CORRECTIONS TO ”SPACE FILLING CURVES OVER FINITE FIELDS”

NICHOLAS M. KATZ

1. Introduction

Ofer Gabber has kindly pointed out to me that the proof of Lemma 5 is wrong. The error occurs in the last paragraph of the proof. The first sentence of that paragraph makes a false statement about Frobenius elements (starting with ”so every...”). This false statement is used in the following sentence (”Therefore...”) to assert that, for r ≥ r_{0}, the conditions (⋆ ⋆ r, D_{r}, C_{r}) and (⋆ ⋆ r, E/W) are equivalent. It is indeed trivially true that the first condition implies the second, but the converse need only hold if in addition r is divisible by N, the order of the cyclic group Γ/Γ_{geom}. The effect of correcting this error is that in Lemmas 4, 5, and 6, and in Corollary 7, what is asserted to hold for r sufficiently large holds only for r sufficiently large and sufficiently divisible. Indeed, Gabber has constructed examples to show that Lemma 6 and Corollary 7 can be false without this extra proviso. In the corrections below, we also modify the statement of Lemma 5 so that its new, weaker conclusion applies in a more general setting.

2. Corrections and modifications to statements of results

page 616, assertion 2) of Lemma 4: should read ”For all sufficiently large and sufficiently divisible r, D_{r}/k is geometrically connected.”.

page 617, lines 7-8 of the statement of Lemma 5 (i.e., lines 13-14 on the page): should read ”r ≥ r_{0}, a smooth, geometrically connected k-scheme C_{r} and a kmorphism i_{r} : C_{r} → W which is surjective on k_{r}-valued points. Form the fibre”

page 617, line 12 of the statement of Lemma 5 (i.e., line 18 on the page): the label of the map in the diagram should be i_{r} and not i_{r,W}.

page 617, last line of the statement of Lemma 5: should read ”Then for r sufficiently large and sufficiently divisible , the fibre product D_{r}/k is geometrically connected.”

page 619, assertion a) of Lemma 6: should read ”For r sufficiently large and sufficiently divisible , we have an equality of images of geometric fundamental groups

ρ_{r}(π_{1}^{geom}(C_{r}, c_{r})) = ρ(π_{1}^{geom}(W, w))

(equality inside G).

Date: July 19, 2001.

page 619, assertion b) of Lemma 6: should read (in its entirety) ”For r sufficiently large and sufficiently divisible, we have an equality of images of fundamental groups

ρ_{r}(π_{1}(C_{r}, c_{r})) = ρ(π_{1}(W, w))

(equality inside G).

page 620, lines -5 and -2 in the statement of Corollary 7 (i.e. lines -10 and -6 on the page): should read respectively ”Then (resp. then) for r sufficiently large and sufficiently divisible, the pullback sheaf (f_{r})^{⋆}(F) on C_{r} has the same...”

3. Corrections to the proof of Lemma 5

page 618, line 15: should read ”We will show that for any r ≥ r_{0} which is divisible by N and which is large enough that (⋆ ⋆ r, E/W) holds, D_{r} is”

page 618, line -2, through page 619, line 2: Delete the last paragraph of the proof. Replace it by the following text.

We will show that for any r ≥ r_{0} which is divisible by N and for which (⋆⋆r, E/W) holds, the condition (⋆ ⋆ r, D_{r}, C_{r}) holds. We now fix one such r.

The subgroup π_{1}(W ⊗_{k}k_{r}, w) of π_{1}(W, w) maps, by ρ, to the subgroup Γ(0) = Γ_{geom}, simply because N divides r . Because this subgroup contains π_{1}^{geom}(W, w), it maps onto Γ_{geom}. This subgroup contains all the Frobenius elements in π_{1}(W, w) attached to k_{r}-valued points of W. By (⋆ ⋆ r, E/W), the images of these elements fill Γ_{geom}. By the spacefilling property, each of these elements is π_{1}(W ⊗_{k} k_{r}, w)-conjugate to the image of a Frobenius element in π_{1}(C_{r} ⊗_{k} k_{r}, c_{r}) attached to a k_{r}-valued point of C_{r}. The group π_{1}(C_{r} ⊗_{k} k_{r}, c_{r}) maps to Γ_{geom}, and the images of its Frobenius elements attached to k_{r}-valued points of C_{r} meet every conjugacy class in Γ_{geom}. So by Jordan’s theorem, we conclude that π_{1}(C_{r} ⊗_{k} k_{r}, c_{r}) maps onto Γ_{geom}. Thus every element in Γ_{geom} is the image of a π_{1}(C_{r}⊗_{k}k_{r}, c_{r})-conjugate of some Frobenius element attached to a k_{r}-valued point of C_{r} (because the images of these Frobenius elements meet every conjugacy class in Γ_{geom}, and π_{1}(C_{r} ⊗_{k} k_{r}, c_{r}) maps onto Γ_{geom}.) But the π_{1}(C_{r} ⊗_{k} k_{r}, c_{r})-conjugates of Frobenius elements attached to k_{r}-valued points of C_{r} are themselves such Frobenius elements, and thus (⋆ ⋆ r, D_{r}, C_{r}) holds.

We remark that from the fact that π_{1}(C_{r} ⊗_{k} k_{r}, c_{r}) maps onto Γ_{geom}, it follows that π_{1}(C_{r}, c_{r}) maps onto Γ, simply because any element of degree one in the source maps onto a generator of the cyclic quotient Γ/Γ_{geom}.

4. Correction to the proof of Lemma 6

page 619, line -3: the sentence should end ”for r divisible by N and r ≫ 0.”

page 619, line -1: the sentence should end ”whence a) and b), cf. the corrected proof of Lemma 5.”

page 620, delete lines 1-4.

page 620, line 10: add the words ”and sufficiently divisible” after the phrase ”for r ≫ 0”.

5. Correction to the proof of Theorem 8

page 621, line 7: should read ”By Lemma 4, for large r which is sufficiently divisible, this closed subscheme D_{rd} of V is a smooth”

6. Counterexamples

In this section, we construct, following ideas of Ofer Gabber, counterexamples to the uncorrected versions of Lemma 6 and Corollary 7. We work over the finite field k = 𝔽_{q}. Let G/k be a smooth, geometrically connected, commutative group-scheme of dimension n ≥ 1, and

ϕ : G → G

a finite etale homomorphism of k-group-schemes of degree deg(ϕ) ≥ 2 with the following property: the finite etale commutative k-group-scheme Ker(ϕ) has no nontrivial k-rational points, i.e., Ker(ϕ)(k) = {e}. Here are three elementary examples of such situations (G, ϕ).

(1) Pick α ≠ 1 in 𝔽_{q}^{×}, take G = 𝔾_{a}, ϕ(x) := x^{q} − αx. Here deg(ϕ) = q.

(2) Pick a prime number l ≥ q + 1, take G = 𝔾_{m}, ϕ(x) := x^{l}. Here deg(ϕ) = l.

(3) Pick a prime number l ≥ 2 + q + 2\sqrt{q} and an elliptic curve E/𝔽_{q}, take G = E, ϕ(P) := lP. Here deg(ϕ) = l^{2}.

The geometric Frobenius F_{k} in Gal(k̄/k) acts as a group-automorphism of the finite group Ker(ϕ)(k̄). Denote by N the order of this automorphism. By hypothesis, F_{k} has no fixed points in Ker(ϕ)(k̄)−{e}. Therefore, for any integer r ≥ 1 with (r, N) = 1, (F_{k})^{r} has no fixed points in Ker(ϕ)(k̄)−{e}. So for any such r, Ker(ϕ)(k_{r}) = {e}, i.e., the map

ϕ : G → G

is injective, and hence bijective, on k_{r}-valued points.

In the notations of Lemma 6 and Corollary 7, take W = G, with base point w = e. For r ≥ 1 with (r, N) = 1, take f_{r} : C_{r} → W to be ϕ : G → G; for other r, take it to be id : G → G. Take c_{r} = e as base point in C_{r} = G, and take as ”chemin” from f_{r}(c_{r}) = e to e the identity. To make counterexamples to Lemma 6 and Corollary 7, we need only exhibit a finite etale covering

π : E → G

which is geometrically nontrivial, but whose pullback by ϕ : G → G is geometrically trivial. For the monodromy representation of such a covering will violate conclusion a) of Lemma 6 for every r ≥ 1 with (r, N) = 1, and the lisse sheaf F := π_{⋆}ℚ̄_{l} on G will violate the first conclusion of Lemma 7 for the same r. Such a π : E → G is given by ϕ : G → G.