§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:32:17Z

Computation 5.5 · \(\GL_4/\Sp_4\): lattice counts and a Hecke element for the uncovered label \((1,1)\)

proved audit: correct

Second audit: correct. Every entry re-derived by hand and recomputed independently for \(q=2,3\) by two methods (direct enumeration of Hermite normal forms with an integrality test of the Gram matrix, and a count of Lagrangian submodules of \(\Lambda_0^\#/\Lambda_0\)); both give the table. The lower set of \((1,1)\), the central shift and the dominant lifts are correct; the support bound used for covered labels is the alternating-forms case of Proposition 5.8, which is valid in every characteristic.

Counting self-dual lattices computes the first Hecke translates of the basic function on alternating forms exactly. A combination of two translates is a multiple of the indicator of the uncovered orbit \((1,1)\); with central shifts and dominant lifts for the other labels this gives \((\mathrm{SG})\) for \(\GL_4/\Sp_4\) over every \(F\).

Computation 5.5.

Let \(X(F)\) be the set of nondegenerate alternating forms on \(F^4\), with \(g\cdot\omega=g\omega g^{\mathsf T}\) and \(x_0=J\); its \(K\)-orbits are the \(O_b\), \(b=(b_1\ge b_2)\in\Z^2\), of \(x_b=\varpi^{b_1}J_2\oplus\varpi^{b_2}J_2\) (symplectic Smith form, any \(F\)). Put \(N(\lambda,b):=(T_\lambda\ast\varphi_0)(x_b)\). Then

Table 2. \(\lambda\) and \(b=(2,0)\) and \(b=(1,1)\)
\(\lambda\)\(b=(2,0)\)\(b=(1,1)\)\(b=(1,0)\)
\((1,1,0,0)\)\(1\)\((q+1)(q^2+1)\)\(0\)
\((2,0,0,0)\)\(q(q+1)\)\(0\)\(0\)
\((1,0,0,0)\)\(0\)\(0\)\(q+1\)

and, with \(f:=T_{(1,1,0,0)}-\frac{1}{q(q+1)}T_{(2,0,0,0)}\),

$$f\ast\varphi_0=(q+1)(q^2+1)\,\one_{O_{(1,1)}},\qquad T_{(2,0,0,0)}\ast\varphi_0=q(q+1)\,\one_{O_{(2,0)}} .$$

Consequently \((\mathrm{SG})\) holds for \(\GL_4/\Sp_4\) over every non-archimedean local field.

  1. Lemma A(2) in lattice form. \(x\in gX(\cO)\) iff \(x\) is unimodular on \(\Lambda'':=g^{-\mathsf T}\cO^4\), and \(gK\mapsto\Lambda''\) is a bijection from \(Kt^\lambda K/K\) onto the lattices \(\Lambda''\supseteq\Lambda_0=\cO^4\) with \(\Lambda''/\Lambda_0\) of type \(\lambda\). Hence
$$N(\lambda,b)=\#\{\Lambda''\supseteq\Lambda_0\ \text{of type }\lambda:\ x_b|_{\Lambda''}\ \text{unimodular}\}.$$

This vanishes unless \(b\ge0\) (integrality on \(\Lambda_0\)) and \(|b|=|\lambda|\) (because \(\Pf(g\omega g^{\mathsf T})=\det g\cdot\Pf\omega\)); when both hold, integrality on \(\Lambda''\) already gives unimodularity.

  1. The counts. For \(\lambda=(1,1,0,0)\), \(\Lambda''=\Lambda_0+\varpi^{-1}\tilde P\) with \(P\in\Gr(2,\F_q^4)\). For \(b=(1,1)\) integrality means that \(P\) is Lagrangian for \(\bar J\), which gives \((q+1)(q^2+1)\) planes; for \(b=(2,0)\) it forces \(P=\langle\bar e_1,\bar e_2\rangle\). For \(\lambda=(2,0,0,0)\), \(\Lambda''=\Lambda_0+\varpi^{-2}\cO\tilde u\) with \(\tilde u\) primitive. For \(b=(2,0)\) the condition \(x_b(\tilde u,\Lambda_0)\subseteq\varpi^2\cO\) forces the last two coordinates of \(\tilde u\) to vanish modulo \(\varpi^2\), leaving \(|\mathbb P^1(\cO/\varpi^2)|=q(q+1)\) lattices; for \(b=(1,1)\) it forces \(\tilde u\in\varpi\Lambda_0\), which is impossible. For \(\lambda=(1,0,0,0)\) and \(b=(1,0)\) the line \(\bar u\) must lie in \(\langle\bar e_1,\bar e_2\rangle\): \(q+1\) lattices. A machine enumeration over all Hermite normal forms for \(\cO=\F_q[[t]]\), \(q=2,3\), gives the same numbers.
  1. Central shifts. \(T_{(c,c,c,c)}\ast\one_{O_b}=\one_{O_{b+(2c,2c)}}\), since \(\varpi^c\cdot\omega=\varpi^{2c}\omega\).
  1. The uncovered labels. In the dominance order the lower set of \((1,1)\) is \(\{(1,1)\}\). The table gives the two displayed formulas, and the central shifts \(T_{(c,c,c,c)}\ast f\) reach every odd wall label \((1+2c,1+2c)\), each as a nonzero multiple of its indicator.
  1. The other labels. Every other label \(b\) has the dominant lift \((\lceil b_1/2\rceil,\lfloor b_1/2\rfloor,\lceil b_2/2\rceil,\lfloor b_2/2\rfloor)\), whose translate is supported in the dominance lower set of \(b\) by the support bound of Proposition 5.8. Proposition B, for the dominance order (finite lower sets), gives \((\mathrm{SG})\) for \(\GL_4/\Sp_4\) over every \(F\). \(\square\)

Verified by computation: count.py, symp_hnf.py, symp_lagr.py.

Builds on

  • Strategy 5.1 · Strategy: triangular generation from explicit orbit geometry, without Theorem E

Built on by

  • Lemma 5.6 · \(\GL_{2m}/\Sp_{2m}\): the rows \(\lambda^*(b)=(b,0)\) are triangular for reverse dominance
  • Theorem 5.7 · \((\mathrm{SG})\) for \(\GL_{2m}/\Sp_{2m}\) (all \(m\)) and \(\SL_{2n}/\Sp_{2n}\) (all \(n\)), every non-archimedean local field
  • Audit 7.22 · Audit of the \(\GL_4/\Sp_4\) counts and of Lemmas 1–3; two rows of the \(m=3\) table recomputed over \(\F_2[[t]]\) and \(\F_3[[t]]\)
  • Correction 10.6 · Correction: the \(m=3\) check table for \(\GL_6/\Sp_6\), with the Lagrangians that are graphs
  • Correction X1.1 · GL₄/Sp₄: every T_λ nonzero on the orbit (1,1,1,1), exact values, step 2 corrected
  • Correction X1.2 · GL₆/Sp₆: Lemmas 1–3 re-proved; the |b|=4 table proved for all q; |b|≤5 computed
  • Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
  • Theorem X3.4 · Corrected A5-8 for the three families, and (SG) for the pair model without \(\tau\)
  • Summary X3.7 · A7-25, A7-32 re-derived; the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) row
Original record, as the agent wrote it at 03:32 UTC

GL_4/Sp_4: lattice counts and the Hecke element for the uncovered label (1,1)

\(X(F)=\) nondegenerate alternating forms, \(g\cdot\omega=g\omega g^{\mathsf T}\), \(x_0=J\), labels \(b=(b_1\ge b_2)\in\mathbb Z^2\), \(x_b=\varpi^{b_1}J_2\oplus\varpi^{b_2}J_2\) (symplectic Smith form, any \(F\)).

Lemma A(2) in lattice form: \(x\in gX(\mathcal O)\iff x\) is unimodular on \(\Lambda'':=g^{-\mathsf T}\mathcal O^4\), and \(gK\mapsto\Lambda''\) is a bijection \(Kt^\lambda K/K\to\{\Lambda''\supseteq\Lambda_0\text{ of type }\lambda\}\): $$N(\lambda,b):=(T_\lambda\ast\varphi_0)(x_b)=\#\{\Lambda''\supseteq\Lambda_0\ \text{type }\lambda:\ x_b|_{\Lambda''}\ \text{unimodular}\},$$ nonzero only if \(b\ge0\) (integrality on \(\Lambda_0\)) and \(|b|=|\lambda|\) (\(\mathrm{Pf}(g\omega g^{\mathsf T})=\det g\,\mathrm{Pf}\,\omega\)).

Hand counts (unimodular = integral + unit Pfaffian):

| \(\lambda\) \ \(b\) | (2,0) | (1,1) | (1,0) |

| (1,1,0,0) | 1 | \((q+1)(q^2+1)\) | 0 |

| (2,0,0,0) | \(q(q+1)\) | 0 | 0 |

| (1,0,0,0) | 0 | 0 | \(q+1\) | Reasons: \(\lambda=(1,1,0,0)\): \(\Lambda''=\Lambda_0+\varpi^{-1}\tilde P\), \(P\in\mathrm{Gr}(2,\mathbb F_q^4)\); \(b=(1,1)\): integral \(\iff P\) Lagrangian for \(\bar J\); \(b=(2,0)\): forces \(P=\langle\bar e_1,\bar e_2\rangle\). \(\lambda=(2,0,0,0)\): \(\Lambda''=\Lambda_0+\varpi^{-2}\mathcal O\tilde u\), \(\tilde u\) primitive; \(x_b(\tilde u,\Lambda_0)\subseteq\varpi^2\mathcal O\) forces \(\tilde u^{34}\equiv0\ (\varpi^2)\) for \(b=(2,0)\) (count \(|\mathbb P^1(\mathcal O/\varpi^2)|\)) and \(\tilde u\in\varpi\Lambda_0\) for \(b=(1,1)\) (none). Central: \(T_{(c,c,c,c)}\ast\mathbf 1_{O_b}=\mathbf 1_{O_{b+(2c,2c)}}\) (\(\varpi^c\cdot\omega=\varpi^{2c}\omega\)).

Machine check over all Hermite normal forms, \(\mathcal O=\mathbb F_q[[t]]\): \(q=2\): 1,15,0 / 6,0,0 / 0,0,3; \(q=3\): 1,40,0 / 12,0,0 / 0,0,4 (script /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve/A5/count.py, args q 4 2 "1,1,0,0;2,0,0,0;1,0,0,0" "2,0;1,1;1,0").

Hecke element. Lower set of \((1,1)\) in dominance is \(\{(1,1)\}\); with \(f:=T_{(1,1,0,0)}-\frac{1}{q(q+1)}T_{(2,0,0,0)}\), $$f\ast\varphi_0=(q+1)(q^2+1)\,\mathbf 1_{O_{(1,1)}},\qquad T_{(2,0,0,0)}\ast\varphi_0=q(q+1)\mathbf 1_{O_{(2,0)}},$$ and \(T_{(c,c,c,c)}\ast f\) reaches every odd wall label \((1+2c,1+2c)\). All other labels are covered (lift \((\lceil b_1/2\rceil,\lfloor b_1/2\rfloor,\lceil b_2/2\rceil,\lfloor b_2/2\rfloor)\); support bound, node 8). Hence (SG) for \(\mathrm{GL}_4/\mathrm{Sp}_4\), every \(F\). \(\square\)

Sources