Hilbert 12: cubic divisor development

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1 \section{Arithmetic conductors and the exact modulus comparison}\label{sec:stark}
2 \noindent\textit{Source anchor A: preserved mathematical development; exact source path and hash are in the claim/source inventory. Retained local equation numbers carry this prefix.}\par
3 \hypertarget{A-fixed-conventions-and-inherited-arithmetic}{%
4 \subsection{Fixed conventions and inherited
5 arithmetic}\label{A-fixed-conventions-and-inherited-arithmetic}}
6 
7 Set
8 
9 \[
10 \omega=e^{2\pi i/3},\quad \beta=7^{1/3}\omega^2,\quad
11 J=2+3\omega,\quad \bar J=-1-3\omega,\quad J\bar J=7.
12 \]
13 
14 Write
15 
16 \[
17 F_0=\mathbb Q(\omega),\qquad K=\mathbb Q(\beta),\qquad
18 \epsilon=2-\beta,\qquad \mathcal R=-\log(2-7^{1/3})>0.
19 \]
20 
21 Here \(\mathcal R\) is the regulator of \(K\), and
22 
23 \[
24 \log|\epsilon|=\mathcal R/2. \tag{A1.1}
25 \]
26 
27 Indeed, \(N_{K/\mathbb Q}(\epsilon)=1\) and its real image is
28 \(2-7^{1/3}\). The class number is three and \(\epsilon\) is a
29 fundamental unit. These arithmetic facts are recorded in BCG, section
30 5.2; the earlier unramified-cubic note also proves the class-number and
31 ray-field identification used here.
32 
33 Take
34 
35 \[
36 \mathfrak f=(3,\beta-1),\qquad
37 E=K(\omega,\theta),\qquad \theta=2\cos(6\pi/7).
38 \]
39 
40 The inherited arithmetic proof identifies \(E\) as the narrow ray class
41 field of modulus \(\mathfrak f\) and the real place. It has degree six
42 over \(K\) and degree eighteen over \(\mathbb Q\).
43 
44 Let
45 
46 \[
47 X=\frac1\beta\sum_{a=1}^{6}\chi_7(a)e^{6\pi ia/7},
48 \]
49 
50 where the character values for \(a=1,\ldots,6\) are
51 \(1,\omega,\omega^2,\omega^2,\omega,1\). Then
52 
53 \[
54 X^3=J,\qquad E=\mathbb Q(\beta,X,\omega). \tag{A1.2}
55 \]
56 
57 This fixes a specific cube-root branch. The preceding CM note proves,
58 with this branch,
59 
60 \[
61 X=\frac{\Omega^2}{\wp_{\Lambda_0}(1/J)},\qquad
62 \Lambda_0=\mathbb Z+\mathbb Z\omega,\qquad
63 \Omega=e^{\pi i/6}\frac{\Gamma_{\rm Euler}(1/3)^3}{2\pi}. \tag{A1.3}
64 \]
65 
66 That proof uses actual seven-torsion and a lattice estimate for the
67 branch, not numerical recognition.
68 
69 The arithmetic Frobenius \(\sigma\) at the norm-two prime acts by
70 
71 \[
72 \sigma(\beta)=\beta,\quad \sigma(\omega)=\omega^2,\quad
73 \sigma(X)=\omega\beta/X. \tag{A1.4}
74 \]
75 
76 The norm-five prime has the same Artin class. In particular, \(\sigma\)
77 generates \(\operatorname{Gal}(E/K)\).
78 
79 \textbf{Actual complex conjugation is a different automorphism.} Denote
80 it by \(\kappa\). In the chosen embedding,
81 
82 \[
83 \kappa(\beta)=\omega^2\beta,\quad \kappa(\omega)=\omega^2,
84 \quad \kappa(X)=\omega\beta/X. \tag{A1.5}
85 \]
86 
87 For the last formula, the conjugate Gauss resolvent is \(7/G\), and
88 \(\bar\beta=\omega^2\beta\). Formula (A1.5) follows directly. It must
89 not be replaced by the involution \(\sigma^3\), which fixes \(K\).
90 
91 \hypertarget{A-the-major-change-of-route-artin-induction}{%
92 \subsection{The major change of route: Artin
93 induction}\label{A-the-major-change-of-route-artin-induction}}
94 
95 The splitting field \(F_0(\beta)/\mathbb Q\) has group \(S_3\), while
96 \(\mathbb Q(\theta)/\mathbb Q\) is cyclic cubic. Their intersection is
97 Galois over \(\mathbb Q\); it cannot have degree three because \(S_3\)
98 has no cyclic cubic quotient. Hence
99 
100 \[
101 \operatorname{Gal}(E/\mathbb Q)=S_3\times C_3. \tag{A2.1}
102 \]
103 
104 Use generators
105 
106 \[
107 r:\beta\mapsto\omega\beta,\qquad
108 c:\omega\mapsto\omega^2,\qquad
109 s_0:\theta\mapsto\theta^2-2,
110 \]
111 
112 where each generator fixes the other specified primitive generators as
113 appropriate. Thus \(r,c\) generate the \(S_3\) factor, \(c\) fixes
114 \(\beta,\theta\), and \(s_0\) generates the second factor. We have
115 
116 \[
117 \operatorname{Gal}(E/K)=\langle c,s_0\rangle,
118 \qquad \sigma=cs_0,
119 \qquad \operatorname{Gal}(E/F_0)=\langle r,s_0\rangle.
120 \]
121 
122 Let \(\varrho(r)=\omega,\varrho(s_0)=1\), and let
123 \(\eta(s_0)=\omega,\eta(r)=1\). The three odd ray characters of \(K\)
124 are
125 
126 \[
127 \chi_j(c)=-1,\qquad \chi_j(s_0)=\omega^j,\qquad j=0,1,2.
128 \]
129 
130 The elementary character identity
131 
132 \[
133 \operatorname{Ind}_{C_2}^{S_3}(\operatorname{sgn})
134 =\operatorname{sgn}\oplus\operatorname{Std},
135 \qquad
136 \operatorname{Std}=\operatorname{Ind}_{C_3}^{S_3}(\varrho)
137 \]
138 
139 and induction invariance of Artin L-functions give
140 
141 \[
142 \boxed{L_K(s,\chi_j)
143 =L_{\mathbb Q}(s,\chi_{-3}\chi_7^j)\,
144 L_{F_0}(s,\varrho\eta^j).} \tag{A2.2}
145 \]
146 
147 For \(j=0\), use the primitive Dirichlet character of conductor three,
148 not its imprimitive extension to modulus twenty-one. All three odd ray
149 characters are ramified at \(\mathfrak f\), so their Artin L-functions
150 agree with the partial-zeta convention omitting that prime. Induction
151 includes the local factors at ramified primes through the
152 inertia-invariant spaces. No unproved Artin holomorphy statement is
153 used.
154 
155 The even characters make no contribution to the derivative at zero. The
156 nontrivial even characters have at least two archimedean zeros. The
157 trivial character has the zero of \(\zeta_K\), together with the extra
158 zero of \(1-3^{-s}\). This also agrees with BCG, section 3.1.
159 
160 Exact finite Dirichlet sums give
161 
162 \[
163 D_0=L(0,\chi_{-3})=1/3,\qquad
164 D_1=L(0,\chi_{-3}\chi_7)=-2\omega,\qquad
165 D_2=-2\omega^2. \tag{A2.3}
166 \]
167 
168 For example, the latter two use
169 \(-21^{-1}\sum_{a=1}^{21}a\chi_{-3}(a)\chi_7(a)^j\).
170 
171 For the untwisted cubic factor,
172 
173 \[
174 L_{F_0}(s,\varrho)=\zeta_K(s)/\zeta_{\mathbb Q}(s).
175 \]
176 
177 The analytic class number formula, \(h_K=3\), and
178 \(\zeta_{\mathbb Q}(0)=-1/2\) give
179 
180 \[
181 L_{F_0}'(0,\varrho)=3\mathcal R. \tag{A2.4}
182 \]
183 
184 Thus the cubic regulator contributes a known normalization; it is not a
185 free scalar left over from a cocycle.
186 
187 \hypertarget{A-a-six-class-imaginary-quadratic-calculation}{%
188 \subsection{A six-class imaginary-quadratic
189 calculation}\label{A-a-six-class-imaginary-quadratic-calculation}}
190 
191 Put \(\lambda=\varrho\eta\). Since both \(r\) and \(s_0\) send \(X\) to
192 \(\omega^2X\), \(\lambda\) is the inverse Kummer character for
193 \(F_0(X)/F_0\). Its conductor is
194 
195 \[
196 \mathfrak m=3J\mathscr O_{F_0}. \tag{A3.1}
197 \]
198 
199 Here is an explicit verification. The Eisenstein integers are principal,
200 and the six units map bijectively to \((\mathscr O_{F_0}/3)^\times\).
201 Every ideal coprime to three therefore has a unique generator
202 \(\alpha\equiv1\pmod3\). Since \(J\equiv-1\pmod3\), cubic reciprocity
203 gives
204 
205 \[
206 \lambda((\alpha))=(J/(\alpha))_3^2=(\alpha/J)_3^2. \tag{A3.2}
207 \]
208 
209 This proves that the conductor divides \(3J\). It must contain \(J\),
210 because the valuation of the Kummer radicand there is one. If its
211 exponent at the prime above three were at most one, its conductor would
212 divide \((1-\omega)J\). The ray class group of that modulus has order
213 two, so it has no nontrivial cubic character. Hence the conductor is
214 exactly (A3.1).
215 
216 The ray group modulo \(3J\) has order
217 
218 \[
219 \frac{63(1-1/3)(1-1/7)}6=6.
220 \]
221 
222 Its six classes have unique primary rational representatives
223 
224 \[
225 1,4,10,13,16,19.
226 \]
227 
228 Reduction modulo \(J\) sends \(\omega\) to \(4\in\mathbb F_7\). Formula
229 (A3.2) groups the representatives as follows:
230 
231 \begin{longtable}[]{@{}lll@{}}
232 \toprule\noalign{}
233 Index \(m\) & Representatives & \(\lambda\)-value \\
234 \midrule\noalign{}
235 \endhead
236 \bottomrule\noalign{}
237 \endlastfoot
238 0 & 1, 13 & 1 \\
239 1 & 4, 10 & \(\omega\) \\
240 2 & 16, 19 & \(\omega^2\) \\
241 \end{longtable}
242 
243 Let \(Z_m(s)\) be the sum of the two ray partial zeta functions in row
244 \(m\), and write \(\ell_m=Z_m'(0)\). Then
245 
246 \[
247 L_{F_0}'(0,\lambda)=\ell_0+\omega\ell_1+\omega^2\ell_2,
248 \qquad
249 L_{F_0}'(0,\lambda^2)=\ell_0+\omega^2\ell_1+\omega\ell_2. \tag{A3.3}
250 \]
251 
252 The sum of all six ray partial zeta functions is
253 
254 \[
255 \zeta_{F_0}(s)(1-3^{-s})(1-7^{-s}),
256 \]
257 
258 whose derivative at zero vanishes. Therefore
259 
260 \[
261 \ell_0+\ell_1+\ell_2=0. \tag{A3.4}
262 \]
263 
264 Conjugation of the base field by \(c\) sends \(\varrho\eta^2\) to
265 \(\varrho^{-1}\eta^2=\lambda^2\), so
266 
267 \[
268 L_{F_0}(s,\varrho\eta^2)=L_{F_0}(s,\lambda^2).
269 \]
270 
271 Combining (A2.2)--(A3.4) and performing inverse Fourier transformation
272 on \(C_2\times C_3\) yields
273 
274 \[
275 \boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0)
276 =(-1)^e\left(\frac{\mathcal R}{6}-\ell_{k-1}\right).} \tag{A3.5}
277 \]
278 
279 All subscripts on \(\ell\) are taken modulo three. To check the
280 constants directly, the two nontrivial Fourier contributions are
281 
282 \[
283 -\frac13\left(\omega^{1-k}L_{F_0}'(0,\lambda)
284 +\omega^{2-2k}L_{F_0}'(0,\lambda^2)\right).
285 \]
286 
287 In (A3.3), the coefficient of \(\ell_{k-1}\) in the parentheses is two
288 and the other two coefficients are minus one. Equation (A3.4) turns the
289 parentheses into \(3\ell_{k-1}\).
290 
291 \textbf{Euler-factor warning.} Replacing \(3J\) by \(21=3J\bar J\) would
292 remove the Euler factor at the other prime over seven. Its
293 \(\lambda\)-value is one, so this inserts \(1-7^{-s}\) and destroys the
294 desired first derivative. That imprimitive derivative is zero. The
295 checker verifies this as a separate normalization control, not as the
296 desired L-value.
297 
298 \hypertarget{A-kronecker-normalization-fixed-explicitly}{%
299 \subsection{Kronecker normalization fixed
300 explicitly}\label{A-kronecker-normalization-fixed-explicitly}}
301 
302 For \(z=a\omega+b\notin\Lambda_0\), define the positive quantity
303 
304 \[
305 H(z)=e^{-\pi\operatorname{Im}(\omega)B_2(a)}
306 |1-e^{2\pi iz}|
307 \prod_{n\ge1}|(1-e^{2\pi i(n\omega+z)})(1-e^{2\pi i(n\omega-z)})|,
308 \tag{A4.1}
309 \]
310 
311 where initially \(0\le a<1\) and \(B_2(a)=a^2-a+1/6\). This is the
312 absolute value of the usual Siegel function. Its value is independent of
313 the representative of \(z\) modulo \(\Lambda_0\).
314 
315 For the shifted Epstein series
316 
317 \[
318 \mathcal E_z(s)=\sum_{m,n\in\mathbb Z}|m+n\omega+z|^{-2s},
319 \]
320 
321 continued from \(\Re s>1\), the precise normalization is
322 
323 \[
324 \mathcal E_z(0)=0,\qquad \mathcal E_z'(0)=-2\log H(z). \tag{A4.2}
325 \]
326 
327 A normalization derivation is given in the normalization subsection below (source A8). In particular, no
328 factor of two is inferred from an unspecified Mellin convention.
329 
330 For the ray class represented by \(\alpha\), all its integral ideals are
331 generated exactly once by the elements of \(\alpha+3J\mathscr O_{F_0}\).
332 The congruence condition eliminates all nontrivial units. Therefore its
333 partial zeta function is
334 
335 \[
336 63^{-s}\mathcal E_{\alpha/(3J)}(s).
337 \]
338 
339 There is no scale contribution to its derivative, because
340 \(\mathcal E_z(0)=0\). Set
341 
342 \[
343 V_0=H(1/(3J))H(13/(3J)),
344 \] \[
345 V_1=H(4/(3J))H(10/(3J)),\qquad
346 V_2=H(16/(3J))H(19/(3J)).
347 \]
348 
349 Then
350 
351 \[
352 \ell_m=-2\log V_m,
353 \qquad V_0V_1V_2=1. \tag{A4.3}
354 \]
355 
356 \hypertarget{A-the-cm-products-collapse-to-three-explicit-units}{%
357 \subsection{The CM products collapse to three explicit
358 units}\label{A-the-cm-products-collapse-to-three-explicit-units}}
359 
360 Let \(u=1/J\) and \(T=\bar J/3\equiv2/3\pmod{\Lambda_0}\). The three
361 pairs of points above are, respectively,
362 
363 \[
364 \{T\pm2u\},\quad\{T\pm u\},\quad\{T\pm3u\}. \tag{A5.1}
365 \]
366 
367 Indeed, \(T=7u/3\), so their numerators over \(3J\) are \(1,13\),
368 \(4,10\), and \(-2,16\). The last numerator \(-2\) may be replaced by
369 \(19\) since \(21/(3J)=\bar J\in\Lambda_0\).
370 
371 On the normalized curve \(E_0:y^2+y=x^3\), set
372 
373 \[
374 x(z)=\wp_{\Lambda_0}(z)/\Omega^2.
375 \]
376 
377 The previous seven-torsion proof gives
378 
379 \[
380 x(u)=X^{-1},\quad x(2u)=\omega^2X^{-1},\quad x(3u)=\omega X^{-1}. \tag{A5.2}
381 \]
382 
383 Also,
384 
385 \[
386 x(T)=-\omega. \tag{A5.3}
387 \]
388 
389 For completeness, \(T\) is nonzero three-torsion and its normalized
390 x-coordinate is a root of \(3x(x^3+1)\). The lattice is invariant under
391 ordinary conjugation, so \(\wp(1/3)\) and its derivative are real. Since
392 \(g_3(\Lambda_0)>0\), the real differential equation
393 \(\wp'^2=4\wp^3-g_3\) implies \(\wp(1/3)>0\). The argument of
394 \(\Omega^2\) is \(\pi/3\), selecting \(-\omega\) among the three roots
395 of \(x^3=-1\).
396 
397 The classical sigma addition identity is
398 
399 \[
400 \frac{\sigma_\Lambda(T+z)\sigma_\Lambda(T-z)}{
401 \sigma_\Lambda(T)^2\sigma_\Lambda(z)^2}
402 =\wp_\Lambda(z)-\wp_\Lambda(T). \tag{A5.4}
403 \]
404 
405 The theta expression for (A4.1), or the sigma--theta relation, turns
406 (A5.4) into the same identity for absolute Green functions up to a
407 constant independent of \(z\). The quadratic exponential corrections
408 cancel by the parallelogram identity. Furthermore,
409 \(H(\omega z)=H(z)=H(-z)\), either from (A4.2) and the isometries of
410 \(\Lambda_0\), or from the theta transformations.
411 
412 Since \(2u\equiv\omega^2u\) and \(3u\equiv-\omega u\), all three factors
413 \(H(ku)\) are equal. Thus for one positive constant \(C\), the same for
414 all three rows,
415 
416 \[
417 V_0=C|\omega^2/X+\omega|,\quad
418 V_1=C|1/X+\omega|,\quad
419 V_2=C|\omega/X+\omega|. \tag{A5.5}
420 \]
421 
422 Multiplication and (A4.3) give \(C^3|X^{-3}+1|=1\). Cubing each equation
423 in (A5.5) now removes the unknown constant exactly:
424 
425 \[
426 \boxed{V_m^3=|a_m|,} \tag{A5.6}
427 \]
428 
429 where the three algebraic values simplify to
430 
431 \[
432 \begin{aligned}
433 a_0&=\frac{(\omega^2+\omega X)^3}{1+J}
434 =(1+\omega)X^2-X+1,\\
435 a_1&=\frac{(1+\omega X)^3}{1+J}
436 =-X^2+(1+\omega)X+1,\\
437 a_2&=\frac{(\omega+\omega X)^3}{1+J}
438 =-\omega X^2-\omega X+1.
439 \end{aligned} \tag{A5.7}
440 \]
441 
442 Exact reduction gives
443 
444 \[
445 a_0+a_1+a_2=3,\qquad
446 \sum_{i<j}a_i a_j=6\omega,\qquad a_0a_1a_2=1.
447 \]
448 
449 In particular, the \(a_m\) are algebraic units, satisfying
450 
451 \[
452 a_m^3-3a_m^2+6\omega a_m-1=0. \tag{A5.8}
453 \]
454 
455 Equations (A4.3) and (A5.6) prove
456 
457 \[
458 \boxed{\ell_m=-\frac23\log|a_m|.} \tag{A5.9}
459 \]
460 
461 This is the evaluated comparison. The unknown logarithmic constants have
462 been replaced by explicit algebraic units without using any
463 elliptic-gamma algebraicity assumption.
464 
465 Substitution into (A3.5) yields the explicit partial-zeta theorem
466 
467 \[
468 \boxed{\zeta_{\mathfrak f}'(c^e s_0^k,0)
469 =(-1)^e\left(\frac{\mathcal R}{6}+\frac23\log|a_{k-1}|\right).} \tag{A5.10}
470 \]
471 
472 \hypertarget{A-identifying-the-previously-constructed-w}{%
473 \subsection{Identifying the previously constructed
474 W}\label{A-identifying-the-previously-constructed-w}}
475 
476 Recall the integral cubic coordinate
477 
478 \[
479 \nu=(X+\beta/X-\beta^2)/3,\quad v=-\beta^2\nu-2\nu^2,
480 \qquad v^3-\beta v^2+1=0.
481 \]
482 
483 The earlier unit is
484 
485 \[
486 W=(A_W+\omega B_W)/3,
487 \]
488 
489 \[
490 \begin{aligned}
491 A_W&=(21\beta-42)v^2+(-15\beta^2+30\beta)v-\beta^2-4\beta+11,\\
492 B_W&=3\beta^2v^2+(18\beta-42)v-11\beta^2+13\beta+7.
493 \end{aligned} \tag{A6.1}
494 \]
495 
496 Let \(N_\kappa(z)=z\kappa(z)=|z|^2\) in the fixed embedding. The six
497 exact identities needed for the comparison are
498 
499 \[
500 \boxed{N_\kappa(\sigma^hW)
501 =N_\kappa(\epsilon a_{h+1}^2)^{(-1)^h},\quad 0\le h<6.} \tag{A6.2}
502 \]
503 
504 These are identities in the explicitly presented degree-eighteen field,
505 not fits to the logarithms. They can be checked by substituting (A1.4),
506 (A1.5), (A5.7), and (A6.1), then reducing using
507 
508 \[
509 \beta^3=7,\quad X^3=2+3\omega,\quad \omega^2+\omega+1=0.
510 \]
511 
512 A basis for the rational calculation is \(\beta^iX^j\omega^k\),
513 \(0\le i,j<3\), \(0\le k<2\). For odd \(h\), multiply the two norms
514 instead of dividing: the residual is their product minus one. Both
515 supplied implementations obtain the exact zero polynomial in all six
516 cases.
517 
518 Taking real logarithms of positive norms gives
519 
520 \[
521 \log|\sigma^h W|=(-1)^h\left(\frac{\mathcal R}{2}+2\log|a_{h+1}|\right)
522 =-3\zeta_{\mathfrak f}'(\sigma^{h+5},0). \tag{A6.3}
523 \]
524 
525 Consequently \(W_{\rm St}=\sigma(W)\) satisfies
526 
527 \[
528 \zeta_{\mathfrak f}'(\sigma^h,0)
529 =-\frac13\log|\sigma^hW_{\rm St}|
530 =-\frac16\log|\sigma^hW_{\rm St}|^2. \tag{A6.4}
531 \]
532 
533 The one-step shift is essential. The unshifted \(W\) belongs to the
534 original raw gamma value's ray-class indexing, not the identity class.
535 
536 \hypertarget{A-completing-the-stark-conditions-for-this-extension}{%
537 \subsubsection{Completing the Stark conditions for this
538 extension}\label{A-completing-the-stark-conditions-for-this-extension}}
539 
540 The earlier note proves that \(W\) is an algebraic unit,
541 \(\mu(E)=\mu_6\), all its embeddings above the real place of \(K\) have
542 modulus one, and adjoining a sixth root produces an abelian extension of
543 \(K\). These statements are preserved by \(\sigma\). Together with
544 (A6.4), this verifies the three conditions stated by BCG in section 3.3
545 for this specific field and modulus.
546 
547 The archived source checks rechecked the following inherited algebraic certificates. In
548 particular,
549 
550 \[
551 W^5=\sigma(W)U_{\rm alg}^6,
552 \]
553 
554 and, for \(c_6=-\omega^2\),
555 
556 \[
557 W^{21}=c_6U_{\rm alg}^{25}\sigma(U_{\rm alg})^5\sigma^2(U_{\rm alg}).
558 \]
559 
560 For \(t^6=W\), the lift \(\varrho(t)=t^5/U_{\rm alg}\) commutes with
561 \(t\mapsto c_6t\), and satisfies \(\varrho^6(t)=c_6^4t\). Irreducibility
562 is checked at the residue assignment \((\beta,v,\omega)=(32,99,48)\)
563 over \(\mathbb F_{181}\), where \(W=23\) and \(T^6-23\) is irreducible.
564 The earlier local-integrality argument validates this reduction. The
565 resulting abelian group is \(C_2\times C_{18}\).
566 
567 These arithmetic statements concern \(W\), not the still unproved
568 algebraicity of the gamma value.
569 
570 \hypertarget{A-closing-the-actual-gamma-modulus-gap}{%
571 \subsection{Closing the actual gamma modulus
572 gap}\label{A-closing-the-actual-gamma-modulus-gap}}
573 
574 The original analytic value is
575 
576 \[
577 U_\Gamma=\frac{\Gamma_{\rm ell}(1/3;\tau,s_{\rm per})^5}
578 {\Gamma_{\rm ell}(5/3;5\tau,5s_{\rm per})},
579 \quad
580 \tau=(\beta^2+2\beta)/15,\quad s_{\rm per}=-(\beta+2)/15.
581 \]
582 
583 BCG section 5.2 identifies this raw expression with the geometric value
584 at ray-class index five. The norm-five smoothing ideal has Artin class
585 \(\sigma\). Their Theorem 3.2 therefore gives, unconditionally,
586 
587 \[
588 \log|U_\Gamma|^2
589 =\zeta_{\mathfrak f}'(1,0)-5\zeta_{\mathfrak f}'(\sigma^5,0). \tag{A7.1}
590 \]
591 
592 Using (A6.3),
593 
594 \[
595 \zeta_{\mathfrak f}'(1,0)-5\zeta_{\mathfrak f}'(\sigma^5,0)
596 =\frac13\left(5\log|W|-\log|\sigma(W)|\right).
597 \]
598 
599 The exact algebraic smoothing identity \(W^5/\sigma(W)=U_{\rm alg}^6\)
600 turns this into \(2\log|U_{\rm alg}|\). Thus
601 
602 \[
603 \boxed{|U_\Gamma|=|U_{\rm alg}|.} \tag{A7.2}
604 \]
605 
606 There is also a smaller explicit algebraic expression:
607 
608 \[
609 \log|U_\Gamma|^2
610 =\mathcal R-\ell_2-5\ell_1
611 =\mathcal R+\frac23\log|a_2a_1^5|,
612 \]
613 
614 or
615 
616 \[
617 \boxed{|U_\Gamma|^3=|\epsilon^3a_2a_1^5|.} \tag{A7.3}
618 \]
619 
620 In particular the positive real number \(|U_\Gamma|\) is algebraic. This
621 conclusion does not extend automatically to the complex number
622 \(U_\Gamma\).
623 
624 \hypertarget{A-the-whole-logarithmic-ray-packet}{%
625 \subsubsection{The whole logarithmic ray
626 packet}\label{A-the-whole-logarithmic-ray-packet}}
627 
628 The same argument evaluates the moduli of the admissible geometric gamma
629 values for every ray class and every smoothing ideal covered by BCG's
630 theorem:
631 
632 \[
633 |\mathbf\Gamma_{\mathfrak a,h}(\mathfrak f\mathfrak b^{-1})|^6
634 =\left|(\sigma_{\mathfrak b}W_{\rm St})^{N(\mathfrak a)-\sigma_{\mathfrak a}}\right|.
635 \tag{A7.4}
636 \]
637 
638 This is an equality of absolute values. It is not an Artin action on an
639 already algebraic gamma value and does not specify the phase of a sixth
640 root.
641 
642 \hypertarget{A-derivation-of-the-shifted-epstein-normalization}{%
643 \subsection{Derivation of the shifted Epstein
644 normalization}\label{A-derivation-of-the-shifted-epstein-normalization}}
645 
646 Let \(\tau=x+iy\) with \(y>0\), and \(z=a\tau+b\), initially \(0<a<1\).
647 Apply Poisson summation in the integer coordinate \(m\) of
648 
649 \[
650 \mathcal E_z(s)=\sum_{n,m\in\mathbb Z}
651 \left((m+b+(n+a)x)^2+(n+a)^2y^2\right)^{-s}.
652 \]
653 
654 The zero Fourier mode is
655 
656 \[
657 \sqrt\pi\,\frac{\Gamma(s-1/2)}{\Gamma(s)}\,
658  y^{1-2s}\bigl[\zeta(2s-1,a)+\zeta(2s-1,1-a)\bigr].
659 \]
660 
661 At zero its value vanishes. Its derivative equals \(2\pi y B_2(a)\),
662 using \(\Gamma(-1/2)=-2\sqrt\pi\), \(1/\Gamma(s)=s+O(s^2)\), and
663 \(\zeta(-1,a)=-B_2(a)/2\).
664 
665 The nonzero Fourier modes, obtained from the usual Gaussian integral,
666 are exponentially convergent near \(s=0\) for \(0<a<1\). Their
667 derivative sums to
668 
669 \[
670 -2\log|1-e^{2\pi iz}|
671 -2\sum_{n\ge1}\log|(1-e^{2\pi i(n\tau+z)})(1-e^{2\pi i(n\tau-z)})|.
672 \]
673 
674 Combining the two terms gives (A4.2). Extension to \(a=0\), away from
675 lattice points, follows by continuation in \(z\). Equivalently,
676 
677 \[
678 H_\tau(z)=e^{-\pi(\Im z)^2/y}
679 \left|\frac{\theta_1(\pi z;e^{\pi i\tau})}{\eta_{\rm Ded}(\tau)}\right|.
680 \]
681 
682 This representation and the sigma--theta relation justify the absolute
683 CM-product addition step above (source A5), including the cancellation of quadratic
684 exponential corrections.

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