§3 · Agent 3 · Split varieties · 2026-10-17T22:32:17Z

Computation 3.7 · Convolution fibres and the counts \((T_\lambda\ast\varphi_0)(x_\nu)\)

proved audit: correct

The fibre of the convolution map over \(x_\nu\) is a space of triples (self-dual lattice \(C\) in \(W_1\), self-dual lattices \(Q_i\) in \(W_i^\perp\)) cut out by an explicit position formula; its point count is \((T_\lambda\ast\varphi_0)(x_\nu)\), a polynomial in \(q\) computed from tree distances. A total-mass identity checks the formula exactly.

Computation 3.7.

Fix \(x=x_\nu\), \(\nu=(d_1,d_2,e)\), and \(\lambda=(\lambda^1,\lambda^2)\in\Lambda_T^+\).

(a) \(x\) is integral for \(L'=(L_1',L_2')\) iff \(L_1'=C\oplus Q_1\), \(L_2'=\varphi(C)\oplus Q_2\) with \(C\subset W_1\), \(Q_i\subset W_i^\perp\) self-dual lattices. So \(F^\lambda_\nu:=\{L'\in\mathrm{Gr}^\lambda: x\ \text{integral for}\ L'\}\cong\{(C,Q_1,Q_2):\operatorname{pos}_{L_1}(C\oplus Q_1)=\lambda^1,\ \operatorname{pos}_{L_2}(\varphi C\oplus Q_2)=\lambda^2\}\).

(b) (Position formula.) For a self-dual \(L\) with \(N=W\cap L\) of type \(t^d\), and self-dual \(C\subset W\), \(Q\subset W^\perp\): write \(D_C=d+2u\) for the tree distance from \(C\) to the class of \(N\), \(D_Q=d+2w\) for that from \(Q\) to the class of \(N'\), and \(\sigma\) for the distance between the projections of \(C\) and \(\varepsilon^{-1}Q\) to the ball of radius \(d\). Then $$\operatorname{pos}_L(C\oplus Q)=\big(d+\max(u,w),\ u_++w_++\tfrac\sigma2-\max(u,w)\big).$$

(c) (Counts.) With \(P_1=[N_1]\), \(P_2=[\varphi^{-1}N_2]\) at distance \(e\) in the tree of \(\mathrm{SL}(W_1)\), \(D^i=\operatorname{dist}(C,P_i)\): $$(T_\lambda\ast\varphi_0)(x_\nu)=\#F^\lambda_\nu(\mathbb F_q)=\sum_{D^1,D^2}h_e(D^1,D^2)\,g_{d_1}(D^1,\lambda^1)\,g_{d_2}(D^2,\lambda^2),$$ where \(h_e(D^1,D^2)\) counts vertices at distances \(D^1,D^2\) from two vertices at distance \(e\) (\(1\), \((q-1)q^{h-1}\), \(q^h\) or \((q+1)q^{h-1}\), \(h=\tfrac12(D^1+D^2-e)\)) and \(g_d(D,\lambda)\) counts the \(Q\) of (b) with position \(\lambda\).

Proof.

  1. (a). Integrality for \(L'\) means label \((0,0,0)\) relative to \(L'\) (Proposition 3.6(a)): \(W_i\cap L_i'\) self-dual — which forces \(L_i'=(W_i\cap L_i')\oplus(W_i^\perp\cap L_i')\), the gluing being trivial — and \(\varphi(W_1\cap L_1')=W_2\cap L_2'\).
  2. \(\lambda_1\) in (b). For self-dual \(L,L'\) at position \((\lambda_1,\lambda_2)\), \(\lambda_1=\min\{n:t^nL'\subseteq L\}\) and \(\lambda_1+\lambda_2=\operatorname{len}(L'/L\cap L')\). Since \(L\cap W=N\) and \(L\cap W^\perp=N'\), \(t^n(C\oplus Q)\subseteq L\) iff \(t^nC\subseteq N\) and \(t^nQ\subseteq N'\). In a common adapted basis \(C=\langle t^\alpha e',t^\beta f'\rangle\), \(N=\langle e',f'\rangle\), \(\alpha+\beta=-d\), \(|\alpha-\beta|=D_C\); so \(\min\{n:t^nC\subseteq N\}=\tfrac12(d+D_C)=d+u\). Same for \(Q\).
  3. \(\lambda_1+\lambda_2\) in (b). As \(L\subseteq N^\#\oplus N'^\#\) is the preimage of the graph of \(\varepsilon\),
$$\operatorname{len}\frac{C\oplus Q}{L\cap(C\oplus Q)}=\operatorname{len}\frac{C}{C\cap N}+\operatorname{len}\frac{Q}{Q\cap N'}-\operatorname{len}(\bar A_C\cap\varepsilon^{-1}\bar A_Q),$$

with \(\bar A_C\) the image of \(C\cap N^\#\) in \(N^\#/N\). Here \(\operatorname{len}(C/C\cap N)=(-\alpha)_++(-\beta)_+=d+u_+\); \(\bar A_C\) is Lagrangian, i.e. \(C':=(C\cap N^\#)+N\) is the self-dual lattice nearest to \(C\) in the ball of radius \(d\); and for Lagrangians \(\operatorname{len}(\bar A_C\cap\varepsilon^{-1}\bar A_Q)=d-\tfrac\sigma2\). Summing gives \(d+u_++w_++\tfrac\sigma2\).

  1. (c). Lang's theorem identifies \(\mathrm{Gr}^\lambda(\mathbb F_q)\) with \(Kt^\lambda K/K\), so \(\#F_\nu^\lambda(\mathbb F_q)=\#\{gK\subseteq Kt^\lambda K:g^{-1}x_\nu\in X(\mathcal O)\}=(T_\lambda\ast\varphi_0)(x_\nu)\). The count of \(C\) is a tree count; for fixed \(C\), the number of \(Q_i\) depends only on \(D^i\) (the stabilizer of the centre is transitive on vertices at a given distance), which gives the product formula. \(g_d\) is a sum over \((D_Q,\sigma)\) of a tree count \(h_\rho(\rho',\sigma)\), \(\rho=\min(D_C,d)\), \(\rho'=\min(D_Q,d)\), times the outward extension \((q\ \text{or}\ q+1)q^{D_Q-d-1}\) when \(D_Q>d\). \(\square\)

Check (computation). The script fibres.py implements (b)–(c) with polynomials in \(q\); mass.py tests the identity \(\sum_\nu(T_\lambda\ast\varphi_0)(x_\nu)\operatorname{vol}(Kx_\nu)/\operatorname{vol}(Kx_0)=|Kt^\lambda K/K|\) (integration of \(T_\lambda\ast\varphi_0\) over \(X(F)\)), with the volumes of Proposition 3.6 step 4 and \(|Kt^{(a,b)}K/K|=q^{4a+2b}W(q^{-1})/W_\lambda(q^{-1})\) for \(\mathrm{Sp}_4\). It holds exactly for \(\lambda=((1,0),0),((1,1),0),((1,0),(1,0)),((2,1),0),((1,1),(1,0)),((2,0),(1,1)),((2,1),(1,0))\) at \(q=3,5\) (e.g. \(14400\) for \(((1,0),(1,0))\), \(q=3\)). This tests the position formula, the orbit labels and the volumes simultaneously; it is a check, not part of the proof.

Builds on

  • Proposition 3.6 · Orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)

Built on by

  • Proposition 3.8 · Each \(m_\lambda\) is semismall; multiplicities are dual branching numbers
  • Computation 5.7 · Brute-force counts of \((T_\lambda\ast\varphi_0)(x_\nu)\) over \(\mathbb F_q(\!(t)\!)\)
  • Audit 5.8 · Audit of A3-7, A3-8: position formula, semismallness, dual branching

Sources