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ANNALES DE
L’INSTITUT FOURIER
Yuri G. Zarhin
Corrigendum to “Tate classes on self-products of Abelian
varieties over finite fields”
Tome 75, n
o
6 (2025), p. 2317-2320.
https://doi.org/10.5802/aif.3724
Article mis à disposition par son auteur selon les termes de la licence
Creative Commons attribution – pas de modification 3.0 France
http://creativecommons.org/licenses/by-nd/3.0/fr/
C EN T R E
MER S ENN E
Les Annales de l’Institut Fourier sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org e-ISSN : 1777-5310
Ann. Inst. Fourier, Grenoble
75, 6 (2025) 2317-2320
CORRIGENDUM TO “TATE CLASSES ON
SELF-PRODUCTS OF ABELIAN VARIETIES OVER
FINITE FIELDS”
Annales de l’Institut Fourier, vol. 72 (2022), no 6, 2339-2383
by Yuri G. ZARHIN
Abstract. — We fix an inaccuracy in the proof of Proposition 6.6 in the original
paper.
Résumé. — Nous corrigeons une imprécision dans la démonstration de la pro-
position 6.6 de l’article original.
Proposition 6.6 of [1] asserts the equivalence of certain two properties,
(a) and (b). Its proof in [1, p. 2369–2371] consists of two parts. First, we
proved that (b) ⇒ (a). Second, we proved that not (a) ⇒ not (b), which
is actually the same! (1) The aim of this note is to provide a missing proof
of the implication (a) ⇒ (b).
We will freely use the notation of [1, §6.3]. In particular, Z+ is the set of
all nonnegative integers, E is a field of characteristic zero, VE is a nonzero
finite-dimensional E-vector space, V ∗E is its dual, AE : VE → VE is an invert-
ible diagonalizable linear operator, spec(A) ⊂ E is the set of its eigenvalues,
and multA : spec(A) → Z+ is the map that assigns to each eigenvalue of AE
its multiplicity. Let us fix an eigenbasis B of VE (with respect to AE) and
choose an order on B. We write π : B → spec(A) for the (surjective) map
that assigns to an eigenvector the corresponding eigenvalue; if α ∈ spec(A)
then π−1(α) consists of multA(α) elements [1, p. 2367].
If j ⩽ dimE(VE) is a positive integer then we assign to each j-element
subset C of B a certain element yC ∈
∧j
E(V ∗E) that is an eigenvector of∧j(A∗E) with eigenvalue ∏x∈C π(x); all such yC ’s consitute an eigenbasis
Keywords: Abelian varieties, Tate classes, Finite fields.
2020 Mathematics Subject Classification: 11G10, 11G25, 14G15.
(1) I am grateful to Sergey Rybakov for pointing it out.
2318 Yuri G. ZARHIN
of
∧j
E(V ∗E) [1, p. 2367]. We will need functions
eC : spec(A) −→ Z+, α 7−→ #{x ∈ C | π(x) = α}.
If λ ∈ E then
∧j
E(V ∗E)(λ) stands for the eigenspace of
∧j(A∗E) attached to
the eigenvalue λ. In order to describe explicitly a basis of
∧j
E(V ∗E)(λ), let
us consider functions e : spec(A) → Z+ that enjoy the following properties:
(i) e(α) ⩽ multA(α) ∀ α ∈ spec(A);
(ii)
∑
α∈spec(A) e(α) = j;
(iii)
∏
α∈spec(A) α
e(α) = λ.
Then the set of yC ’s such that eC enjoys the properties (i)–(iii), is a basis
of
∧j
E(V ∗E)(λ) [1, p. 2367].
Let λ1, λ2 ∈ K where K is a subfield of E. Assuming that j ⩾ 2, let us
consider the E-linear map
(6.10)
j−2∧
E
(V ∗E)(λ1)⊗E
2∧
E
(V ∗E)(λ2) −→
j∧
E
(V ∗E)(λ1λ2), ψ⊗ϕ 7−→ ψ∧ϕ.
By [1, Lem. 6.9], the image of (6.10) is generated by all yC ’s where C is
any j-element subset of B that enjoys the following properties.
The set C is a disjoint union of a (j−2)-element subset S and a 2-element
subset T such that the corresponding functions
eS : spec(A) −→ Z+, eT : spec(A) −→ Z+
enjoy the following properties.
(6.11)
∏
α∈spec(A)
αeS(α) = λ1,
∏
α∈spec(A)
αeT (α) = λ2.
The condition (a) of [1, Prop. 6.6] is equivalent to the non-surjectiveness
of (6.10).
The condition (b) of [1, Prop. 6.6] is the existence of a function e : spec(A) →
Z+ such that:
(bi) e(α) ⩽ multA(α) ∀ α ∈ spec(A);
(bii)
∑
α∈spec(A) e(α) = j;
(biii)
∏
α∈spec(A) α
e(α) = λ1λ2;
(biv) If α ∈ spec(A) and e(α) ̸= 0 then e(α) ⩾ 1 and one of the following
conditions holds.
(1) λ2/α ̸∈ spec(A);
(2) λ2/α ∈ spec(A) but e(λ2/α) = 0;
(3) α = λ2/α (i.e., α2 = λ2) and e(α) = 1.
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM TO “TATE CLASSES ON ABELIAN VARIETIES” 2319
Proposition 6.6 may be restated (see Remark 6.8 of [1]) as an equivalence
of the non-surjectiveness of (6.10) and property (b). Our proof is based on
the following assertion.
Lemma 6.9 of [1]. — The image of the map (6.10) is generated by
all yC ’s where C is any j-element subset of B that enjoys the following
properties.
The set C is a disjoint union of a (j−2)-element subset S and a 2-element
subset T such that the corresponding functions
eS : spec(A) −→ Z+, eT : spec(A) −→ Z+
enjoy the following properties.
(6.11)
∏
α∈spec(A)
αeS(α) = λ1,
∏
α∈spec(A)
αeT (α) = λ2.
Proof of Proposition 6.6 of [1]. — The implication (b) ⇒ (a) is proven
in [1, p. 2370]. Conversely, suppose that (a) holds, i.e., (6.10) is not surjec-
tive. Then there is a basis vector yC ∈
∧j
E(V ∗E)(λ1λ2) that does not belong
to the image where C is a certain j-element subset of B such that∏
x∈C
π(x) = λ1λ2.
I claim that the function e := eC enjoys the properties (b). Indeed, by
definition, eC enjoys the propetries (bi), (bii), (biii). In particular,∏
α∈C
αeC(α) =
∏
x∈C
π(x) = λ1λ2.
Suppose that eC does not enjoy the property (biv). This means that
there exists α ∈ spec(A) such that eC(α) ̸= 0 (i.e., α ∈ π(C)) and
λ2
α
∈ spec(A), eC
(
λ2
α
)
̸= 0
(i.e., λ2/α ∈ π(C) ⊂ spec(A)). In addition, if λ2/α = α then eC(α) ⩾ 2
(i.e., π−1(α) contains at least two elements of C).
This implies that there are distinct elements x1 and x2 of C such that
π(x1) = α, π(x2) =
λ2
α
.
Then C coincides with the disjoint union of the 2-element subset T =
{x1, x2} and the (j − 2)-element subset S = C \ T . By definition of T ,∏
α∈T
αeT (α) =
∏
x∈T
π(x) = π(x1) · π(x2) = α ·
λ2
α
= λ2.
TOME 75 (2025), FASCICULE 6
2320 Yuri G. ZARHIN
Hence, ∏
α∈T
αeT (α) = λ2.
Since C is the disjoint union of S and T , we get∏
α∈S
αeS(α) =
∏
α∈C α
eC(α)∏
α∈T α
eT (α)
= λ1λ2
λ2
= λ1.
Hence, ∏
α∈S
αeS(α) = λ1.
It follows from Lemma 6.9 that yC lies in the image of the map (6.10),
which is not true. The obtained contradiction implies that eC enjoys the
property (biv). This ends the proof of Proposition 6.6. □
BIBLIOGRAPHY
[1] Y. G. Zarhin, “Tate classes on self-products of abelian varieties over finite fields”,
Ann. Inst. Fourier 72 (2022), no. 6, p. 2339-2383.
Manuscrit reçu le 7 janvier 2025,
révisé le 20 février 2025,
accepté le 20 mars 2025.
Yuri G. ZARHIN
Department of Mathematics
Pennsylvania State University
University Park, PA 16802, USA
zarhin@math.psu.edu
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