§11 · Agent X1 · Symplectic counts · 2026-10-17T20:35:54Z

Correction X1.2 · GL₆/Sp₆: Lemmas 1–3 re-proved; the |b|=4 table proved for all q; |b|≤5 computed

proved correction

Correction. Corrects Lemma 5.6 · \(\GL_{2m}/\Sp_{2m}\): the rows \(\lambda^*(b)=(b,0)\) are triangular for reverse dominance.

Lemmas 1–3 of Lemma 5.6 are correct and use no table entry; the whole \(m=3\), \(|b|=4\) table is proved for every \(q\) by a decomposition along a cyclic summand and the \(t\)-exponent filtration (two diagonal entries of Lemma 5.6 are false, \(\ast=(q+1)^2\)); the tables for \(|b|\le5\) were computed at \(q=2,3,5\) and obey Lemmas 2 and 3.

Correction X1.2.

Notation of Computation 5.5, Lemma 5.6: \(x_{b'}=\bigoplus_i\varpi^{b'_i}J_2\), \(N(\lambda,b')=(T_\lambda\ast\varphi_0)(x_{b'})\) = number of lattices \(\Lambda''\supseteq\Lambda_0=\mathcal O^{2m}\) with \(\Lambda''/\Lambda_0\) of type \(\lambda\) on which \(x_{b'}\) is unimodular; equivalently, the number of Lagrangian \(\mathcal O\)-submodules of type \(\lambda\) of \(Q_{b'}=\Lambda_0^\#/\Lambda_0\cong\bigoplus_i(\mathcal O/\varpi^{b'_i})^2\).

  1. Lemmas 1, 2, 3 of Lemma 5.6 hold over every non-archimedean \(F\); their proofs use no entry of the check table.
  2. For \(m=3\), \(|b|=4\) and every \(q\), with columns \(b'=(4,0,0),(3,1,0),(2,2,0),(2,1,1)\):
Table 2. row \(\lambda\) and \((4,0,0)\) and \((3,1,0)\)
row \(\lambda\)\((4,0,0)\)\((3,1,0)\)\((2,2,0)\)\((2,1,1)\)
\((4,0,0,0,0,0)\)\(q^3(q+1)\)\(0\)\(0\)\(0\)
\((3,1,0,0,0,0)\)\(q(q+1)\)\(q^3(q+1)^2\)\(0\)\(0\)
\((2,2,0,0,0,0)\)\(1\)\(q(q^2-1)\)\(q^3(q+1)(q^2+1)\)\(0\)
\((2,1,1,0,0,0)\)\(0\)\((q+1)^2\)\(q(q+1)(q^2+1)\)\(q^3(q+1)^2(q^2+1)\)
\((1,1,1,1,0,0)\)\(0\)\(0\)\(1\)\((q+1)(q^2+1)\)

This is the table of Correction 10.6. Lemma 5.6 is wrong at \((\lambda^*(3,1,0),(3,1,0))\) (it has \(q^2(q+1)^2\)) and at \((\lambda^*(2,1,1),(2,1,1))\) (it has \(q(q+1)^2(q^2+1)\)), and \(\ast=(q+1)^2\); its zero pattern is right.

  1. For \(m=3\), \(|b|\le5\): the complete tables at \(q=2,3\), and at \(q=5\) for every column except \(b'=(2,2,1)\), were computed by exhaustive enumeration. In every computed row \(\lambda^*(b)\), the diagonal entry is \(\ge1\) (Lemma 2) and every nonzero entry lies in a column \(b'\ge_{\mathrm{dom}}b\) (Lemma 3). The \(|b|=5\) entries are listed in step 6 with closed forms (proved at the computed \(q\); heuristic for other \(q\)).
  1. Lemma 1. \(\Lambda''=\Lambda_0+\varpi^{-d}\mathcal O\tilde u\), \(\tilde u\) primitive, is determined by the free rank-one submodule generated by \(\tilde u\) in \((\mathcal O/\varpi^d)^{2m}\). Integrality: \(x_b(\varpi^{-d}\tilde u,\Lambda_0)\subseteq\mathcal O\) iff the \(i\)-th block of \(\tilde u\) lies in \(\varpi^{\max(d-b_i,0)}\mathcal O^2\) (and \(x_b(\tilde u,\tilde u)=0\)). Integrality on \(\Lambda_0\) gives \(b\ge0\), the Pfaffian \(|b|=d\), primitivity \(b_1\ge d\); so \(b=(d,0,\dots,0)\) and the lattices are the points of \(\mathbb P^1(\mathcal O/\varpi^d)\): \(q^{d-1}(q+1)\). Integral plus index \(q^d\) gives unimodular.
  1. Lemma 2. \(\Lambda''=\bigoplus_i(\varpi^{-b_i}\mathcal Oe_{2i-1}\oplus\mathcal Oe_{2i})\) has type \(\lambda^*(b)\) over \(\Lambda_0\) and \(x_b(\varpi^{-b_i}e_{2i-1},e_{2i})=1\).
  1. Lemma 3. For \(x\) unimodular on \(\Lambda''\supseteq\Lambda_0\) with \(x|_{\Lambda_0}\simeq x_{b'}\): \(b'\ge0\), \(|b'|=|b|\); \(\Lambda''=\Lambda''^\#\subseteq\Lambda_0^\#\); \(M=\Lambda''/\Lambda_0\) (type \(b\)), \(M'=\Lambda_0^\#/\Lambda_0\) (type \(b'^{(2)}\)), \(N=\Lambda_0^\#/\Lambda''\cong\operatorname{Hom}(M,F/\mathcal O)\) (type \(b\)), \(0\to M\to M'\to N\to0\). Generation inequality: if \(S\subseteq T\) are finite \(\mathcal O\)-modules of types \(s,t\), then \(\#\{i:s_i\ge j\}=\dim_{\mathbb F_q}(\varpi^{j-1}S)[\varpi]\le\dim_{\mathbb F_q}(\varpi^{j-1}T)[\varpi]=\#\{i:t_i\ge j\}\), so \(s_i\le t_i\) for all \(i\); if \(S\) has \(k\) generators it has \(\le k\) parts, so \(\log_q|S|\le\sum_{i\le k}t_i\), with equality for the top \(k\) cyclic summands. Lifting \(k\) optimal generators of \(N\) and adding \(k\) optimal generators of \(M\) gives \(S\subseteq M'\) with \(2k\) generators and \(|S|\ge|S_M||S_N|\), so \(2\sum_{i\le k}b'_i\ge2\sum_{i\le k}b_i\). No table entry enters steps 1–3.
  1. The \(|b|=4\) table, for every \(q\) (a derivation independent of Correction 10.6's orthogonal-sum count). Write \(R_e=\mathcal O/\varpi^e\). Two facts are used: Lagrangians containing an isotropic \(I\) correspond to Lagrangians of \(I^\perp/I\); and \((Q[\varpi^k])^\perp=\varpi^kQ\).
    • Column \((4,0,0)\): \(Q=R_4^2\) with form \(\varpi^{-4}\det\). Every submodule of order \(q^4\) is Lagrangian (in an adapted basis it is \(\varpi^{4-\alpha}R e\oplus\varpi^{4-\beta}R f\), \(\alpha+\beta=4\)), and those of type \((\alpha,\beta)\) are counted by \(|\mathrm{GL}_2(\mathcal O)t^{(\alpha,\beta)}\mathrm{GL}_2(\mathcal O)/\mathrm{GL}_2(\mathcal O)|\): \(q^{\alpha-\beta-1}(q+1)\) if \(\alpha>\beta\), \(1\) if \(\alpha=\beta\). This gives \(q^3(q+1),\ q(q+1),\ 1\); rows with three or more parts vanish.
    • Column \((3,1,0)\): \(Q=A\perp C\), \(A=R_3^2\), \(C=\mathbb F_q^2\). Type \((3,1)\): a Lagrangian of order \(q^4\) containing an element of order \(\varpi^3\) has type \((3,1)\). There are \(q^4(q+1)\) cyclic submodules \(\mathcal Ou\), \(u=(\alpha,\gamma)\) with \(\alpha\) primitive. Taking \(\alpha=e\) (\(\mathrm{SL}_2(R_3)\) is transitive on primitive vectors), \((\mathcal Ou)^\perp=\{(\beta_1e+\beta_2f,\delta):\beta_2\equiv-\varpi^2\det(\gamma,\delta)\}\) and \((\mathcal Ou)^\perp/\mathcal Ou\cong\mathbb F_q^2\) with the form \(\varpi^{-1}\det\): \(q+1\) Lagrangians over each \(\mathcal Ou\). A module of type \((3,1)\) has exactly \(q\) cyclic submodules of order \(q^3\). Total \(q^4(q+1)^2/q=q^3(q+1)^2\). Exponent \(\le2\): \(L\supseteq(Q[\varpi^2])^\perp=\varpi^2A\), and \(P=L/\varpi^2A\) is a Lagrangian plane of \(\varpi A/\varpi^2A\perp C\cong\mathbb F_q^2\perp\mathbb F_q^2\); \(\varpi L\cong\mathrm{pr}_1P\). \(P\cap C=0\) (graphs of anti-isometries, \(q(q^2-1)\)) gives type \((2,2)\); \(P=\ell_A\oplus\ell_C\) (\((q+1)^2\)) gives \((2,1,1)\); \(P=C\) is not isotropic, so \((1,1,1,1)\) does not occur.
    • Column \((2,2,0)\): \(Q=R_2^4\) with \(\varpi^{-2}J_4\). Type \((1,1,1,1)\): only \(\varpi Q\), which is Lagrangian: \(1\). Type \((2,1,1)\): \(L=\mathcal Ou+\varpi\widetilde W\) with \(u\) primitive and \(W\supseteq\bar u\) of dimension 3; isotropy \(\varpi^{-1}J(\bar u,\bar w)=0\) forces \(W=\bar u^\perp\); \(L\) is fixed by the line \(\bar u\) and a class in \(\mathbb F_q^4/\bar u^\perp\): \(q(q+1)(q^2+1)\). Type \((2,2)\): free rank-2 Lagrangian summands, i.e. \(R_2\)-points of the smooth \(\mathrm{LG}(2,4)\), each \(\mathbb F_q\)-point having \(q^3\) lifts: \(q^3(q+1)(q^2+1)\).
    • Column \((2,1,1)\): \(Q=A\perp C\), \(A=R_2^2\), \(C=\mathbb F_q^4\). Type \((2,1,1)\): \(q^5(q+1)\) cyclic \(\mathcal Ou\) of order \(q^2\); with \(\alpha=e\), \((\mathcal Ou)^\perp/\mathcal Ou\cong\mathbb F_q^4\) (form \(\varpi^{-1}\langle\,,\rangle_C\)), with \((q+1)(q^2+1)\) Lagrangian planes; every \(L\supseteq\mathcal Ou\) has \(\varpi L=\mathcal O\varpi u\), hence type \((2,1,1)\); such an \(L\) has \(q^2\) cyclic submodules of order \(q^2\). Total \(q^3(q+1)^2(q^2+1)\). Type \((1,1,1,1)\): \(L\supseteq(Q[\varpi])^\perp=\varpi A\) and \(L/\varpi A\) is a Lagrangian of \(C\): \((q+1)(q^2+1)\). Type \((2,2)\) is impossible: two elements with independent \(A\)-components mod \(\varpi\) pair to \(\varpi^{-2}\cdot\)unit.
  1. Computation (different method from Correction 10.6's Lagrangian enumeration). For \(m=3\) and every \(b'\ge0\) with \(|b'|\le5\), all lattices \(\Lambda_0\subseteq\Lambda''\subseteq\Lambda_0^\#\) on which \(x_{b'}\) is integral were enumerated in Hermite normal form over \(\mathbb F_q[\![t]\!]\) (integrality conditions solved column by column as affine systems), and their types computed by Smith reduction. At \(q=2,3,5\) the \(|b|=4\) entries equal the table of step 4, and the \(|b|\le3\) entries are: \(N(\lambda^*(1),(1))=q+1\); column \((2,0,0)\): \(q(q+1)\) and \(1\) (rows \(\lambda^*(2)\), \(\lambda^*(1,1)\)); column \((1,1,0)\): \((q+1)(q^2+1)\); column \((3,0,0)\): \(q^2(q+1)\), \(q+1\); column \((2,1,0)\): \(q^2(q+1)^2\), \(q+1\) (rows \(\lambda^*(2,1)\), \(\lambda^*(1,1,1)\)); column \((1,1,1)\): \((q+1)(q^2+1)(q^3+1)\); all other entries \(0\). Lemma 2 and the zero pattern of Lemma 3 hold in every computed row.
  1. The \(|b|=5\) table (columns \((5,0,0),(4,1,0),(3,2,0),(3,1,1),(2,2,1)\)), equal to the computed values at \(q=2,3\) and, except the last column, \(q=5\):
    • \(\lambda^*(5,0,0)\): \(q^4(q+1),0,0,0,0\);
    • \(\lambda^*(4,1,0)\): \(q^2(q+1),\ q^4(q+1)^2,0,0,0\);
    • \(\lambda^*(3,2,0)\): \(q+1,\ q(q-1)(q+1)^2,\ q^5(q+1)^2,0,0\);
    • \(\lambda^*(3,1,1)\): \(0,\ q(q+1)^2,\ q^4(q+1),\ q^4(q+1)^2(q^2+1),\ 0\);
    • \(\lambda^*(2,2,1)\): \(0,\ q+1,\ q^2(q+1)^2,\ q(q-1)(q+1)^2(q^2+1),\ q^5(q+1)^2(q^2+1)\);
    • \((2,1,1,1,0,0)\): \(0,0,\ q+1,\ (q+1)^2(q^2+1),\ q^2(q+1)^2(q^2+1)\); \((1,1,1,1,1,0)\): \(0,0,0,0,\ q+1\). The triangular shape of Lemma 3 is visible (row \(\lambda^*(3,1,1)\) vanishes at \((5,0,0)\) although \((5,0,0)\ge_{\mathrm{dom}}(3,1,1)\): Lemma 3 is only a necessary condition). Heuristic: every computed diagonal entry equals \(q^{\sum_jjb_j-\sum_vk_v(k_v+1)/2}\prod_v\prod_{i=1}^{k_v}(q^i+1)\), \(k_v\) the multiplicity of the part \(v>0\).
  1. What the error affected. Exactly the two diagonal cells and \(\ast\): Lemma 5.6 kept only Lagrangians split along \(A\perp C\). The zero pattern, Lemmas 1–3 and therefore Theorem 5.7 are unaffected; Correction 10.6 is confirmed by an independent derivation and computation.

Builds on

  • Lemma 5.6 · \(\GL_{2m}/\Sp_{2m}\): the rows \(\lambda^*(b)=(b,0)\) are triangular for reverse dominance
  • Computation 5.5 · \(\GL_4/\Sp_4\): lattice counts and a Hecke element for the uncovered label \((1,1)\)
  • Correction 10.6 · Correction: the \(m=3\) check table for \(\GL_6/\Sp_6\), with the Lagrangians that are graphs

Built on by

  • Correction X1.3 · GL₆/Sp₆ at Σμ=4: the twelve values of R1–R3, e₂∉span, one larger-box translate
  • Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
  • Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes
  • Re-check V1 · Re-check by V1
  • Summary 9.6 · Verdict after the repairs: every input held or repaired; \(E_6/F_4\) still the one divergence

Sources