Computation X2.3 · Stalks for \(\mu=\alpha^\vee,2\alpha^\vee\): \(\mathcal L_{\mathrm{sgn}}\) is Kummer on each slice branch

1. The cover. For \(A\in\mathcal O_m\), \(m\ge1\), \(B=t^mA\) is integral with \(\det B=t^{2m}\) and \(B(0)\ne0\), so \(B(0)\) is a rank-one symmetric matrix. Over \(k\), with \(p\) odd, \(B(0)=ww^{\mathsf T}\) with \(w\) unique up to sign. The action \(g\cdot(A,w)=(gAg^{\mathsf T},g(0)w)\) makes \(\widetilde{\mathcal O}_m\to\mathcal O_m\) equivariant. Over \(x_m=\operatorname{diag}(t^m,t^{-m})\), \(B(0)=\operatorname{diag}(0,1)\) and \(w=\pm e_2\), and \(-I\in\operatorname{Stab}(x_m)\) exchanges the two points. So \(\widetilde{\mathcal O}_m\) is one orbit with stabilizer \(\{g_{a,b}:a(0)=1\}=\operatorname{Stab}(x_m)^\circ\) (Computation 2.1), and \(\pi_*\overline{\mathbb Q}_\ell=\mathcal L_{\mathrm{triv}}\oplus\mathcal L_{\mathrm{sgn}}\). Over \(\mathbb F_q\), Frobenius fixes the fibre \(\{\pm\sqrt\lambda\,v\}\) pointwise iff \(\lambda\in\mathbb F_q^{\times2}\). The class of \(\lambda\) is \(K\)-invariant, because \(g(0)\in\mathrm{SL}_2(\mathbb F_q)\). Both values occur, at \(x_m\) and at \(hx_m\) of Lemma X2.1 (\(\lambda=1,u\)), and Lang's count gives exactly two \(K\)-orbits.

2. Slice at \(x_0\) of \(\overline{\mathcal O_1}\). Chen–Yi's slice \(S^0\) consists of the symmetric \(h\in L^{<0}\mathrm{SL}_2\), and \(\overline{\mathcal O_1}=\{\min\operatorname{val}\ge-1\}\). So \(h=I+t^{-1}h_1\) with \(h_1=\begin{pmatrix}a&b\\b&d\end{pmatrix}\), and \(\det h=1+(a+d)t^{-1}+(ad-b^2)t^{-2}=1\) forces \(d=-a\) and \(a^2+b^2=0\). On the punctured line \(b=\pm ia\), \(a\ne0\), one has \((tA)(0)=a\,(1,\pm i)^{\mathsf T}(1,\pm i)\), checked exactly in slice_leading.py. So by step 1 the cover is \(w=\pm\sqrt a\,(1,\pm i)\): the Kummer cover, and \(\mathcal L_{\mathrm{sgn}}\) restricts to the nontrivial Kummer sheaf of each punctured line.

3. IC of the slice. \(L^+G\times S^\lambda\to LX\) is formally smooth (Chen–Yi, Prop. 47), so \(\mathrm{IC}_{\chi,\mu}\) restricted to the slice is the IC of the slice with the restricted local system, shifted by the orbit dimension. For a curve with smooth branches, the IC is the pushforward from the normalisation of \(j_*\mathcal L[1]\). Its stalk at the singular point is \(\bigoplus_{\text{branches}}(j_*\mathcal L)_0\) in degree \(-1\): \(\overline{\mathbb Q}_\ell\) per branch for constant \(\mathcal L\), and \(0\) for Kummer \(\mathcal L\) (no invariants under the monodromy \(-1\)). This gives (ii). The \(\pi_0\)-action is trivial because \(-I\) acts trivially on the slice and on the constant sheaf.

4. Slice at \(x_1\) of \(\overline{\mathcal O_2}\). By Lemma 2.6 (rechecked exactly by the second audit), the points are \(A=hx_1\) with \(h=\begin{pmatrix}1+c/t+c^2/t^2+c^3/t^3&e/t\\ e/t^3&1-c/t\end{pmatrix}\) and \(e^2=-c^4\). For \(c\ne0\), \(\min\operatorname{val}A=-2\) and \((t^2A)(0)=c\,(c,\pm i)^{\mathsf T}(c,\pm i)\) on the branch \(e=\pm ic^2\) (checked exactly in slice_leading.py). So \(\mathcal L_{\mathrm{sgn}}\) is Kummer in \(c\) on each punctured branch, and step 3 gives a zero stalk at \(x_1\) for \(\mathrm{IC}_{\mathrm{sgn},2}\) and \(\overline{\mathbb Q}_\ell^{\,2}\) for \(\mathrm{IC}_{\mathrm{triv},2}\). At \(x_0\), Lemma 2.6 gives the cone \(\cong\mathbb A^2/\pm1\) (rationally smooth) and two planes \(P_\pm\), normalised by their disjoint union; the stalk is \(\overline{\mathbb Q}_\ell^{\,3}\), in degree \(-\delta(2)\) only. \(\mathrm{IC}_{\mathrm{sgn},2}|_{\mathcal O_0}=0\) by Lemma X2.2(b). By equivariance, stalks at \(x_\nu\) determine the restrictions to \(\mathcal O_\nu\).

5. Frobenius and fibre counts. In the \(\mathrm{SO}_2\)-form, the branches \(e=\pm ic^2\) and the planes \(P_\pm=\{(a_1,\pm ia_1,\nu,\pm i\nu,-\nu)\}\) are exchanged by \(i\mapsto i^q\), so they are exchanged iff \(q\equiv3\bmod4\); the cone is defined over \(\mathbb F_q\). The convolution map \(m\) for \(V_{\alpha^\vee}\ast\mathrm{IC}_0\) is finite, with one fibre point per local branch or component (Lemma 2.6), and \(\mathrm{Gr}_{\le\alpha^\vee}\) is rationally smooth. So the trace of \(V_{\alpha^\vee}\ast\mathrm{IC}_0\) is the fibre count \(((T+1)\varphi_0)(x)\) (Lemma 4.3), and it equals the trace of the permutation action of Frobenius on branches. fibre_counts.py counts lattices \(L=t^{-1}M\), with \(\mathcal O^2/M\cong\mathcal O/t^2\), on which \(A^{-1}\) is unimodular, for \(X_1\) and for \(X_{-1}\), giving the values in (iv). Thus the cone plus planes give \(3=1+2\) (rational) or \(1=1+0\) (swapped) at \(x_0\), and the branches give \(2\) or \(0\) at \(x_1\). The split column reproduces \(Tf_0=2f_0+2f_1+f_2\) of Computation 2.8. In every case the value is the same on both \(K\)-orbits of \(\mathcal O_1\), as Lemma X2.2(c) predicts for \(\mathrm{IC}_{\mathrm{triv},2}\). In the \(q\equiv3\bmod4\) column, \(T\varphi_0\) vanishes on \(\mathcal O_1\), which lies in the other \(\mathrm{SL}_2(F)\)-orbit (Proposition X2.6). \(\square\)

Status. Proved, from the audited slices of Lemma 2.6, Chen–Yi's Prop. 47, and exact checks (slice_leading.py, fibre_counts.py). These are illustrations and checks; the general statement is Lemma X2.2, which does not depend on them.