Realization Calculus

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \clearpage
2 \section{Dyadic full characters and the odd critical endpoint}\label{app:dyadic_character}
3 
4 Source record: Paper\_XI\_Dyadic167. Dependency:
5 Paper\_XI\_Attachment166.
6 
7 Evidence: ordinary mathematical proof using the actual Artin--Schreier
8 matrix tower and the retained Bushnell--Kutzko character and transfer
9 theorems. No Lean verification is claimed. This note proves no dyadic
10 quotient-capacity, quadratic-phase, packet-attachment, or H5 result.
11 
12 \hypertarget{dyadic_character-exact-theorem}{%
13 \subsection{1. Exact theorem}\label{dyadic_character-exact-theorem}}
14 
15 Let F=F\_2((t)), let psi\_F be an additive character of conductor t O\_F
16 (trivial there and nontrivial on O\_F), and let
17 
18 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
19 0<h_1<...<h_d,       each h_i odd,
20 q_i=3h_i,           N=2^d.
21 \end{Verbatim}
22 
23 Use the dyadic Artin--Schreier recursion specified in Section~2. It gives
24 the actual maximal simple parameter beta\_d with its native arithmetic
25 defining sequence. There is an actual full simple character theta of its
26 maximal simple stratum such that
27 
28 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
29 theta restricted to U_F^1 is trivial.                         (1)
30 \end{Verbatim}
31 
32 The construction retains the actual canonical beta extension and can be
33 completed with any prescribed central quasicharacter omega\_F trivial on
34 U\_F\^{}1. Because \(k_F^\times=\FF_2^\times\) is trivial,
35 such an omega\_F is unramified. Thus this covers every tamely ramified
36 central quasicharacter for this particular residue field.
37 
38 The corresponding compact induction is an actual irreducible admissible
39 supercuspidal of rank N, by the retained maximal-simple-type theorem. No
40 claim is made here about its Heisenberg quotient size or about
41 realization of a supplied dyadic finite packet.
42 
43 The trace-zero argument from Block 166 does not apply verbatim: at p=2 the
44 primitive Artin--Schreier parameter has trace one. The correct invariant
45 is \textbf{integral trace}, not necessarily zero trace. Since psi\_F
46 kills t O\_F, integral correction trace is sufficient on scalar
47 principal units. The recursion below preserves the stronger exact value
48 one.
49 
50 \hypertarget{dyadic_character-actual-dyadic-recursion-and-exact-native-traces}{%
51 \subsection{2. Actual dyadic recursion and exact native
52 traces}\label{dyadic_character-actual-dyadic-recursion-and-exact-native-traces}}
53 
54 At a recursive step let L be a characteristic-two local field with
55 residue field F\_2, uniformizer t\_L, and a positive odd q. Define
56 
57 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
58 E=L(alpha),       alpha^2-alpha=t_L^(-q).
59 \end{Verbatim}
60 
61 The right side is not an Artin--Schreier coboundary: a negative
62 valuation of x\^{}2-x is even, whereas -q is odd. The degree-two
63 polynomial is therefore irreducible and separable. Its valuation
64 equation gives total ramification and v\_E(alpha)=-q. Put
65 
66 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
67 a=(q+1)/2,       z=t_L^a alpha.
68 \end{Verbatim}
69 
70 Then z is an actual uniformizer and its Eisenstein polynomial is
71 
72 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
73 f(Z)=Z^2+t_L^a Z+t_L.                                      (2)
74 \end{Verbatim}
75 
76 Its two conjugate alpha values are alpha and alpha+1, so
77 
78 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
79 Tr_(E/L)(alpha)=1.                                         (3)
80 \end{Verbatim}
81 
82 Use the coefficient basis 1,z, the trace corestriction pr, and the same
83 normalized corestriction convention as in Block 166:
84 
85 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
86 kappa=-f'(z)/z = t_L^a/z = alpha^(-1),
87 s=kappa pr.
88 \end{Verbatim}
89 
90 Minus and plus agree in characteristic two. If P0 extracts the constant
91 coefficient on every E-coordinate, then
92 
93 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
94 u_0=s(P0)=t_L/z^2 is a unit,
95 Lift_E(C)=u_0^(-1) C P0,
96 s(Lift_E(C))=C.
97 \end{Verbatim}
98 
99 The trace-dual calculation in Block 166 gives, as an identity of L-matrix
100 traces for every matrix C over E,
101 
102 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
103 tr_L Lift_E(C)
104    =Tr_(E/L)(kappa^(-1) tr_E(C))
105    =Tr_(E/L)(alpha tr_E(C)).                               (4)
106 \end{Verbatim}
107 
108 The lift preserves the complete-chain filtration valuation, by expansion
109 in 1,z. Thus it is the actual filtered right inverse used in the field
110 constructor, not an arbitrary inverse on vector spaces.
111 
112 Define B(L,t\_L;h\_1,\ldots,h\_s) recursively. At s=1 use the L-matrix
113 of multiplication by alpha with q=q\_1. At s\textgreater=2 use (2) with
114 q=q\_s, m=2\^{}(s-1), and
115 
116 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
117 C=B(E,z;h_1,...,h_(s-1)),
118 B(L,t_L;h_1,...,h_s)=alpha I_m+Lift_E(C).                    (5)
119 \end{Verbatim}
120 
121 The base trace is one by (3). If tr\_E(C)=1, then (4) gives lift trace
122 one. Since m is even, the trace of alpha I\_m as an L-matrix is m
123 Tr\_(E/L)(alpha)=0. Consequently
124 
125 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
126 tr_L B(L,t_L;h_1,...,h_s)=1                                (6)
127 \end{Verbatim}
128 
129 for every native recursion, by induction. There is no division by the
130 rank and no confusion between a native trace and its amplified trace.
131 
132 The actual field proof from Block 166 applies with this degree-two extension
133 and normalized lift. At a nontrivial step its hypotheses are
134 n=2\^{}(s-1)q\_s, R=2\^{}(s-2)q\_(s-1), s(Lift(C))=C, and
135 
136 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
137 n>2R,
138 -2R>-n-q_1.
139 \end{Verbatim}
140 
141 The first strict inequality follows from h\_s\textgreater h\_(s-1).
142 These are the conditions used in BK Remark 2.2.7. The displayed beta
143 itself generates the totally ramified degree-2\^{}s field normalizing
144 the complete chain, and its final critical exponent is -q\_1. This
145 statement does not identify the final field with its auxiliary derived
146 field.
147 
148 For the final length d write the native shorter matrices as
149 
150 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
151 b_j=B(F,t;h_(d-j+1),...,h_d),       1<=j<=d.
152 \end{Verbatim}
153 
154 They are the compatible native truncations of (5), because the outer
155 extension, uniformizer, and lift depend on the same retained largest h
156 at each stage. For \(1\le j<d\), in rank 2\^{}(j+1), put
157 
158 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
159 gamma_j=b_j tensor I_2,
160 delta_j=b_(j+1)-gamma_j.
161 \end{Verbatim}
162 
163 The defining-sequence calculation of Block 166 gives gamma\_j as a simple
164 approximation at the critical level r\_j=q\_(d-j) of b\_(j+1), and
165 delta\_j has grade -r\_j. Equation (6) gives the exact scalar identity
166 
167 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
168 tr_F(gamma_j)=0,
169 tr_F(delta_j)=1.                                          (7)
170 \end{Verbatim}
171 
172 Thus for every scalar u in U\_F\^{}1 on which its additive correction is
173 defined,
174 
175 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
176 psi_(delta_j)(u I)=psi_F((u-1)tr_F(delta_j))
177                  =psi_F(u-1)=1.                           (8)
178 \end{Verbatim}
179 
180 It is (8), not a false trace-zero assertion, that drives the full
181 character construction.
182 
183 \hypertarget{dyadic_character-the-odd-critical-level-rounding}{%
184 \subsection{3. The odd-critical-level
185 rounding}\label{dyadic_character-the-odd-critical-level-rounding}}
186 
187 The critical levels r=q\_i are odd. Write r=2s+1, and set
188 
189 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
190 m=floor(r/2)=s,        m+1=(r+1)/2.
191 \end{Verbatim}
192 
193 The relevant BK statements use the floor, with no assumption that r or
194 the residue characteristic is odd or even:
195 
196 \begin{itemize}
197 \tightlist
198 \item
199   BK 3.1.9(ii) identifies H\^{}k(beta) and H\^{}k(gamma) for
200   k\textgreater=floor(r/2)+1.
201 \item
202   BK 3.3.18 gives the correction bijection at
203   floor(r/2)\textless=m\textless=r-1.
204 \item
205   BK 3.2.5 extends the resulting character at floor(r/2) to a full
206   character, its fibers being the indicated determinant characters.
207 \end{itemize}
208 
209 At our exact endpoint m=s, the correction character is legitimate. For
210 x,y in the radical power P\^{}(s+1),
211 
212 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
213 delta*x*y belongs to P^(2(s+1)-r)=P^1.
214 \end{Verbatim}
215 
216 The level-one additive matrix character is trivial on P\^{}1, so the
217 multiplicative cross term contributes one. Therefore psi\_delta is a
218 character on U\^{}(s+1), and hence on the recursive subgroup used below.
219 This checks the borderline odd-level rounding directly. Replacing m by
220 r/2 as if it were an integer would be incorrect.
221 
222 The same check applies at the native minimal base level q, which is odd:
223 its forced additive character begins at floor(q/2)+1=(q+1)/2, and
224 2*(q+1)/2-q=1.
225 
226 \hypertarget{dyadic_character-finite-character-extension-at-maximal-rank}{%
227 \subsection{4. Finite character extension at maximal
228 rank}\label{dyadic_character-finite-character-extension-at-maximal-rank}}
229 
230 We reuse the elementary extension procedure with its compatibility
231 condition explicit. If K/F is a finite totally ramified field, and chi
232 is a smooth character of U\_K\^{}a trivial on U\_K\^{}a intersect
233 U\_F\^{}1, define a character on U\_K\^{}a U\_F\^{}1 by
234 
235 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
236 chi_0(ku)=chi(k).
237 \end{Verbatim}
238 
239 It is well defined by the intersection condition. Choose M\textgreater=a
240 with chi\textbar U\_K\^{}M=1. Then chi\_0 is a character of a subgroup
241 of the finite abelian group \(U_K^1/U_K^M\). Extend it by
242 adjoining generators and choosing the required complex roots, and pull
243 it back. The resulting chi\_K extends chi and is trivial on U\_F\^{}1.
244 
245 All freely chosen character extensions below take place at maximal
246 native rank, where the centralizer is K itself. We do not apply this
247 scalar-intersection argument directly inside a nonmaximal matrix
248 centralizer, which would require a separate determinant-image check.
249 
250 \hypertarget{dyadic_character-base-full-character-with-trace-one}{%
251 \subsection{5. Base full character with trace
252 one}\label{dyadic_character-base-full-character-with-trace-one}}
253 
254 At native rank two, let E=F{[}alpha{]}, with q=q\_d and a=floor(q/2)+1.
255 The forced character on U\_E\^{}a is
256 
257 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
258 phi(v)=psi_F(Tr_(E/F)(alpha(v-1))).
259 \end{Verbatim}
260 
261 For scalar u in U\_E\^{}a intersect U\_F\^{}1, equation (3) gives
262 
263 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
264 phi(u)=psi_F((u-1)Tr_(E/F)(alpha))=psi_F(u-1)=1.
265 \end{Verbatim}
266 
267 Section~4 constructs chi\_E on U\_E\^{}1 extending phi and trivial on
268 U\_F\^{}1. Define theta\_1 on \(H^1=U_E^1\) U\_(A\_1)\^{}a
269 by chi\_E on the first factor and psi\_alpha on the second. The
270 restrictions agree on the intersection. BK 3.2.1 gives the actual full
271 simple character, with theta\_1\textbar U\_F\^{}1=1.
272 
273 This uses the true trace value one and only the stated conductor
274 condition on psi\_F. It does not require an artificial zero-trace
275 primitive parameter in a quadratic extension.
276 
277 \hypertarget{dyadic_character-native-rank-induction-via-full-beta-determinant-transfer}{%
278 \subsection{6. Native-rank induction via full beta-determinant
279 transfer}\label{dyadic_character-native-rank-induction-via-full-beta-determinant-transfer}}
280 
281 Assume theta\_j is a full simple character for native b\_j in rank
282 2\^{}j, with theta\_j\textbar U\_F\^{}1=1. Let K\_j=F{[}b\_j{]}. Since
283 K\_j is maximal, the character chi\_j=theta\_j\textbar U\_(K\_j)\^{}1 is
284 its full beta-determinant character.
285 
286 Transfer theta\_j to the multiplicity-two realization gamma\_j=b\_j
287 tensor I\_2 in rank 2\^{}(j+1), using BK 3.6.14. Denote the full
288 transferred character by Theta\_j. BK 3.6.13(ii) preserves the entire
289 beta-determinant character on U\_(K\_j)\^{}1 at level m0=0. Hence for
290 every literal scalar u in U\_F\^{}1,
291 
292 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
293 Theta_j(u I)=chi_j(det_(K_j)(u I_2))
294             =chi_j(u^2)=theta_j(u^2)=1.                    (9)
295 \end{Verbatim}
296 
297 This statement includes nonminimal b\_j. The determinant exponent two is
298 essential and has not been suppressed.
299 
300 Now set beta=b\_(j+1), gamma=gamma\_j, delta=delta\_j, r=r\_j,
301 m=floor(r/2), and
302 
303 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
304 D=H^(m+1)(beta)=H^(m+1)(gamma).
305 \end{Verbatim}
306 
307 The equality and endpoint m were verified in Section~3. BK 3.3.18
308 constructs the actual recursive simple character
309 
310 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
311 vartheta=(Theta_j restricted to D)*psi_delta.
312 \end{Verbatim}
313 
314 For u in D intersect U\_F\^{}1, equations (8) and (9) prove
315 
316 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
317 vartheta(u)=1.                                             (10)
318 \end{Verbatim}
319 
320 The new field K=F{[}beta{]} is maximal and totally ramified in native
321 rank 2\^{}(j+1). Its chain has K-period one, so
322 
323 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
324 D intersect U_K^1=U_K^(m+1),
325 D intersect U_F^1=U_F^ceil((m+1)/2^(j+1)).
326 \end{Verbatim}
327 
328 Thus (10) is exactly the compatibility required by Section~4. Extend
329 vartheta\textbar U\_K\^{}(m+1) to chi\_K on U\_K\^{}1 trivial on
330 U\_F\^{}1. BK 3.2.5 constructs the full simple character by
331 
332 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
333 theta_(j+1)(kd)=chi_K(k)*vartheta(d),
334 k in U_K^1, d in D.
335 \end{Verbatim}
336 
337 That proposition supplies normalization and well-definedness. Its proof is an argument extending characters and imposes no exclusion of residue
338 characteristic two. The new theta\_(j+1) is trivial on U\_F\^{}1.
339 Induction proves (1) in every native rank 2\^{}d.
340 
341 More generally the character argument needs only integral base field
342 trace and integral traces for the native corrections. The dyadic tower
343 establishes their exact value one, while the odd-prime tower of Block 166
344 establishes zero.
345 
346 \hypertarget{dyadic_character-canonical-beta-extension-and-prescribed-tame-center}{%
347 \subsection{7. Canonical beta extension and prescribed tame
348 center}\label{dyadic_character-canonical-beta-extension-and-prescribed-tame-center}}
349 
350 Let K/F be the final totally ramified degree-N parameter field and let
351 eta be the Heisenberg representation for the constructed theta. The
352 compact type group is
353 
354 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
355 J=O_K^times J^1.
356 \end{Verbatim}
357 
358 Here \(\OO_K^\times=U_K^1\) because the residue field is
359 F\_2. Since U\_K\^{}1 is contained in J\^{}1, we have
360 
361 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
362 J=J^1,       J/J^1=F_2^times={1}.                           (11)
363 \end{Verbatim}
364 
365 Thus the actual canonical beta extension on J equals eta. There is no
366 residue-character choice and no change of that extension. It acts
367 trivially on the central U\_F\^{}1 by (1). Equation (11) does not
368 identify any externally prescribed dyadic quadratic phases.
369 
370 Let omega\_F be any smooth quasicharacter trivial on U\_F\^{}1. Since
371 \(\OO_F^\times=U_F^1\), it is determined by its value
372 omega\_F(t). The actual maximal type extends to K\^{}times J.
373 Concretely, (K\^{}times J)/J is infinite cyclic, generated by a
374 K-uniformizer. An intertwiner for its conjugation action on eta gives an
375 extension Lambda\_0. Denote its central character by omega\_0.
376 
377 Choose a complex scalar s with
378 
379 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
380 s^N=omega_F(t)/omega_0(t).
381 \end{Verbatim}
382 
383 Multiply Lambda\_0 by the character of K\^{}times J trivial on J and
384 taking the K-uniformizer to s. Because t has valuation N in K, this
385 changes its scalar by s\^{}N and gives the requested central
386 quasicharacter on all F\^{}times. The value at every element of J,
387 including the canonical beta extension, is unchanged. This does not
388 assume that a chosen K-uniformizer raised to N equals t.
389 
390 The retained maximal-simple-type theorem now gives the actual
391 irreducible admissible supercuspidal pi=c-Ind\_(K\^{}times
392 J)\^{}GL\_N(F) Lambda. The noncompact scalar is fixed before any H5
393 comparison. An incompatible immutably prescribed extension is not overwritten. As in Block 166, its value must be free or already compatible.
394 
395 \hypertarget{dyadic_character-what-this-closes-and-what-it-does-not}{%
396 \subsection{8. What this closes and what it does
397 not}\label{dyadic_character-what-this-closes-and-what-it-does-not}}
398 
399 This closes the full-character and actual tame-central-character
400 construction for the explicit dyadic arithmetic tower. The failure of
401 trace zero at p=2 is repaired by the correct integral trace invariant.
402 Both the odd-level floor and the multiplicity-two determinant transfer
403 are accounted for explicitly.
404 
405 This note does not compute \(J^1/H^1\) or prove that it
406 contains a particular dyadic source packet. It does not establish a
407 quadratic refinement, canonical phase matching, nontrivial stabilizer
408 action, general transport graph, source admission, or H5. In particular,
409 an actual supercuspidal host with the required central character is not
410 asserted to be a completed realization of the original dyadic packet
411 without those separate proofs.
412 
413 \hypertarget{dyadic_character-exact-source-checks-and-input-pins}{%
414 \subsection{9. Exact source checks and input
415 pins}\label{dyadic_character-exact-source-checks-and-input-pins}}
416 
417 The following final Block 166 files were read and reused without editing:
418 
419 \begin{itemize}
420 \tightlist
421 \item
422   \path{attachment166/central/TRACE_ZERO_CHARACTER_INDUCTION166.md}
423   SHA256
424   \path{99cb6e80c419e0bf7325e2b2b3fbaba1391903448a3b6a0c4cd9f3944ed406a1}.
425 \item
426   \path{attachment166/tower/TRACE_ZERO_AS_TOWER166.md} SHA256
427   \path{49a18fb733f0209b3e38118f5132839a060961edd0bdc0211e1f7b441a4e5653}.
428 \item
429   \path{attachment166/tower/RECURSIVE_TOWER166.md} SHA256
430   \path{6f39a077df18eb03f9fab2b6e426529dbe14cf4f7e1cfce95ddb0fcd187d7b33}.
431 \item
432   \path{attachment166/audit/TOWER_CENTRAL_REVIEW166.md} SHA256
433   \path{4bc71dd050ddb321a253105c8a4ee7a86185af255532d5605caa7a9de383e26b}.
434 \end{itemize}
435 
436 Primary retained source: Bushnell--Kutzko, \emph{The admissible dual of
437 GL(N) via compact open subgroups}, text SHA256
438 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8}.
439 The current dyadic check reread Definitions~3.1.7--8 and
440 Proposition 3.1.9(ii) (printed pp. 90--91), Definition~3.2.1 and
441 Proposition 3.2.5 (printed pp. 98--100), and Corollary 3.3.18
442 (printed pp. 112--113). Their level thresholds are floor(r/2)+1 or
443 floor(r/2), as used above, and their statements do not exclude p=2. The
444 transfer's preservation of the full beta-determinant character is
445 BK 3.6.13(ii), with existence Theorem~3.6.14 (printed pp. 136--137), already
446 directly checked in Block 166 and reread for this application.
447 
448 The character-extension and correction proofs use neither division by
449 two nor an odd-order averaging argument. The exact dyadic trace
450 calculation and the borderline correction-character check are given in
451 Sections~2--3. No new publication lookup, external
452 arithmetic-induction/descent constructor, or inverse LLC is used.

Back to source controls

Back to Realization Calculus