Realization Calculus

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1 \clearpage
2 \section{The trace-zero Artin--Schreier tower}\label{app:odd_tower}
3 
4 
5 \hypertarget{odd_tower-exact-statement}{%
6 \subsection{Exact statement}\label{odd_tower-exact-statement}}
7 
8 Let p be odd, F=F\_p((t)), and
9 0\textless h\_1\textless\ldots\textless h\_d with p not dividing any
10 h\_i. Put q\_i=(p+1)h\_i. The constructor below gives an actual matrix
11 B\_d over F of size p\^{}d with exactly the field, simplicity,
12 approximation-level, odd-prime quotient, and conductor conclusions of
13 \path{RECURSIVE_TOWER166.md}. In addition, its matrix trace is exactly zero.
14 Every native shorter constructor and every amplified arithmetic
15 approximation also has matrix trace zero. Therefore every difference
16 between consecutive arithmetic approximations has matrix trace zero.
17 
18 The trace statement is an identity over the current coefficient field,
19 not merely a congruence, a field trace inferred from characteristic
20 zero, or a division by the matrix size. All coefficient fields have
21 characteristic p, so division by p\^{}j would be invalid.
22 
23 \hypertarget{odd_tower-an-explicit-separable-minimal-extension-with-a-traceless-parameter}{%
24 \subsection{1. An explicit separable minimal extension with a traceless
25 parameter}\label{odd_tower-an-explicit-separable-minimal-extension-with-a-traceless-parameter}}
26 
27 At an arbitrary recursive step let L have residue field F\_p and
28 uniformizer t\_L. Given q\textgreater0 prime to p, set
29 
30 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
31 E=L(alpha),  alpha^p-alpha=t_L^(-q).
32 \end{Verbatim}
33 
34 The right side is not an Artin--Schreier coboundary in L: the negative
35 valuation of x\^{}p-x, if x has negative valuation, is divisible by p.
36 Thus the polynomial is irreducible of degree p.~The extension is
37 separable, and the valuation equation forces ramification index p and
38 
39 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
40 v_E(alpha)=-q.
41 \end{Verbatim}
42 
43 It is therefore totally ramified and alpha is minimal over L. For odd p
44 its field trace is
45 
46 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
47 Tr_(E/L)(alpha)=sum_(c in F_p)(alpha+c)=0.
48 \end{Verbatim}
49 
50 Choose the unique b in \{1,\ldots,p-1\} with qb congruent to -1 modulo
51 p, put a=(qb+1)/p, and define
52 
53 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
54 z=t_L^a alpha^b.
55 \end{Verbatim}
56 
57 Then v\_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic
58 minimal polynomial \(f(X)=X^p+a_{p-1}X^{p-1}+\cdots+a_0\)
59 is Eisenstein. It is an explicit polynomial: compute the product over
60 the p conjugates
61 
62 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
63 f(X)=product_(c in F_p) (X-t_L^a(alpha+c)^b)
64 \end{Verbatim}
65 
66 and reduce coefficients using \(\alpha^p-\alpha=t_L^{-q}\).
67 The invariant coefficients belong to L. This is a finite algebraic
68 instruction, with no choice of an unknown extension or matrix.
69 
70 \hypertarget{odd_tower-normalized-corestriction-and-an-explicit-filtered-right-inverse}{%
71 \subsection{2. Normalized corestriction and an explicit filtered right
72 inverse}\label{odd_tower-normalized-corestriction-and-an-explicit-filtered-right-inverse}}
73 
74 Use the basis 1,z,\ldots,z\^{}(p-1) of E over L. If T\_ij is its
75 coefficient matrix unit and pr is the trace corestriction, then
76 
77 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
78 pr(T_ij)=z^i z_j^*,
79 z_j^*=sum_(k=j+1)^p a_k z^(k-j-1) / f'(z),  a_p=1.
80 \end{Verbatim}
81 
82 Set
83 
84 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
85 kappa=-f'(z)/z^(p-1),  s=kappa pr.
86 \end{Verbatim}
87 
88 The displayed dual-basis formula gives
89 
90 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
91 s(T_ij)=-z^(i-j) + terms of strictly larger E-valuation.
92 \end{Verbatim}
93 
94 Indeed v\_E(a\_k)\textgreater=p for 1\textless=k\textless p, so the
95 contribution indexed by a\_k has valuation at least k greater than the
96 first contribution. Applying s entrywise on E\^{}m gives a B-bimodule
97 corestriction from End\_L(E\^{}m) to B=End\_E(E\^{}m). With the complete
98 interleaved period-pm L-chain and period-m E-chain, it preserves grades
99 and satisfies \(s(P^r)=Q^r\).
100 
101 Let P0 extract the constant coefficient on every E-coordinate. The
102 constant dual coefficient satisfies
103 
104 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
105 z_0^*=-a_0/(z f'(z)),
106 u_0=s(P0)=kappa z_0^*=a_0/z^p in O_E^times.
107 \end{Verbatim}
108 
109 For any C in M\_m(E), define the actual L-linear matrix
110 
111 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
112 Lift_E(C)=u_0^(-1) C P0.
113 \end{Verbatim}
114 
115 Then
116 
117 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
118 s(Lift_E(C))=C.
119 \end{Verbatim}
120 
121 This lift preserves the exact filtration valuation. To see this, first
122 multiply C by the unit u\_0\^{}(-1), which does not change its valuation
123 in the E-chain. Expand each entry in the basis 1,z,\ldots,z\^{}(p-1).
124 The only nonzero columns of Lift are its constant-coefficient columns.
125 The L-grade of a coefficient c\_i z\^{}i E\_uv is
126 
127 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
128 pm*v_L(c_i)+mi+u-v,
129 \end{Verbatim}
130 
131 equal to its E-grade, and distinct i have valuations distinct modulo p.
132 Thus the minimum cannot cancel. This also proves the stated surjectivity
133 \(s(P^r)=Q^r\) without a compactness assumption or an assumption that a lift exists.
134 
135 \hypertarget{odd_tower-exact-trace-identity}{%
136 \subsection{3. Exact trace identity}\label{odd_tower-exact-trace-identity}}
137 
138 For y in E, extraction of its constant coefficient is
139 
140 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
141 coeff_0(y)=Tr_(E/L)(z_0^* y).
142 \end{Verbatim}
143 
144 The ordinary L-matrix trace of the rank-one operator yP0 is coeff\_0(y).
145 Consequently, for every C in M\_m(E),
146 
147 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
148 tr_L(Lift_E(C))
149    =Tr_(E/L)(z_0^* u_0^(-1) tr_E(C))
150    =Tr_(E/L)(kappa^(-1) tr_E(C)).             (T)
151 \end{Verbatim}
152 
153 Here tr\_E(C) is the ordinary m-by-m matrix trace over E. In particular,
154 if tr\_E(C)=0, then the lift has ordinary L-matrix trace zero exactly.
155 
156 \hypertarget{odd_tower-the-recursive-actual-matrix}{%
157 \subsection{4. The recursive actual
158 matrix}\label{odd_tower-the-recursive-actual-matrix}}
159 
160 Define B(L,t\_L;h\_1,\ldots,h\_d) by induction. For d=1 use the
161 multiplication matrix of the alpha from Section~1 with q=q\_1. For
162 d\textgreater=2 use Section~1 with q=q\_d, put m=p\^{}(d-1), recursively
163 construct
164 
165 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
166 C=B(E,z;h_1,...,h_(d-1)),
167 \end{Verbatim}
168 
169 and set
170 
171 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
172 B(L,t_L;h_1,...,h_d)=alpha I_m+Lift_E(C).       (ASR)
173 \end{Verbatim}
174 
175 Everything in (ASR) is explicit in finite field operations and the
176 displayed degree-p polynomial extensions. It differs from the old
177 conjugacy-preserving two-summand construction. Preserving old lower
178 summand conjugacy classes is not claimed.
179 
180 Inductively tr\_E(C)=0. Equation (T) and the Artin--Schreier trace give
181 
182 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
183 tr_L(B)=m Tr_(E/L)(alpha)+tr_L(Lift_E(C))=0.
184 \end{Verbatim}
185 
186 For the induction base the trace is Tr\_(E/L)(alpha)=0. Thus this
187 argument proves tracelessness in the current base field for every native
188 rank, not merely tracelessness after an amplification whose size is
189 divisible by p.
190 
191 \hypertarget{odd_tower-actual-fields-and-profiles}{%
192 \subsection{5. Actual fields and
193 profiles}\label{odd_tower-actual-fields-and-profiles}}
194 
195 Set gamma=alpha I\_m and b=Lift\_E(C). The minimal outer stratum has
196 
197 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
198 n=p^(d-1)q_d,  k_0(gamma)=-n.
199 \end{Verbatim}
200 
201 The inner actual maximal field supplied by induction has
202 
203 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
204 R=p^(d-2)q_(d-1),  v(C)=-R,  k_0(C)=-q_1.
205 \end{Verbatim}
206 
207 We have s(b)=C, v(b)=-R, and n\textgreater pR\textgreater2R because p is
208 odd and the h\_i increase. Hence
209 
210 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
211 -2R > -n-q_1 = k_0(gamma)+k_0(C).
212 \end{Verbatim}
213 
214 BK Remark 2.2.7 applies to this pure derived maximal field. It gives the
215 field generated by the actual displayed gamma+b, with degree and
216 ramification p\^{}d, residue degree one, normalization of the complete
217 chain, and k\_0=-q\_1. Its valuation remains -n.~Thus {[}A,n,0,B{]} is
218 simple.
219 
220 This repeats the precise argument of \path{RECURSIVE_TOWER166.md} with a new
221 outer minimal extension and normalized right inverse. No step of that
222 argument required the old polynomial e\^{}p+t e+t.
223 
224 \hypertarget{odd_tower-native-approximations-amplification-and-traceless-corrections}{%
225 \subsection{6. Native approximations, amplification, and traceless
226 corrections}\label{odd_tower-native-approximations-amplification-and-traceless-corrections}}
227 
228 For the full rank p\^{}d matrix put, for 1\textless=j\textless=d,
229 
230 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
231 B^[j]=B(F,t;h_(d-j+1),...,h_d) tensor I_(p^(d-j)).
232 \end{Verbatim}
233 
234 These are literal truncations of (ASR). At the outer level the same
235 largest q\_d selects the same Artin--Schreier extension and uniformizer
236 in all truncations. At the next level the same q\_(d-1) does the same,
237 and so on. Thus no hidden change of bases or incompatible parameter
238 fields enters the truncations. Lift\_E commutes with amplification by an
239 identity because P0 acts separately on each coordinate.
240 
241 Every native shorter B has trace zero by Section~4. Its amplification
242 therefore has trace zero. In particular
243 
244 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
245 tr_F(B^[j])=0,
246 tr_F(B^[j+1]-B^[j])=0.                        (Z)
247 \end{Verbatim}
248 
249 The same is true at native rank p\^{}(j+1): the correction between
250 B(F,t;h\_(d-j),\ldots,h\_d) and the identity-p amplification of
251 B(F,t;h\_(d-j+1),\ldots,h\_d) has trace zero. This native assertion is
252 the one relevant when constructing characters before each transfer.
253 
254 The field degrees and critical levels of \(B^{[j]}\) are exactly
255 
256 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
257 degree p^j,  n_j=p^(d-j)q_(d-j+1).
258 \end{Verbatim}
259 
260 Valuation preservation by every Lift gives
261 
262 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
263 v_A(B^[j+1]-B^[j])=-n_(j+1).
264 \end{Verbatim}
265 
266 Thus the approximation and defining-sequence proof of the first note
267 applies unchanged. The odd-prime J/H dimensions, grade-index formula,
268 Swan conductor and self-pair conductor depend on these same levels and
269 centralizer degrees, so all their formulas carry over unchanged.
270 
271 \hypertarget{odd_tower-precisely-what-the-trace-supplies}{%
272 \subsection{7. Precisely what the trace
273 supplies}\label{odd_tower-precisely-what-the-trace-supplies}}
274 
275 For any displayed correction delta and any scalar u in L\^{}times for
276 which its additive correction character is defined,
277 
278 \begin{Verbatim}[fontsize=\small,breaklines=true,breakanywhere=true]
279 psi_delta(u I)=psi_L((u-1)tr_L(delta))=1.
280 \end{Verbatim}
281 
282 This proves triviality of the actual additive correction on the scalar
283 subgroup of its domain. It does not, by itself, construct a simple
284 character with prescribed values on larger scalar groups. Compatible
285 extension, transfer, and intersection arguments for the full simple
286 character are separate obligations.
287 
288 The note also does not identify the constructed candidates with any Weil
289 source, choose them by inverse LLC, realize arbitrary stabilizer
290 operators, or prove same-candidate H5. Its new output is an explicit
291 arithmetic defining sequence whose trace vanishes at every native and
292 amplified stage, suitable as input to that separate character argument.
293 
294 \hypertarget{odd_tower-primary-dependency}{%
295 \subsection{Primary dependency}\label{odd_tower-primary-dependency}}
296 
297 Bushnell--Kutzko, The admissible dual of GL(N) via compact open
298 subgroups, Remark 2.2.7 (printed p. 55), with the defining-sequence and
299 ring recursion already pinned in \path{RECURSIVE_TOWER166.md}. Retained source
300 SHA256:
301 \path{d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8}. The
302 trace-dual and Artin--Schreier calculations above are given in full.

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