1 Introduction
Let k be a field, and let X be a smooth and projective variety over k. Write
$Z^{n}(X)$
for the group of codimension n algebraic cycles on X, and let
$Z^{n}_{\mathrm {rat}}(X)\subseteq Z^{n}_{\mathrm {alg}}(X)\subseteq Z^{n}_{\mathrm {hom}}(X)$
be the subgroups of
$Z^{n}(X)$
consisting of those cycles which are rationally (resp. algebraically, resp. homologically (with respect to a fixed Weil cohomology theory; see Remark 2.1)) equivalent to zero. Let
$\mathrm {CH}^{n}(X)\supseteq \mathrm {CH}^{n}_{\mathrm {hom}}(X)\supseteq \mathrm {CH}^{n}_{\mathrm {alg}}(X)$
denote the quotients of
$Z^{n}(X)\supseteq Z^{n}_{\mathrm {hom}}(X)\supseteq Z^{n}_{\mathrm {alg}}(X)$
by
$Z_{\mathrm {rat}}(X)$
. Let
$\mathrm {Griff}^{n}(X):=\mathrm {CH}_{\mathrm {hom}}^{n}(X)/\mathrm {CH}_{\mathrm {alg}}^{n}(X)$
be the Griffiths group of codimension n cycles on X.
Griffiths [Reference GriffithsGri69] was the first to show that smooth projective varieties can have nontrivial Griffiths groups—
$\mathrm {Griff}^{2}(X)\otimes \mathbb {Q}$
is nontrivial for a very general quintic hypersurface
$X\subset \mathbb {P}_{\mathbb {C}}^{4}$
. Clemens [Reference ClemensCle83] later showed that such hypersurfaces have
$\mathrm {dim}_{\mathbb {Q}}(\mathrm {Griff}^{2}(X)\otimes \mathbb {Q})=\infty $
, and then Voisin [Reference VoisinVoi00] generalized this by proving that
$\mathrm {dim}_{\mathbb {Q}}(\mathrm {Griff}^{2}(X)\otimes \mathbb {Q})=\infty $
for any very general Calabi–Yau threefold X over
$\mathbb {C}$
. Since we are interested in abelian varieties, let us also point out that Ceresa [Reference CeresaCer83] has shown that, for a very general curve C of genus
$\geq 3$
, the Ceresa cycle is a nontrivial element in
$\mathrm {Griff}^{2}(J(C))$
. In fact, it was shown in [Reference ShiodaSch22] that the torsion subgroup of
$\mathrm {Griff}^{n}$
need not even be finitely generated (at least for
$n\geq 3$
, with the situation for
$n=2$
currently open). This phenomena is not specific to “large” base fields either: Harris [Reference HarrisHar83] has given an explicit abelian threefold defined over
$\mathbb {Q}$
with nontrivial Griffiths group—the Ceresa cycle on the Jacobian of the Fermat quartic is not algebraically equivalent to zero. Bloch [Reference BlochBl84, Theorem 4.1] gave a different proof of Harris’ result on the Jacobian of the Fermat quartic, and showed moreover that the Ceresa cycle is nontorsion. In general, for a smooth projective variety defined over a number field, it is part of the Bloch–Beilinson conjectures that the dimensions of the Griffiths groups tensor
$\mathbb {Q}$
are finite, and the dimensions are controlled by the orders of vanishing at integers of appropriate L-functions [Reference BlochBl85, p. 381].
In this note, we are interested in the situation where the base field has positive characteristic. In this setting, again the Griffiths groups of smooth projective varieties can be nontrivial and even infinite. For example, Schoen [Reference SchoenSch95, Theorem 0.1] showed that if k is a finite field of characteristic
$p\equiv 1\mod 3$
and E denotes the Fermat cubic, then
$\mathrm {Griff}^{2}(E^{3}_{\overline {k}})$
is nontrivial and has a nontrivial divisible part. Here,
$\overline {k}$
denotes the algebraic closure of k, and subscript
$\overline {k}$
means the base change to
$\overline {k}$
.
Now, let k be a perfect field with
$\mathrm {char}(k)=p>0$
. The positive characteristic analog of the Bloch-Beilinson philosophy says that cycles on varieties, at least after tensoring with
$\mathbb {Q}$
, are controlled by the slopes of the Frobenius on crystalline cohomology. Recall that a smooth proper variety X over k is called ordinary if
$H^{m}(X,d\Omega ^{r}_{X/k})=0$
for all
$m,r$
. When the crystalline cohomology groups
$H_{\mathrm {cris}}^{n}(X/W(k))$
of X are torsion-free, X is ordinary if and only if, for each n, the Newton polygon of X coincides with the Hodge polygon [Reference Bloch and KatoBK86, Proposition 7.3]. If A is an abelian variety, then A is ordinary if and only if
$A(\overline {k})[p]=(\mathbb {Z}/p\mathbb {Z})^{\dim A}$
. For example, the condition that
$p\equiv 1\mod 3$
in Schoen’s theorem forces
$E_{\overline {k}}^{3}$
to be an ordinary abelian threefold. Generalizing Schoen’s result, under the rubric of the Tate conjecture for surfaces over finite fields, Brent Gordon and Joshi [Reference Brent Gordon and JoshiBGJ02, Proposition 6.2] proved that the codimension
$2$
Griffiths group of ordinary abelian threefolds over the algebraic closure of a finite field are nontrivial, and contain a nontrivial divisible part.
At the opposite extreme to ordinarity is supersingularity, and in this situation, the Bloch–Beilinson philosophy suggests that Griffiths group should be smaller because of the extreme degeneracy in the slopes of Frobenius. Recall that a smooth projective variety X over k is said to be supersingular if the Newton polygons of X are isoclinic. If A is an abelian variety, then A is supersingular if and only if
$A_{\overline {k}}$
is isogenous to the self-product of an (any!) supersingular elliptic curve, where an elliptic curve E is supersingular if and only if
$E(\overline {k})[p]=0$
(see [Reference OortOor74, Theorem 4.2]). Schoen [Reference SchoenSch95, Theorem 14.4] showed that if k is a finite field of characteristic
$p\equiv 2\mod 3$
and E denotes the Fermat cubic, then
$\mathrm {Griff}^{2}(E^{3}_{\overline {k}})$
is at most a p-primary torsion group. The condition that
$p\equiv 2\mod 3$
implies that
$E^{3}$
is a supersingular abelian threefold. Using the work of Fakhruddin [Reference FakhruddinFak02], Brent Gordon and Joshi [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1] generalized Schoen’s result to all supersingular abelian varieties—the codimension
$2$
Griffiths groups of supersingular abelian varieties defined over the algebraic closure of a finite field are at most p-primary torsion. The question of whether these groups possess nontrivial p-torsion was left open.
In this note, we prove the following (see Theorem 4.1(1)).
Theorem 1.1 Let k be an algebraically closed field of characteristic
$p>0$
, and let A be a supersingular abelian variety over k. Then the inclusions
$$ \begin{align*} \mathrm{CH}^{2}_{\mathrm{alg}}(A)_{\mathrm{tors}}\subseteq\mathrm{CH}^{2}_{\mathrm{hom}}(A)_{\mathrm{tors}}\subseteq\mathrm{CH}^{2}(A)_{\mathrm{tors}} \end{align*} $$
are equalities. (Here,
$G_{\mathrm{tors}}$
denotes the torsion subgroup of the group G.)
It was already shown in the proof of [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1] that
$\mathrm {CH}^{2}_{\mathrm {alg}}(A)[\ell ^{\infty }]=\mathrm {CH}^{2}_{\mathrm {hom}}(A)[\ell ^{\infty }]=\mathrm {CH}^{2}(A)[\ell ^{\infty }]$
for each prime
$\ell \neq p$
(where
$G[\ell ^{\infty }]$
denotes the
$\ell $
-primary torsion subgroup of the group G). Our only new result is that this is also true for p-primary torsion. To handle the
$\ell =p$
case, we initially follow the proof of Brent Gordon and Joshi, but then conclude using an inductive argument based on the Bloch–Srinivas method [Reference Bloch and SrinivasBS83].
As a consequence of Theorem 1.1, we settle the p-primary torsion case of [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1]. Indeed, we have the following corollary (see Theorem 4.1(2)).
Theorem 1.2 Let k be a finite field of characteristic
$p>0$
, and let A be a supersingular abelian variety over
$\overline {k}$
. Then
$\mathrm {Griff}^{2}(A)[p^{\infty }]$
is trivial.
Together with [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1], this shows that
$\mathrm {Griff}^{2}(A)$
is trivial. That is, homological equivalence coincides with algebraic equivalence for codimension
$2$
cycles on supersingular abelian varieties over the algebraic closure of finite fields.
2 Chow groups of supersingular abelian varieties
We repeat the discussion from [Reference Brent Gordon and JoshiBGJ02, Sections 2 and 3]. Let A be an abelian variety of dimension g over a field k, and let n be a nonnegative integer. Then, by the work of Mukai [Reference MukaiMuk81], Beauville [Reference BeauvilleBea86], and Deninger–Murre [Reference Deninger and MurreDM91], the rational Chow groups of A admit a direct sum decomposition
$$ \begin{align*} \mathrm{CH}^{n}(A)\otimes\mathbb{Q}=\bigoplus_{i}\mathrm{CH}^{n}_{i}(A), \end{align*} $$
where
$\mathrm {CH}^{n}_{i}(A):=\{Z\in \mathrm {CH}^{n}(A)\otimes \mathbb {Q}\,:\,m_{A}^{\ast }(Z)=m^{2n-i}Z\text { for all }m\in \mathbb {Z}\}$
and
$m_{A}^{\ast }$
denotes the flat pullback by multiplication-by-m on A.
Now, suppose that k is an algebraically closed field of characteristic
$p>0$
. Then Fakhruddin [Reference FakhruddinFak02] has proved that if A is a supersingular abelian variety over k,
$\mathrm {CH}^{n}_{i}(A)=0$
, for
$i\neq 0,1$
. Moreover, the
$\ell $
-adic cycle class map induces an isomorphism
$$ \begin{align*} \mathrm{CH}^{n}_{0}(A)\otimes\mathbb{Q}_{\ell}\xrightarrow{\sim}H^{2n}_{\mathrm{\acute{e}t}}(A,\mathbb{Q}_{\ell}(n)) \end{align*} $$
for all primes
$\ell \neq p$
. The same proof shows that the crystalline cycle class map induces an isomorphism
$$ \begin{align*} \mathrm{CH}^{n}_{0}(A)\otimes K\xrightarrow{\sim}H_{\mathrm{cris}}^{2n}(A/W(k))\otimes_{W(k)}K, \end{align*} $$
where
$K=W(k)[1/p]$
is the fraction field of the Witt vectors
$W(k)$
of k. In particular,
$\mathrm {CH}^{n}_{1}(A)=\mathrm {CH}^{n}_{\mathrm {hom}}(A)\otimes \mathbb {Q}$
, where
$\mathrm {CH}^{n}_{\mathrm {hom}}(A)$
is the kernel of the cycle class map.
Remark 2.1 A priori, the definition of the group
$\mathrm {CH}^{n}_{\mathrm {hom}}(X)$
of codimension n cycles homologically equivalent to zero on a smooth projective variety X depends on the choice of Weil cohomology theory for X. Of course, it is a consequence of the standard conjectures (specifically that homological equivalence coincides with numerical equivalence) that
$\mathrm {CH}^{n}_{\mathrm {hom}}(X)$
is independent of the choice of Weil cohomology theory. Notice, though, that Fakhruddin’s result shows that
$\mathrm {CH}^{n}_{\mathrm {hom}}(A)$
is independent of any choice when A is a supersingular variety over an algebraically closed field of characteristic
$p>0$
. Since this is the setting that we are interested in, there is no ambiguity in the definition.
As pointed out in [Reference Brent Gordon and JoshiBGJ02, Section 2], if k is moreover the algebraic closure of a finite field, then it is known by the results of Soulé [Reference SouléSou84] and Künnemann [Reference KünnemannKün93] that
$\mathrm {CH}^{n}_{1}(A)=0$
. In particular,
$\mathrm {CH}^{n}_{\mathrm {hom}}(A)$
is torsion.
Remark 2.2 Beilinson [Reference BeilinsonBei87, 1.0] has conjectured that
$\mathrm {CH}^{n}_{\mathrm {hom}}(X)$
is torsion for any smooth projective variety X over the algebraic closure of a finite field.
3 Abel–Jacobi maps
In this section, we fix notation involving
$\ell $
-adic Abel–Jacobi maps, for primes
$\ell $
(including
$\ell =p$
).
Let X be a smooth projective variety over an algebraically closed field k of characteristic
$p\geq 0$
. Let
$\ell $
be a prime. Define
$$ \begin{align*} H^{i}(X,\mathbb{Z}_{\ell}(j)):=\begin{cases} H^{i}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(j)), & \text{if }\ell\neq p \\ H^{i-j}(X_{\mathrm{\acute{e}t}},W\Omega^{j}_{X,\log}), & \text{if }\ell= p \end{cases} \end{align*} $$
and
$$ \begin{align*} H^{i}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(j)):=\begin{cases} H^{i}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(j)), & \text{if }\ell\neq p, \\ \displaystyle\varinjlim_{r}H^{i-j}(X_{\mathrm{\acute{e}t}},W_{r}\Omega^{j}_{X,\log}), & \text{if }\ell= p, \end{cases} \end{align*} $$
where
$W_{r}\Omega ^{j}_{X,\log }$
denotes the logarithmic Hodge–Witt sheaf of X (see [Reference IllusieIll79, Chapter I, 5.7]) and the limit is taken over the maps
$\underline {p}:W_{r}\Omega ^{j}_{X}\rightarrow W_{r+1}\Omega ^{j}_{X}$
[Reference IllusieIll79, Chapter I, Proposition 3.4]. Let
$$ \begin{align*} \lambda_{n,X}:\mathrm{CH}^{n}(X)[\ell^{\infty}]\rightarrow H^{2n-1}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(n)) \end{align*} $$
be Bloch’s
$\ell $
-adic Abel–Jacobi map [Reference BlochBlo79] if
$\ell \neq p$
, and the Gros–Suwa p-adic Abel–Jacobi map [Reference Gros and SuwaGS88] if
$\ell =p$
. Here,
$\mathrm {CH}^{n}(X)[\ell ^{\infty }]$
denotes the
$\ell $
-primary torsion subgroup of
$\mathrm {CH}^{n}(X)$
.
4 The result
Theorem 4.1 Let k be an algebraically closed field of characteristic
$p>0$
, and let A be a supersingular abelian variety over k. Then:
-
(1) We have
$$ \begin{align*} \mathrm{CH}^{2}_{\mathrm{alg}}(A)_{\mathrm{tors}}=\mathrm{CH}^{2}_{\mathrm{hom}}(A)_{\mathrm{tors}}=\mathrm{CH}^{2}(A)_{\mathrm{tors}}. \end{align*} $$
-
(2) If k is the algebraic closure of a finite field, then
$\mathrm {Griff}^{2}(A)$
is trivial, i.e.,
$$ \begin{align*} \mathrm{CH}^{2}_{\mathrm{alg}}(A)=\mathrm{CH}^{2}_{\mathrm{hom}}(A). \end{align*} $$
Proof (1) Let
$\ell $
be a prime. Consider the following commutative diagram:
where
$\lambda ^{\prime }_{2,A}$
denotes the restriction of
$\lambda _{2,A}$
. It is known that
$\lambda _{2,A}$
is injective (as a consequence of the Merkurjev–Suslin theorem [Reference Colliot-Thélène, Sansuc and SouléCTSS83, Corollary 4] for
$\ell \neq p$
, [Reference Gros and SuwaGS88, Section III, Proposition 3.4] for
$\ell =p$
), and hence
$\lambda ^{\prime }_{2,A}$
is injective. It therefore suffices to show that
$\lambda ^{\prime }_{2,A}$
is surjective, since then all maps in (4.1) are isomorphisms and in particular
$$ \begin{align*} \mathrm{CH}^{2}_{\mathrm{alg}}(A)[\ell^{\infty}]=\mathrm{CH}^{2}_{\mathrm{hom}}(A)[\ell^{\infty}]=\mathrm{CH}^{2}(A)[\ell^{\infty}] \end{align*} $$
for each prime
$\ell $
.
For any
$n\geq 0$
, consider the following diagram:
Up to a sign, the induced map
$\mathrm {CH}^{n}(A)[\ell ^{\infty }]\rightarrow H^{2n}(A,\mathbb {Z}_{\ell }(n))$
is the restriction of the cycle class map ([Reference Colliot-Thélène, Sansuc and SouléCTSS83, Corollary 4] for
$\ell \neq p$
, [Reference Gros and SuwaGS88, Section III, Propositions 1.16 and 1.21] for
$\ell =p$
). The bottom row of the diagram is exact (see [Reference Gros and SuwaGS88, (3.33)] for exactness when
$\ell =p$
). Therefore, the restriction of
$\lambda _{n,A}$
to
$\mathrm {CH}^{n}_{\mathrm {hom}}(A)[\ell ^{\infty }]$
has image in
$H^{2n-1}(A,\mathbb {Z}_{\ell }(n))\otimes _{\mathbb {Z}_{\ell }}\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
. In particular,
$\lambda ^{\prime }_{2,A}$
has image in
$H^{3}(A,\mathbb {Z}_{\ell }(2))\otimes _{\mathbb {Z}_{\ell }}\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
. Therefore, the cokernel of
$\lambda ^{\prime }_{2,A}$
is divisible. Note that
$H^{3}(A,\mathbb {Z}_{\ell }(2))$
is torsion-free (the nontrivial case when
$\ell =p$
follows from
$H^{3}_{\mathrm {cris}}(A/W(k))$
being torsion-free [Reference Gros and SuwaGS88, Lemme 3.12], which is because
$H^{1}_{\mathrm {cris}}$
is always torsion-free and
$H^{3}_{\mathrm {cris}}=\wedge ^{3}H^{1}_{\mathrm {cris}}$
for abelian varieties), and hence
$H^{3}(A,\mathbb {Z}_{\ell }(2))\otimes _{\mathbb {Z}_{\ell }}\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
is a direct sum of a finite number of copies of
$\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
.
We are therefore reduced to showing that
$\mathrm {coker}(\lambda ^{\prime }_{2,A})$
is annihilated by a positive integer. (Indeed,
$\mathrm {coker}(\lambda ^{\prime }_{2,A})$
is a quotient of the divisible group
$H^{3}(A,\mathbb {Z}_{\ell }(2))\otimes _{\mathbb {Z}_{\ell }}\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }\cong \left (\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }\right )^{r}$
for some r. If
$\mathrm {coker}(\lambda ^{\prime }_{2,A})$
is finite, then it must be trivial since any finite divisible group is trivial. So we must rule out the case that
$\mathrm {coker}(\lambda ^{\prime }_{2,A})$
is infinite, in which case it is a finite number of copies of
$\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
. However, this group is not annihilated by a positive integer.) We shall prove this by induction on the dimension g of A. Of course, the entire theorem is trivial if
$g=1$
, so suppose that
$g>1$
and suppose that
$\lambda ^{\prime }_{2,B}$
is surjective for supersingular abelian varieties of dimension
$\leq g-1$
. By [Reference FakhruddinFak02, Lemma 3] and its proof, there exist g-dimensional abelian subvarieties
$Y_{1},\ldots , Y_{n}\subset A\times A$
such that the class of the diagonal
$\Delta _{A}$
decomposes as
$$ \begin{align*} \Delta_{A}=\sum_{i}c_{i}[Y_{i}]\in\mathrm{CH}^{g}(A\times A)\otimes\mathbb{Q} \end{align*} $$
for some
$c_{i}\in \mathbb {Q}$
, and such that, for each i, the image of
$Y_{i}$
under at least one of the projections
$\mathrm {pr}_{1},\mathrm {pr}_{2}:A\times A\rightarrow A$
has dimension
$\leq g-1$
. By clearing denominators, we see that there exists an integer
$N>0$
such that
$$ \begin{align*} N\Delta_{A}=\sum_{i}d_{i}[Y_{i}]\in\mathrm{CH}^{g}(A\times A) \end{align*} $$
for some
$d_{i}\in \mathbb {Z}$
. Label the
$Y_{1},\ldots , Y_{n}$
so that
$A_{i}:=\mathrm {pr}_{1}(Y_{i})$
has dimension
$\leq g-1$
for
$i=1,\ldots , m$
, and
$A_{i}:=\mathrm {pr}_{2}(Y_{i})$
has dimension
$\leq g-1$
for
$i=m+1,\ldots , n$
. Let
$V_{1}:=A_{1}\cup \cdots \cup A_{m}$
and
$V_{2}:=A_{m+1}\cup \cdots \cup A_{n}$
, and let
$j_{1},j_{2}:V_{1},V_{2}\hookrightarrow A$
be the natural inclusions. Then
$$ \begin{align*} N\Delta_{A}=Z_{1}+Z_{2}\in\mathrm{CH}^{g}(A\times A), \end{align*} $$
where
$\mathrm {Supp}(Z_{1})\subset V_{1}\times A$
and
$\mathrm {Supp}(Z_{2})\subset A\times V_{2}$
. Let
$\widetilde {V}_{1}:=A_{1}\sqcup \cdots \sqcup A_{m}$
and
$\widetilde {V}_{2}:=A_{m+1}\sqcup \cdots \sqcup A_{n}$
be the disjoint unions, and let
$\tau _{1}:\widetilde {V}_{1}\rightarrow V_{1}$
,
$\tau _{2}:\widetilde {V}_{2}\rightarrow V_{2}$
be the natural morphisms. We claim that there is a correspondence
$\widetilde {Z}_{1}\in \mathrm {CH}^{g-1}(\widetilde {V}_{1}\times A)$
such that
$$ \begin{align*} Z_{1}=\widetilde{Z}_{1}\circ\gamma_{1}, \end{align*} $$
where
$\gamma _{1}\in \mathrm {CH}^{g}(A\times \widetilde {V}_{1})$
is the correspondence given by the transpose of the graph of
$\widetilde {j}_{1}:=j_{1}\circ \tau _{1}$
. Indeed, let
$V_{1}^{\mathrm {sm}}$
denote the smooth locus of
$V_{1}$
, and consider the pullback square
Since
$V_{1}^{\mathrm {sm}}$
,
$\tau _{1}^{-1}(V_{1}^{\mathrm {sm}})$
, and
$\widetilde {V}_{1}$
are smooth, the morphisms in the diagram admit refined Gysin pullbacks (see [Reference FultonFul84, Section 6.6]). Set
$\widetilde {Z}_{1}$
to be the closure in
$\widetilde {V}_{1}\times A$
of the pullback of
$Z_{1}$
along
$\tau _{1}^{-1}(V_{1}^{\mathrm {sm}})\times A\rightarrow V_{1}^{\mathrm {sm}}\times A\rightarrow V_{1}\times A$
, where consider
$Z_{1}$
as a cycle on
$V_{1}\times A$
. Then
$Z_{1}=(j_{1}\circ \tau _{1}\times \mathrm {id}_{A})_{\ast }\widetilde {Z}_{1}=\widetilde {Z}_{1}\circ \gamma _{1}$
by [Reference FultonFul84, Proposition 16.1.1], as desired. The same argument applied to the transpose of
$Z_{2}$
shows that there exists a correspondence
$\widetilde {Z}_{2}\in \mathrm {CH}^{g-1}(A\times \widetilde {V}_{2})$
such that
$$ \begin{align*} Z_{2}=\Gamma_{\widetilde{j}_{2}}\circ\widetilde{Z}_{2}, \end{align*} $$
where
$\Gamma _{\widetilde {j}_{2}}\in \mathrm {CH}^{g}(\widetilde {V}_{2}\times A)$
is the correspondence given by the graph of
$\widetilde {j}_{1}:=j_{2}\circ \tau _{2}$
. Hence,
$$ \begin{align*} N\Delta_{A}=Z_{1}+Z_{2}=\widetilde{Z}_{1}\circ\gamma_{1}+\Gamma_{\widetilde{j}_{2}}\circ\widetilde{Z}_{2} \end{align*} $$
and the self-correspondence
$N\Delta _{A}^{\ast }:\mathrm {CH}^{2}(A)\rightarrow \mathrm {CH}^{2}(A)$
factors as
$$ \begin{align*} \mathrm{CH}^{2}(A)\xrightarrow{\widetilde{Z}_{1}^{\ast}\oplus\widetilde{j}_{2}^{\ast}}\mathrm{CH}^{1}(\widetilde{V}_{1})\oplus\mathrm{CH}^{1}(\widetilde{V}_{2})\xrightarrow{\widetilde{j}_{1\ast}+\widetilde{Z}_{2}^{\ast}}\mathrm{CH}^{2}(A). \end{align*} $$
Since the
$\ell $
-adic Abel–Jacobi maps are compatible with correspondences ([Reference BlochBlo79, Proposition 3.5] for
$\ell \neq p$
, [Reference Gros and SuwaGS88, Proposition 2.9] for
$\ell =p$
), we get a commutative diagram
where the composition of the vertical arrows is
$N\Delta _{A}^{\ast }$
. However,
$\Delta _{A}^{\ast }$
is the identity, so
$N\Delta _{A}^{\ast }$
is multiplication-by-N.
The map
$\lambda ^{\prime }_{1,\widetilde {V}_{1}}:\mathrm {CH}^{1}_{\mathrm {alg}}(\widetilde {V}_{1})[\ell ^{\infty }]\rightarrow H^{1}(\widetilde {V}_{1},\mathbb {Z}_{\ell }(1))\otimes _{\mathbb {Z}_{\ell }}\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
is a bijection [Reference Achter, Casalaina-Martin and VialACMV21, Proposition A.28]. We claim that the map
$\lambda ^{\prime }_{2,\widetilde {V}_{2}}:\mathrm {CH}^{2}_{\mathrm {alg}}(\widetilde {V}_{2})[\ell ^{\infty }]\rightarrow H^{3}(\widetilde {V}_{2},\mathbb {Z}_{\ell }(2))\otimes _{\mathbb {Z}_{\ell }}\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }$
is also a bijection. Indeed, it is injective by the same reasoning that showed that
$\lambda ^{\prime }_{2,A}$
is injective. To see that
$\lambda ^{\prime }_{2,\widetilde {V}_{2}}$
is surjective, recall that
$\widetilde {V}_{2}:=A_{m+1}\sqcup \cdots \sqcup A_{n}$
and
$\lambda ^{\prime }_{2,\widetilde {V}_{2}}$
is the direct sum
$$ \begin{align*} \bigoplus_{i=m+1}^{n}\mathrm{CH}^{2}_{\mathrm{alg}}(A_{i})[\ell^{\infty}]\xrightarrow{\oplus\lambda^{\prime}_{2,A_{i}}}\bigoplus_{i=m+1}^{n}H^{3}(A_{i},\mathbb{Z}_{\ell}(2))\otimes{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}. \end{align*} $$
The
$A_{i}$
are supersingular abelian varieties of dimension
$\leq g-1$
(they are supersingular since they are subvarieties of A), so the induction hypothesis implies that
$\lambda ^{\prime }_{2,\widetilde {V}_{2}}=\oplus \lambda ^{\prime }_{2,A_{i}}$
is surjective as claimed.
In particular, we see that the middle horizontal arrow in (4.2) is a bijection. A diagram chase shows that
$\mathrm {coker}(\lambda ^{\prime }_{2,A})$
is annihilated by N.
(2) We have seen in Section 2 that
$\mathrm {CH}^{2}_{\mathrm {hom}}(A)$
is a torsion group when k is the algebraic closure of a finite field. Therefore, the subgroup
$\mathrm {CH}^{2}_{\mathrm {alg}}(A)$
is also torsion. We may then conclude by part (1).
Remark 4.2 It was already shown in [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1] that
$\mathrm {Griff}^{2}(A)[\ell ^{\infty }]=0$
for all primes
$\ell \neq p$
, so the only new result is that
$\mathrm {Griff}^{2}(A)[p^{\infty }]=0$
as well. The proof strategy in Theorem 4.1 of reducing to showing surjectivity of
$\lambda ^{\prime }_{2,A}$
is the same as [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1]. When
$\ell \neq p$
, surjectivity of
$\lambda ^{\prime }_{2,A}$
is due to Suwa [Reference SuwaSuw88, Théorème 4.7.1]. Suwa’s proof proceeds by considering the following commutative diagram:
The cycle class map
$\mathrm {cl}$
is surjective for supersingular abelian varieties by [Reference SchreiederShi75, Appendix], and we have already seen that
$\lambda ^{\prime }_{1,A}$
is a bijection. The cup-product map along the bottom of the square is a surjection as a consequence of
$H^{i}(A,\mathbb {Z}/\ell \mathbb {Z})=\bigwedge ^{i}H^{1}(A,\mathbb {Z}/\ell \mathbb {Z})$
for abelian varieties. This forces
$\lambda ^{\prime }_{2,A}$
to be surjective.
In the case
$\ell =p$
, we have the commutative square analogous to (4.3). Unravelling notation, the square is as follows:
The cup-product map along the bottom horizontal is rarely surjective when A is not an ordinary abelian variety. Indeed, we have
$\mathrm {CH}^{1}_{\mathrm {alg}}(A)=\mathrm {Pic}^{0}_{A/k}(k)$
, so if A is an abelian variety with p-rank
$0$
(if A is supersingular, for example), then
$\mathrm {CH}^{1}_{\mathrm {alg}}(A)[p^{\infty }]\cong \varinjlim _{r}H^{0}(A_{\mathrm {\acute {e}t}},W_{r}\Omega _{A,\log }^{1})$
is the trivial group. This is why we must use a different argument for the
$\ell =p$
case of Theorem 4.1 than the argument for
$\ell \neq p$
used in [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1]. Notice that the proof of Theorem 4.1 treats all primes
$\ell $
(including
$\ell =p$
), and in particular gives a new proof of [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1].
Remark 4.3 The proof of Theorem 4.1 shows that the inclusion
$$ \begin{align*} H^{3}(A,\mathbb{Z}_{\ell}(2))\otimes_{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}\hookrightarrow H^{3}(A,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2)) \end{align*} $$
is an equality for all primes
$\ell $
. This was known for
$\ell \neq p$
as a consequence of the proof of [Reference Brent Gordon and JoshiBGJ02, Theorem 5.1].
Acknowledgment
The author would like to thank Kirti Joshi for his generosity in sharing his ideas, and for enlightening discussions.
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