S1-3 · 2026-10-21T15:11:19Z
The correct replacement for the even-extension assertion
For an invertible finite-dimensional endomorphism \(A\), \(\operatorname{tr}A=0\) implies neither \(\operatorname{tr}A^{2}\ne0\) nor a zero row of a multiplicity matrix. For a finite-type \(V/\mathbb F_q\) with \(\dim V\le f\), if
$$\#V(\mathbb F_{q^n})=c q^{nf}+O(q^{n(f-1)})\quad(n\to\infty),\qquad c>0,$$
then \(\dim V=f\), it has exactly \(c\) geometric top components, and all are Frobenius fixed. Thus Frobenius acts by \(q^f\) on \(H_c^{2f}(V)\). This second assertion uses the uniform-in-extensions point-count estimate, not a parity argument.
- A three-cycle permutation matrix has eigenvalues \(1,\zeta_3,\zeta_3^2\) and trace \(0\) at both \(n=1\) and \(n=2\); its power trace is \(3\) exactly when \(3\mid n\). A transposition has trace \(0\) at odd powers and \(2\) at even powers. The first trace alone does not distinguish these behaviors.
- A coefficient is \(m^{(n)}_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}(A_{\nu,i}(\lambda)^n)\). Vanishing of one summand does not force even that coefficient to vanish, since other degrees may contribute. Even vanishing of that coefficient says nothing about other columns in its row. Only the separately checked vanishing of all its entries forces a zero row.
- For the point-count assertion, let \(P\) be Frobenius's finite permutation on the geometric \(f\)-dimensional components. The finite-type Lang–Weil estimate used in A5-8, applied along each extension-field congruence class fixing the components, gives
$$q^{-nf}\#V(\mathbb F_{q^n})=\#\operatorname{Fix}(P^n)+O(q^{-n/2}).$$
Together with the assumed uniform leading coefficient, the integer \(\#\operatorname{Fix}(P^n)\) is eventually constantly \(c\). Let \(h\) be the order of \(P\). Arbitrarily large \(n\equiv0\pmod h\) show that the total number of components is \(c\); arbitrarily large \(n\equiv1\pmod h\) show that \(P\) fixes all of them. Positive \(c\) also forces \(\dim V=f\). Now apply S1-1.
- Accordingly delete all generic “the row becomes nonzero at even \(n\)” language from A5-11 (and the originating remark in A3-11). The conditional split example still has scalar top Frobenius: its relevant finite-dimensional fibres and uniform extension-field counts are precisely the hypotheses independently checked in A5-8, and the identification of the multiplicity space with their top compact cohomology is the separate conditional support argument in A5-9.