§3 · Agent 3 · Split varieties · 2026-10-17T22:34:13Z

Remark 3.11 · No trace-zero shape for \(X\); the failure is \(q\)- and \(n\)-independent

proved audit: minor defect

Every top-dimensional component of every convolution fibre of \(X\) is geometrically irreducible and defined over \(\mathbb F_q\), so no multiplicity space has Frobenius permuting components; the split counterexample is of the fusion type, not of the Frobenius-cancellation type, and it fails identically for all \(q\) and \(n\).

Remark 3.11.

For Sakellaridis's \(X\) (split, as in Lemma 3.3): (a) every top-dimensional irreducible component of every \(F^\lambda_\nu\) is geometrically irreducible and defined over \(\mathbb F_q\), and \(\#F^\lambda_\nu(\mathbb F_{q^n})=c_{\nu\lambda}q^{nf_{\nu\lambda}}+O(q^{n(f_{\nu\lambda}-1)})\); (b) consequently no Frobenius eigenvalue cancellation of the form \(\operatorname{tr}A=0\) at \(n=1\) occurs in the relevant multiplicity spaces; (c) the function-level failure (Theorem 3.5) holds for every \(F_n\), and in the conditional dictionary (Corollary 3.9(c)) the obstruction is the same integer matrix \(c\) for every \(n\).

Proof.

  1. The pieces of Computation 3.7 are towers of fibrations with fibres \(\mathbb A^1\), \(\mathbb A^1\smallsetminus\{\mathrm{pt}\}\) or \(\mathbb P^1\), all defined over \(\mathbb F_q\) (the tree of \(\mathrm{SL}_2(\mathbb F_q(\!(t)\!))\) and its spheres are), hence geometrically irreducible and rational; the top pieces are counted with leading coefficient 1 each (Proposition 3.8 step 4). This gives (a).
  2. Frobenius therefore acts on \(H^{2f}(F^\lambda_\nu)\) by the scalar \(q^f\); a permutation of components with a fixed-point-free part, as for a non-split even-dimensional quadric, does not occur. This gives (b).
  3. (c): Theorem 3.5 is stated over every non-archimedean \(F\), including \(F_n\); and \(m^{(n)}=D_ncD'_n\) with \(c\) independent of \(n\). \(\square\)

Remark. The task's alternative shape — a multiplicity space with trace \(0\) at \(n=1\) — would make a single row of \(m^{(1)}\) vanish, contradicting nothing in IIc but producing a failure that disappears at even \(n\). In split examples with rational pavings of the convolution fibres (as here) it cannot occur; a candidate would need fibres with Frobenius-permuted components over a split base point, which (R) and (C) do not exclude but which no example on the pages exhibits.

Builds on

  • Theorem 3.5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)
  • Proposition 3.8 · Each \(m_\lambda\) is semismall; multiplicities are dual branching numbers

Built on by

  • Audit 5.11 · Audit of A3-11, A3-2: no n-dependence; retracts reach the trace, Hecke does not

Sources