Paper XI: Packets, Arithmetic Realization, and Conditional Comparison

Download original file

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

Read the typeset paper

Read this chapter with formatted equations

1 \clearpage
2 \section{Actual scalar inertia phases and finite-cover descent}\label{app:odd_scalar}
3 
4 Evidence: ordinary proof using rank-one reciprocity, the given central
5 character, and elementary finite group arguments. The result constructs
6 actual central matrix operators through a finite cover and proves their
7 descent. It does not construct a new arithmetic candidate or alter a
8 canonical beta extension.
9 
10 \hypertarget{odd_scalar-exact-statement}{%
11 \subsection{1. Exact statement}\label{odd_scalar-exact-statement}}
12 
13 Let F be a nonarchimedean local field with residue field k of
14 cardinality q and residue characteristic p.~Put Q=q-1 and n=p\^{}r,
15 allowing r=0. Let rho:W\_F -\textgreater{} GL\_n(C) be a smooth Weil
16 representation with finite inertia image. Use the fixed reciprocity
17 convention identifying its determinant with a character omega:F×
18 -\textgreater{} C×.
19 
20 Let C be a finite group of order prime to p, and suppose its prescribed
21 operators on the \textbf{full source carrier} are
22 
23 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
24 Omega_src(c)=chi(c)I_n,
25 \end{Verbatim}
26 
27 where chi is a character and each Omega\_src(c) belongs to the actual
28 matrix image rho(I\_F). A finite cyclic scalar tame image is the
29 intended example. This hypothesis refers to actual source operators, not
30 freely chosen phases on an unrelated multiplicity-space factor.
31 
32 Let Pi be any already constructed irreducible smooth GL\_n(F)
33 representation with central character omega. Suppose it has its supplied
34 actual inducing representation Lambda on N, with F× contained in N, and
35 an attached finite sector E=eLambda realizing the prescribed source
36 Heisenberg packet. Assume the prescribed C action on the source scalar
37 extension is trivial, and that the honest scalar phase on the packet is
38 the same chi above.
39 
40 \textbf{Theorem.} There is a finite group C\_tilde surjecting onto C and
41 a genuine homomorphism
42 
43 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
44 g:C_tilde -> F× I_n contained in N
45 \end{Verbatim}
46 
47 whose action on Lambda and on Pi descends to the prescribed honest C
48 action chi(c)I. Every finite sector is stable under this action. Any
49 center-preserving Heisenberg attachment and its intertwiner are
50 C-equivariant. The retained sector-induction recovery maps recover the
51 same Lambda and Pi equivariantly. All actual inducing data, including
52 the fixed canonical beta extension, remain unchanged.
53 
54 The arithmetic action is supplied by actual central matrices, with an
55 explicit finite-cover descent. The theorem does not assert that C itself
56 embeds as a matrix subgroup. Such an embedding can fail even for a
57 genuine scalar tame source, as Section4 shows.
58 
59 \hypertarget{odd_scalar-construction-and-proof}{%
60 \subsection{2. Construction and proof}\label{odd_scalar-construction-and-proof}}
61 
62 Let D=k×, and let Teich:D -\textgreater{} O\_F× be the canonical
63 multiplicative lift. For p=2 this is still the prime-to-p root-of-unity
64 subgroup. When q=2, D is trivial and the argument below forces chi
65 trivial as well.
66 
67 Local reciprocity identifies the determinant image of inertia with
68 
69 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
70 det rho(I_F)=omega(O_F×).
71 \end{Verbatim}
72 
73 The unit group decomposes as
74 
75 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
76 O_F× = Teich(D) times U_F¹.
77 \end{Verbatim}
78 
79 The smooth character image omega(U\_F¹) is a finite p-group, while
80 omega(Teich(D)) has order prime to p.~Therefore the prime-to-p elements
81 of omega(O\_F×) are exactly omega(Teich(D)).
82 
83 For every c in C, the actual-source hypothesis implies
84 
85 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
86 chi(c)^n=det(Omega_src(c)) in det rho(I_F).
87 \end{Verbatim}
88 
89 This value has order prime to p, and hence lies in omega(Teich(D)).
90 Since n is a p-power and chi(C) has order prime to p, the map z
91 -\textgreater{} z\^{}n permutes chi(C). Consequently
92 
93 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
94 chi(C) is contained in omega(Teich(D)).                 (1)
95 \end{Verbatim}
96 
97 In particular every source phase has order dividing Q. This conclusion
98 does not require selecting a splitting of the tame quotient or choosing
99 Frobenius representatives.
100 
101 Choose an integer k\_0 with n k\_0 congruent to1 modulo Q. When Q=1 take
102 k\_0=1. Define
103 
104 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
105 delta(u)=omega(Teich(u))^(k_0),       u in D.
106 \end{Verbatim}
107 
108 Because exponentiation by k\_0 is an automorphism on every group of
109 roots of unity of order dividing Q, delta(D)=omega(Teich(D)). Define the
110 finite fiber product
111 
112 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
113 C_tilde={(c,u) in C times D : chi(c)=delta(u)}.          (2)
114 \end{Verbatim}
115 
116 It is a subgroup, and projection pr:C\_tilde -\textgreater{} C is
117 surjective by (1). Define the actual central matrix homomorphism
118 
119 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
120 g(c,u)=Teich(u^(k_0)) I_n.                             (3)
121 \end{Verbatim}
122 
123 Its action on Lambda and Pi is the central-character scalar
124 
125 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
126 Lambda(g(c,u))=omega(Teich(u^(k_0)))I
127               =delta(u)I=chi(c)I.                     (4)
128 \end{Verbatim}
129 
130 If (1,u) belongs to ker pr, (4) is the identity. Hence the represented
131 action factors uniquely through C. It is exactly the prescribed source
132 action. Although selecting a particular matrix representative for c may
133 require a choice of u, the represented operator is independent of that
134 choice. Changing k\_0 by a multiple of Q leaves (3) unchanged.
135 
136 All matrices (3) are central. Their conjugation on actual groups,
137 characters, and scalar pushouts is therefore the identity, and their
138 Lambda operators are scalar. Thus they preserve every e and every sector
139 E. The source extension action is also the identity by hypothesis, so
140 any attachment of those scalar extensions is equivariant. Both honest
141 carrier actions are chi(c)I, so every Heisenberg intertwiner intertwines
142 C without a new phase choice.
143 
144 For the recovery step, put P=\{a in N:Lambda(a)eLambda(a)\^{}(-1)=e\}.
145 Then F× is contained in P. The retained maps
146 
147 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
148 Phi:Ind_P^N E -> Lambda,
149 j:Lambda -> Ind_P^N E
150 \end{Verbatim}
151 
152 are N-equivariant. They therefore commute with all actual matrices (3),
153 and ker pr also acts trivially on Ind\_P\^{}N E: its matrices are
154 central elements of P acting trivially on E, hence trivially on every
155 induced translate. Thus the equality descends through pr to C. The
156 recovering idempotent, and then compact induction to GL\_n(F), preserve
157 this equivariance. They recover the same fixed Lambda and Pi. QED.
158 
159 \hypertarget{odd_scalar-what-this-closes-and-what-it-does-not}{%
160 \subsection{3. What this closes and what it does
161 not}\label{odd_scalar-what-this-closes-and-what-it-does-not}}
162 
163 For a source packet whose marked tensor carrier is the entire rank-n
164 source, this proves the stabilizer step whenever its genuine tame
165 operators are scalar and an actual host with the correct central
166 character and full-character attachment is already present. There is no
167 remaining scalar-phase equation to solve in that situation: the
168 determinant and central character supply it.
169 
170 If only the source symplectic C action is known to be trivial, the
171 preceding \path{STABILIZER_CONTRACT166.md} proves triviality of its
172 full scalar-extension action for prime-to-p C. Schur then gives a scalar
173 packet character. To apply the present theorem one must still identify
174 that character with the action of actual inertia on the full source.
175 That identification holds when the supplied honest packet carrier is the
176 whole source carrier. It is not automatic for a proper
177 multiplicity-space factor, whose omitted factor can contribute to the
178 determinant.
179 
180 For several factors the total packet character is the product of the
181 supplied factor characters. The theorem matches that product on the
182 whole attached packet. It does not represent independent factor actions
183 by distinct central matrices unless those additional marked actions are
184 separately specified and verified.
185 
186 The result uses the fixed full inducing representation Lambda. If
187 Lambda\textbar J=kappa\_can tensor sigma, then kappa\_can and sigma are
188 both unchanged. Equation(4) concerns Lambda, and does not assert that
189 kappa\_can alone has the same scalar character. A contract prescribing
190 an action specifically on the canonical beta extension requires that
191 distinction to remain explicit.
192 
193 No nontrivial source symplectic action is realized here. No new base
194 candidate, new H5 comparison, dyadic quadratic refinement, or general
195 rank theorem is claimed. Existing quadratic refinements are preserved
196 because the actual central conjugation action is the identity.
197 
198 \hypertarget{odd_scalar-why-a-direct-map-c-into-residue-units-is-too-strong}{%
199 \subsection{4. Why a direct map C into residue units is too
200 strong}\label{odd_scalar-why-a-direct-map-c-into-residue-units-is-too-strong}}
201 
202 Take q=5, n=5\^{}r, and let omega on k×=F5× be the quadratic character.
203 Let C=C2 and let chi be its nontrivial character. Such scalar tame
204 behavior is obtained by twisting a tame-trivial rank-5\^{}r source by
205 the quadratic tame character, whenever such a source is supplied; the
206 retained odd-prime two-factor family supplies rank25 examples.
207 
208 A group homomorphism a:C2 -\textgreater{} F5× must send its generator
209 to1 or -1. Both have quadratic character1. Consequently
210 
211 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
212 omega(a(c)) = chi(c)^n
213 \end{Verbatim}
214 
215 is impossible at the nontrivial c, since n is odd. The proposed direct
216 section is not forced by genuine scalar tame source data. Reciprocity
217 supplies a quotient of residue units; that quotient need not split.
218 
219 The construction above still works. Here k\_0=1 modulo4, C\_tilde is the
220 cyclic group of order4, and its central matrices act by the quadratic
221 character. The kernel of C\_tilde -\textgreater{} C2 acts trivially on
222 the fixed candidate, so the actual operator action descends correctly.
223 This example explains why finite-cover descent is necessary rather than
224 a cosmetic reformulation.
225 
226 \hypertarget{odd_scalar-dependencies-and-review-status}{%
227 \subsection{Dependencies and review
228 status}\label{odd_scalar-dependencies-and-review-status}}
229 
230 External input: rank-one local class field theory, with the same fixed
231 normalization used for omega=det rho; the existing actual candidate's
232 central character identity; the retained sector recovery theorem. The
233 finite-cover construction and its descent are proved above and use no
234 local Langlands candidate selection. No computation is needed. This
235 version has undergone author self-review; a parent audit is requested.
236 No Lean run is claimed.

Back to source controls

Back to Paper XI: Packets, Arithmetic Realization, and Conditional Comparison