-norm-exponents}}
90 
91 Suppose two nonzero elliptic functions with the same divisor differ by a
92 scalar \(r\), and both satisfy the exactly normalized degree-two and
93 degree-three multiplication laws. On the degree-four and degree-nine
94 fibers those laws imply
95 
96 \[
97  r^4=r,\qquad r^9=r.
98 \]
99 
100 Thus \(r^3=r^8=1\), and
101 
102 \[
103  r=(r^3)^3/r^8=1.                                       \tag{N2.1}
104 \]
105 
106 More generally, if \(c^3=A\) and \(c^8=B\) with \(c\ne0\), then
107 
108 \[
109  \boxed{c=A^3/B.}                                       \tag{N2.2}
110 \]
111 
112 This is a uniqueness and recovery statement within a fixed divisor
113 class. It does NOT say that arbitrary functions are determined by two
114 norms. In particular it does not contradict the nonconstant norm-one
115 factor established in the preceding note.
116 
117 The commutative-group statements (N2.1)-(N2.2), including uniqueness and
118 the fixed-norm conditions, are formalized in
119 \texttt{H12CoprimeNormalization}.
120 
121 \hypertarget{N-the-fifth-torsion-ordinate-identity}{%
122 \subsection{The fifth-torsion ordinate
123 identity}\label{N-the-fifth-torsion-ordinate-identity}}
124 
125 Since \(2P\) doubles to \(4P=-P\), let \(\ell,m\) be the full-y tangent
126 slopes at \(P,2P\). The addition law gives
127 
128 \[
129  \ell^2=4(2a+b),\quad m^2=4(a+2b),
130 \]
131 
132 \[
133  \ell d=y_a+y_b,\qquad md=y_a-y_b.                       \tag{N3.1}
134 \]
135 
136 Subtract the squares of the last two equalities:
137 
138 \[
139  4y_ay_b=(\ell^2-m^2)d^2=4d^3.
140 \]
141 
142 Therefore
143 
144 \[
145  \boxed{y_ay_b=d^3,\qquad f(a)f(b)=d^6.}                 \tag{N3.2}
146 \]
147 
148 The polynomial implication from (N3.1) to (N3.2) is formalized. The fact
149 that the actual elliptic points satisfy (N3.1) is supplied by the
150 ordinary group-law argument here, not by an undeclared formal
151 assumption.
152 
153 Define the third division polynomial in this convention:
154 
155 \[
156  \psi_3(X)=3X^4-\tfrac32g_2X^2-3g_3X-\tfrac1{16}g_2^2,
157 \]
158 
159 and the duplication numerator
160 
161 \[
162  \phi_2(X)=X^4+\tfrac12g_2X^2+2g_3X+\tfrac1{16}g_2^2.
163 \]
164 
165 The identities
166 
167 \[
168  Xf(X)-\phi_2(X)=\psi_3(X),\qquad
169  b=\phi_2(a)/f(a),\quad a=\phi_2(b)/f(b)
170 \]
171 
172 give
173 
174 \[
175  \psi_3(a)=f(a)d,\qquad \psi_3(b)=-f(b)d.
176 \]
177 
178 Consequently
179 
180 \[
181  \boxed{\psi_3(a)\psi_3(b)=-d^8.}                        \tag{N3.3}
182 \]
183 
184 The minus sign in (N3.3) is retained. It disappears only after the
185 square used below.
186 
187 \hypertarget{N-evaluate-both-scalar-powers-without-ambiguity}{%
188 \subsection{Evaluate both scalar powers without
189 ambiguity}\label{N-evaluate-both-scalar-powers-without-ambiguity}}
190 
191 Let \(z\to0\) in (N1.1). The fourth-order zero gives
192 
193 \[
194  \boxed{\prod_{\eta\in E_t[n]\setminus\{0\}}q_t(\eta)=n^4}
195  \quad(5\nmid n).                                      \tag{N4.1}
196 \]
197 
198 \hypertarget{N-degree-two}{%
199 \subsubsection{Degree two}\label{N-degree-two}}
200 
201 The three x-coordinates of nonzero two-torsion are the roots of
202 \(f(X)/4\). Hence
203 
204 \[
205  \prod_{\eta\in E_t[2]\setminus\{0\}}Q_5(x(\eta))
206  =\frac{f(a)f(b)}{16}.
207 \]
208 
209 Using (N4.1),
210 
211 \[
212  c_t^3=16\prod_{\eta\ne0}Q_5(x(\eta))=f(a)f(b)=d^6.       \tag{N4.2}
213 \]
214 
215 Degree two alone determines \(c_t/d^2\) only up to a cube root of unity.
216 
217 \hypertarget{N-degree-three}{%
218 \subsubsection{Degree three}\label{N-degree-three}}
219 
220 The eight nonzero three-torsion points have four x-coordinates, each
221 repeated twice; those coordinates are the roots of \(\psi_3(X)/3\).
222 Therefore
223 
224 \[
225  \prod_{\eta\in E_t[3]\setminus\{0\}}Q_5(x(\eta))
226  =\left(\frac{\psi_3(a)\psi_3(b)}9\right)^2.
227 \]
228 
229 The \(3^4=81\) in (N4.1) cancels the denominator:
230 
231 \[
232  c_t^8=(\psi_3(a)\psi_3(b))^2=d^{16}.                    \tag{N4.3}
233 \]
234 
235 Combining the coprime exponents in (N4.2)-(N4.3),
236 
237 \[
238  \boxed{c_t=(c_t^3)^3/c_t^8=d^{18}/d^{16}=d^2.}           \tag{N4.4}
239 \]
240 
241 This closes the scalar-normalization subproblem completely.
242 
243 \hypertarget{N-canonical-theta-normal-form}{%
244 \subsection{Canonical theta normal
245 form}\label{N-canonical-theta-normal-form}}
246 
247 Equations (N1.2) and (N4.4) prove
248 
249 \[
250  \boxed{
251  q_t(z)=\frac{(a-b)^2}{(x(z)-a)(x(z)-b)}
252        =\frac{\operatorname{disc}(Q_5)}{Q_5(x(z))}.
253  }                                                       \tag{N5.1}
254 \]
255 
256 Writing \(S_5=a+b\) and \(P_5=ab\), the symmetric form is
257 
258 \[
259  q_t(z)=\frac{S_5^2-4P_5}{x(z)^2-S_5x(z)+P_5}.            \tag{N5.2}
260 \]
261 
262 Thus the formula depends on the unordered fifth-torsion kernel, not a
263 choice between its two x-coordinates. Under a change of generator
264 \(P\mapsto2P\), \(a,b\) swap and the formula is unchanged.
265 
266 Define a dimensionless elliptic coordinate
267 
268 \[
269  \mathsf X_t(z)=\frac{x(z)-a}{a-b}.
270 \]
271 
272 Then
273 
274 \[
275  \boxed{q_t(z)=\frac1{\mathsf X_t(z)(\mathsf X_t(z)+1)}.}   \tag{N5.3}
276 \]
277 
278 Swapping \(a,b\) changes \(\mathsf X\) to \(-1-\mathsf X\), leaving
279 (N5.3) fixed. Scaling all Weierstrass x-coordinates by the same nonzero
280 factor also leaves the formula fixed. These field identities are
281 formalized in \texttt{H12CanonicalTheta}.
282 
283 The identities are interpreted meromorphically at their zeros and poles.
284 Numerical pointwise evaluations and group-valued Lean applications use
285 nonzero sampled values.
286 
287 \hypertarget{N-an-explicit-order-five-plane-curve-normalization}{%
288 \subsection{An explicit order-five plane-curve
289 normalization}\label{N-an-explicit-order-five-plane-curve-normalization}}
290 
291 The same calculation places the entire elliptic curve in an order-five
292 normal form without adjoining square roots. Put
293 
294 \[
295  u=-y_b/y_a,\qquad \lambda=y_a/d,
296 \]
297 
298 and for an arbitrary point \((x,y)\), define
299 
300 \[
301  X=\frac{x-a}{\lambda^2},\qquad
302  Y=\frac{y-y_a-(1-u)\lambda(x-a)}{2\lambda^3}.
303 \]
304 
305 Then
306 
307 \[
308  \boxed{Y^2+(1-u)XY-uY=X^3-uX^2,}                        \tag{N6.1}
309 \]
310 
311 with \(P=(0,0)\), \(2P=(u,u^2)\), and
312 
313 \[
314  \boxed{q_t(z)=\frac{u^2}{X(z)(X(z)-u)}.}                  \tag{N6.2}
315 \]
316 
317 Here is a derivation rather than an assumption of that model. From
318 (N3.2),
319 
320 \[
321  d=-u\lambda^2,\quad y_a=-u\lambda^3,\quad y_b=u^2\lambda^3.
322 \]
323 
324 The first tangent slope is \((1-u)\lambda\). Equation (N3.1) therefore
325 gives
326 
327 \[
328  a=\lambda^2(1-6u+u^2)/12.
329 \]
330 
331 Writing \(b_2=1-6u+u^2\), \(b_4=u^2-u\), the derivative and curve
332 equation imply
333 
334 \[
335  g_2=\lambda^4(b_2^2/12-2b_4),\quad
336  g_3=\lambda^6(-b_2^3/216+b_2b_4/6-u^2).
337 \]
338 
339 Substitution proves (N6.1). This coefficient calculation is
340 independently checked symbolically over \(\mathbb Q(u)\) and numerically
341 at the specified analytic lattices. The full geometric
342 change-of-variables theorem is not Lean-formalized in the corresponding
343 source checkpoint.
344 
345 The parameter \(u\) remains analytically defined. Calling the curve a
346 rational normal form does not assert that \(u\) is an algebraic number
347 at the cubic period.
348 
349 \hypertarget{N-the-actual-cubic-inter-chart-covariance}{%
350 \subsection{The actual cubic inter-chart
351 covariance}\label{N-the-actual-cubic-inter-chart-covariance}}
352 
353 Return to
354 
355 \[
356  \beta^3=7,\quad\epsilon=2-\beta,\quad\rho=\epsilon^{-1},
357  \quad t=-(\beta^2+2\beta)/15,\quad s=-(\beta+2)/15.
358 \]
359 
360 The retained modular relation gives \(\rho\Lambda_s=\Lambda_t\). Since
361 
362 \[
363  \rho=4-15t,\quad\rho/5\equiv-1/5\pmod{\Lambda_t},
364 \]
365 
366 scaling of the Weierstrass function gives
367 
368 \[
369  a_s=\rho^2a_t,\quad b_s=\rho^2b_t,\quad c_s=\rho^4c_t,
370 \]
371 
372 and hence
373 
374 \[
375  \mathsf X_t(\rho z)=\mathsf X_s(z),\qquad
376  \boxed{q_t(\rho z)=q_s(z).}                              \tag{N7.1}
377 \]
378 
379 Thus the phase-free theta covariance has a proof through the canonical
380 fifth-torsion normal form as well as its earlier eta-multiplier proof.
381 This statement is about two homothetic elliptic lattices, not an
382 endomorphism of the single lattice \(\Lambda_t\).
383 
384 \hypertarget{N-recovery-of-every-signed-unit-carry}{%
385 \subsection{Recovery of every signed unit
386 carry}\label{N-recovery-of-every-signed-unit-carry}}
387 
388 Use the predecessor's conventions
389 
390 \[
391  w_v=\tfrac13+(b\tau+cs+a)/2,\quad v=(b,c,a)\in\{0,1\}^3,
392  \quad T(b,c,a)=(a,b,c),\quad \tau=-t.
393 \]
394 
395 Its proved unit carry is
396 
397 \[
398  F(\rho w_v)=F(w_{Tv})B_v,\qquad
399  B_v=\prod_{j=0}^{A_v-1}q_t(w_{T^2v}+js),
400  \quad A_v=5+4b-2c+7a.
401 \]
402 
403 By (N5.3) this now has the exact expression
404 
405 \[
406  \boxed{
407  B_v=\prod_{j=0}^{A_v-1}
408  \frac1{\mathsf X_t(w_{T^2v}+js)
409              (\mathsf X_t(w_{T^2v}+js)+1)}.
410  }                                                       \tag{N8.1}
411 \]
412 
413 This includes the full normalization of every factor. It does not
414 discard the scalar.
415 
416 The predecessor's signed gains were, in bit-index order,
417 
418 \[
419  (76,-28,8,0,-16,0,0,0).
420 \]
421 
422 When written with the unnormalized denominators \(Q_5\), the
423 corresponding known powers of \(d=a-b\) are
424 
425 \[
426  (152,-56,16,0,-32,0,0,0).
427 \]
428 
429 In particular, the aggregate of the three first-order carries retains
430 the factor
431 
432 \[
433  c_t^{-36}=d^{-72}.                                      \tag{N8.2}
434 \]
435 
436 It would be false to set this factor to one. Formula (N8.1) incorporates
437 it automatically.
438 
439 The same substitution works for every oriented three-step return product
440 \(J_{n,v}\) from the predecessor. Consequently ALL seven nontrivial
441 signed-channel limits have finite rational elliptic expressions without
442 a separately supplied eta constant. The first-order limits still have
443 only their guaranteed first-order local cancellation; this normalization
444 change does not upgrade their convergence rate.
445 
446 These facts use the inherited analytic return theorem. Its limit remains
447 the actual signed gamma value; it has not been independently evaluated
448 in the ray field.

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