§12 · Agent X2 · SL₂/SO₂ traces · 2026-10-17T20:27:22Z

Proposition X2.6 · (SG) fails for $\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)$ over every $F$, $q$ odd

proved audit: correct

Second audit: correct. I re-derived that there are two SL2(F)-orbits, since H¹(F,SO2)=F^×/N(E^×)≅Z/2 for −1∉F^{×2}. The parity criterion holds because E/F is unramified with norms of even valuation. X(O)=K·I for p odd, so Hecke translates are supported in G(F)·X(O)=X^ev. For q≡1 mod 4 the reduction to SL2/T goes through P∈GL2(O), which needs i∈O (Hensel). Numerical checks for p=3,7: random g·I always has even m and random g·x_1 always has odd m, and T^kφ0 vanishes on every odd O_m. For p=5,13 both parities occur. The scope note on residue characteristic 2 is correct: x²+y² does not represent 3 over Z_2.

The function-level failure, proved in the run only for the split form $\mathrm{SL}_2/T$, holds for the $\mathrm{SO}_2$-form as well. For $q\equiv1\bmod4$ the two forms are integrally isomorphic. For $q\equiv3\bmod4$, $X(F)$ has two $\mathrm{SL}_2(F)$-orbits, distinguished by the parity of the minimal valuation, and Hecke translates of $\varphi_0$ never leave the even one. This closes the gap left open in Summary 2.9 and Audit 7.18.

Proposition X2.6.

Let $F$ be a non-archimedean local field of any characteristic, with residue field $\mathbb F_q$, $q$ odd, and uniformizer $\varpi$. Let $K=\mathrm{SL}_2(\mathcal O)$, $X=X_1=\{A\in\mathrm{Sym}_2(F):\det A=1\}$ (the $\mathrm{SO}_2$-form), and $\varphi_0=\mathbf 1_{X(\mathcal O)}$. Then (SG) $\mathcal H_K\varphi_0=C_K$ fails.

(a) If $q\equiv1\bmod4$, the isomorphism of Correction (repair of A7-16) X2.5(d) identifies $(X,K,\varphi_0,\mathcal H_K)$ with the split form $\mathrm{SL}_2/T$, where (SG) fails (Proposition J(2), Computation 2.8, Proposition 1.2).

(b) If $q\equiv3\bmod4$, then $X(F)=X^{\mathrm{ev}}\sqcup X^{\mathrm{odd}}$, where $X^{\mathrm{ev}}=\{A:\min\operatorname{val}A_{ij}\text{ even}\}=\mathrm{SL}_2(F)\cdot I$ and $X^{\mathrm{odd}}=\mathrm{SL}_2(F)\cdot\operatorname{diag}(\varpi,\varpi^{-1})$ are open. Every element of $\mathcal H_K\varphi_0$ is supported in $X^{\mathrm{ev}}$, so $\mathbf 1_{K\operatorname{diag}(\varpi,\varpi^{-1})}\in C_K\setminus\mathcal H_K\varphi_0$.

(c) For $F=\mathbb F_q(\!(t)\!)$ and every odd $q$, also $\mathcal H_K\varphi_0\subseteq C_K^{\sigma}\subsetneq C_K$, with $\sigma$ from Lemma X2.1.

(d) Remark. Here $G^\theta=\mathrm{SO}_2$ is connected, yet $X(F)\ne G(F)/G^\theta(F)$ when $-1\notin F^{\times2}$, because $H^1(F,\mathrm{SO}_2)=F^\times/N_{E/F}E^\times\cong\mathbb Z/2$ with $E=F(i)$. The second audit's gloss on Objection 1.8, that $X(F)$ is one $G(F)$-orbit for the rows with $G^\theta$ connected, needs this exception. It is harmless for the run, which used the split form.

(a). By Hensel ($p$ odd), $q\equiv1\bmod4$ gives $i\in\mathcal O$. By Correction (repair of A7-16) X2.5(d), the map $g\,\mathrm{SO}_2\mapsto P^{-1}gP\,T$, $P\in\mathrm{GL}_2(\mathcal O)$, is an isomorphism $X_1=\mathrm{SL}_2/\mathrm{SO}_2\to\mathrm{SL}_2/T$ over $\mathcal O$, equivariant along $\operatorname{Ad}(P^{-1})$. It maps $K$ to $K$, $X_1(\mathcal O)=Kx_0$ to $Kx_0$, and each $Kt^\lambda K$ to itself. So it identifies $C_K$, $\varphi_0$ and the $\mathcal H_K$-action, $f\mapsto f\circ\operatorname{Ad}(P)=f$. (SG) fails for $\mathrm{SL}_2/T$ for every $q$ and $F$ (Computation 2.8(c), Proposition 1.2).

(b), normal form. Let $A\in X(F)$ with $\min\operatorname{val}A_{ij}=-m$. As in Computation 2.1, step 1 (only $2\in\mathcal O^\times$ is used), $\varpi^mA$ is an integral binary form with a unit entry, so it represents a unit and splits. Using $\det A=1$, this gives $A\simeq_{\mathrm{GL}_2(\mathcal O)}\operatorname{diag}(v\varpi^{-m},v'\varpi^{m})$ with $vv'=c^2$, $c\in\mathcal O^\times$.

(b), orbits. $\mathrm{SL}_2(F)$-orbits on $X(F)$ are isometry classes of binary quadratic spaces with Gram determinant exactly $1$. If $B=gAg^{\mathsf T}$ with $g\in\mathrm{GL}_2(F)$, then $\det g=\pm1$, and $\det g=-1$ is corrected by a reflection in $\mathrm O(A)$. Such a space is $\cong\langle1,1\rangle$ iff it represents $1$. Now

$$v\varpi^{-m}x^2+v'\varpi^my^2=v\varpi^{-m}\big(x^2+(cv^{-1}\varpi^my)^2\big)=v\varpi^{-m}\,N_{E/F}(x+icv^{-1}\varpi^my),\qquad E=F(i).$$

For $q\equiv3\bmod4$, $E/F$ is unramified quadratic, norms have even valuation, and units are norms. So the form represents $1$ iff $\varpi^m v^{-1}\in N(E^\times)$, iff $m$ is even. Hence the orbit of $I$ is $X^{\mathrm{ev}}$, and the complement is one orbit, that of $\operatorname{diag}(\varpi,\varpi^{-1})$: there are two classes, since $H^1(F,\mathrm{SO}_2)\cong\mathbb Z/2$. Both are open (the orbit map is smooth).

(b), Hecke support. $(f\ast\varphi_0)(A)=\sum_{g\in G/K}f(g)\varphi_0(g^{-1}\cdot A)\ne0$ forces $A\in G(F)\cdot X(\mathcal O)\subseteq X^{\mathrm{ev}}$, since $X(\mathcal O)=K\cdot I$ (unimodular binary forms of determinant $1$ over $\mathcal O$, $p$ odd). The indicator of the open compact set $K\operatorname{diag}(\varpi,\varpi^{-1})\subseteq X^{\mathrm{odd}}$ lies in $C_K$ and is nonzero.

(c). This is Lemma X2.1(a),(c) with $n=1$: $\sigma$ fixes $\varphi_0$, commutes with $\mathcal H_K$, and exchanges the two $K$-orbits of each $\mathcal O_m(\mathbb F_q)$, $m\ge1$.

Illustration. For $F=\mathbb F_q(\!(t)\!)$ with $q=3,7,11$, the lattice count gives $T\varphi_0=\mathbf 1_{\mathcal O_2(\mathbb F_q)}$ for $T=\mathbf 1_{Kt^{\alpha^\vee}K}$ (fibre_counts.py; Computation X2.3(iv)). It vanishes on $\mathcal O_1(\mathbb F_q)\subseteq X^{\mathrm{odd}}$, as step 4 requires. $\square$

Scope. Residue characteristic $2$ for the $\mathrm{SO}_2$-form is not treated: $X(\mathcal O)$ is then not one $K$-orbit; for example, over $\mathbb Z_2$, $\operatorname{diag}(3,3^{-1})\not\simeq I$, since $x^2+y^2$ does not represent $3$. The run's "every $q$" refers to the split form $\mathrm{SL}_2/T$, which is how Table 1 defines the row.

Builds on

Lemma X2.1 · $\sigma_n$ is translation by $h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$ with $h^{-1}\mathrm{Fr}^n(h)=-I$
Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the $\mathrm{SO}_2$-form iff $q\equiv1\bmod4$; Lang’s count holds in both
Computation 2.8 · Function level for $\SL_2/T$: Lang count of $K$-orbits, the sign-class dictionary, and a Hecke dimension count for every $q$
Proposition 1.2 · $\SL_2/T$ versus $\GL_2/T$: one variety, two Hecke algebras
Computation 2.1 · Stabilizers for $\SL_2/\SO_2$: $\mu_2$ times a pro-unipotent group
Summary 2.9 · Verdict for $\SL_2/\SO_2$: both levels fail, and the sheaf argument is the geometric form of Proposition J(2)
Audit 7.18 · Audit of the remaining checklist items: heuristic labels are honest, and two verdict sentences need the field caveat
Objection 1.8 · Comparability: the same pair $(G,X)$ in every row, but three different fields

Built on by

Audit X2.7 · No $\mathrm{SL}_2/\mathrm{SO}_2$ conclusion of the run rests on the false §8.2 remark
Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
Re-check V2 · Re-check by V2

Sources

classical.html Proposition J(2)
summary.html Table 1 (row SL2/SO2 = SL2/T)
Computation 2.1 step 1
Computation 2.8(c)
scripts: fibre_counts.py

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Proposition X2.6 · (SG) fails for \(\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)\) over every \(F\), \(q\) odd

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