Tate Conjecture and Higher Brauer Groups of Abelian Varieties in Characteristic Zero

Let $X$ be a smooth projective variety over a field $k$. The cohomological Brauer groups $\mathrm{Br}(X)=\mathrm{H}_{\operatorname{et}}^{2}(X,\mathbb{G}_{m})$ appears in several different settings in algebraic geometry. For example it obstructs the Hasse principle on $X$ (see [5]) and it appears in a formula describing Zeta functions (see [10, Conjecture C]).

The list of applications also includes obstructions to the Tate conjecture: Let $k$ be finitely generated over its prime field and let $X$ be a smooth projective $k$-variety. The Tate conjecture $\mathrm{TC}^{r}(X)_{\mathbb{Q}_{\ell}}$ in codimension $r$ at a prime $\ell\neq\operatorname{char}(k)$ is the statement that the canonical cycle map $c^{r}_{\mathbb{Q}_{\ell}}:\mathrm{CH}^{r}(X)\otimes\mathbb{Q}_{\ell}\to\mathrm{H}_{\operatorname{et}}^{2r}(\overline{X},\mathbb{Q}_{\ell}(r))^{G_{k}}$ is surjective [11]. Here $\overline{X}=X\times_{k}\overline{k}$ denotes the base change to a separable closure $\overline{k}$ of $k$ and $G_{k}=\mathrm{Gal}(\overline{k}/k)$ is the absolute Galois group. For a surface $X$ over a finite field $k$ Tate has shown that $\mathrm{TC}^{1}(X)_{\mathbb{Q}_{\ell}}$ holds if and only if the $\ell$-primary torsion subgroup $\mathrm{Br}(X)\{\ell\}$ of the cohomological Brauer group is finite [10, Theorem 5.2].

This has been generalised using higher Brauer groups which are defined using Lichtenbaum cohomology. The Lichtenbaum cohomology groups $\mathrm{H}_{\textrm{L}}^{m}(X,\mathbb{Z}(n))$ are defined to be the hyper-cohomology groups of the (unbounded) complex of étale sheaves $\mathbb{Z}_{X}(n)_{\textrm{et}}$ given by Bloch’s cycle complex [1] on the small étale site $X_{\textrm{et}}$. The higher Brauer groups considered in this note are $\mathrm{Br}^{r}(X)=\mathrm{H}_{\textrm{L}}^{2r+1}(X,\mathbb{Z}(r))$; as there exists a quasi-isomorphism $\mathbb{Z}_{X}(1)_{\textrm{et}}\sim\mathbb{G}_{m}[-1]$, we retrieve the Brauer group $\mathrm{Br}(X)=\mathrm{H}_{\operatorname{et}}^{2}(X,\mathbb{G}_{m})\cong\mathrm{H}_{\textrm{L}}^{3}(X,\mathbb{Z}(1))=\mathrm{Br}^{1}(X)$. For any smooth projective, geometrically integral variety $X$ over a finite field $k$ and any prime $\ell\neq\operatorname{char}(k)$, the Tate conjecture $\mathrm{TC}^{r}(X)_{\mathbb{Q}_{\ell}}$ holds precisely when the $\ell$-primary subgroup of $\mathrm{Br}^{r}(X)$ is finite [7, Theorem 1.4].

We give a partial answer to the question whether higher Brauer groups provide a similar obstruction to the Tate conjecture when $k$ is finitely generated over $\mathbb{Q}$ in Section 3 where we prove the following

For any smooth projective variety $X$ over a field $k$ there exists a natural homomorphism $\mathrm{Br}^{r}(X)\to\mathrm{Br}^{r}(\overline{X})$ where again $\overline{X}=X\times_{k}\overline{k}$ denotes the base change to a separable closure $\overline{k}$ of $k$. The kernel $\mathrm{Br}_{\textrm{alg}}^{r}(X):=\ker\left(\mathrm{Br}^{r}(X)\to\mathrm{Br}^{r}\left(\overline{X}\right)\right)$ is referred to as algebraic Brauer group while the quotient $\mathrm{Br}^{r}(X)/\mathrm{Br}^{r}_{\textrm{alg}}(X)$ is called transcendental Brauer group.

We remark that if the characteristic if $k$ is zero and $X$ is geometrically integral, the cokernel of $\mathrm{Br}(X)\to\mathrm{Br}(\overline{X})$ is finite [2, Théorème 2.1]. The question whether this is also true for $r\geq 2$ is open.

If furthermore $X=A$ is an abelian variety, Skorobogatov and Zarhin have proven (in the classical case, i.e. $r=1$) the formula

for any odd integer $n$ [8, Corollary 2.3], relating the transcendental Brauer group of $A$ with étale cohomology and the Chow group of the base change $\overline{A}$. This formula allows some computations of transcendental Brauer groups, see for example [8].

While we cannot say anything about $\mathrm{Br}^{r}(A)$ itself, we prove an analogue of the above formula for higher Brauer groups and discuss how some of the computations of transcendental Brauer groups carry over to higher Brauer groups.

We start with the definitions of Lichtenbaum cohomology, higher Brauer groups and Chow-L groups as used in this paper.

For the entire section let $X$ be an equi-dimensional smooth quasi-projective variety over a field $k$ and denote by $z^{n}(X,\bullet)$ the cycle complex of abelian groups defined by Bloch [1]. As the assertion

defines a complex of sheaves on the flat site of $X$, it also defines complexes $z^{n}(-,\bullet)_{\textrm{et}}$ resp. $z^{n}(-,\bullet)_{\mathrm{Zar}}$ of sheaves on the étale resp. Zariski site of $X$. We remark, that these complexes are unbounded on the left. For an abelian group $A$, we set $A_{X}(n)_{\textrm{et}}=(z^{n}(-,\bullet)_{\textrm{et}}\otimes A)[-2n]$ and define the Lichtenbaum respectively motivic cohomology groups with coefficients in $A$ by letting

The definition of the hypercohomology of an unbounded complex was given in [9, p. 121].

For Chow groups we have the isomorphisms $\mathrm{CH}^{r}(A)\cong\mathrm{H}_{\textrm{M}}^{2r}(X,\mathbb{Z}(r))$ which motivates the definition $\mathrm{CH}_{\mathrm{L}}^{r}(A):=\mathrm{H}_{\textrm{L}}^{2r}(X,\mathbb{Z}(r))$ of Chow-L groups. With rational coefficients motivic cohomology and Lichtenbaum cohomology agree and therefore $\mathrm{CH}_{\mathrm{L}}^{r}(A)\otimes\mathbb{Q}\cong\mathrm{CH}^{r}(A)\otimes\mathbb{Q}$.

We define higher Brauer groups as $\mathrm{Br}^{r}(X):=\mathrm{H}_{\textrm{L}}^{2r+1}(X,\mathbb{Z}(r))$. The term ‘higher Brauer groups’ is justified by the fact that in weight $n=1$ we have a quasi-isomorphism $\mathbb{Z}_{X}(1)_{\textrm{et}}\sim\mathbb{G}_{m}[-1]$ which implies that $\mathrm{Br}^{1}(X)$ and $\mathrm{Br}(X)$ are isomorphic.

We remark that with rational coefficients Lichtenbaum cohomology groups and motivic cohomology groups coincide and therefore $\mathrm{H}_{\textrm{L}}^{2r+1}(X,\mathbb{Q}(r))$ is trivial. This implies that the higher Brauer groups $\mathrm{Br}^{r}(X)$ is a quotient of $\mathrm{H}_{\textrm{L}}^{2r}(X,\mathbb{Q}/\mathbb{Z}(r))$ and thus a torsion group.

For each prime $\ell$ multiplication by $\ell^{m}$ gives us the exact sequence

For primes $\ell$ different from the characteristic of $k$ this sequence together with the quasi-isomorphism $(\mathbb{Z}/\ell^{m}\mathbb{Z})_{X}(n)_{\textrm{et}}\sim\mu_{l^{m}}^{\otimes n}$ constructed by Geisser and Levine [4, Theorem 1.5] furnishes us with short exact universal coefficient sequences

In particular, in bidegree $(i,n)=(2r,r)$ this sequence reads

and generalises the exact sequence for the classical Brauer group induced by the Kummer sequence.

Let $k$ be a field finitely generated over $\mathbb{Q}$. Fix a separable closure $\overline{k}$ of $k$ and denote by $G_{k}:=\mathrm{Gal}(\overline{k}/k)$ the absolute Galois group. For a smooth projective geometrically integral $k$-variety we consider the composition of canonical homomorphisms

which induces a bunch of homomorphisms

where $\mathrm{CH}_{\mathrm{L}}^{r}(X)^{\wedge}:=\varprojlim_{n}\mathrm{CH}_{\mathrm{L}}^{r}(X)\otimes\mathbb{Z}/\ell^{n}\mathbb{Z}$. The map $\varprojlim_{n}c^{r}_{\ell^{n}}$ induces a map

and the image of $\varprojlim_{n}c^{r}_{\ell^{n}}\otimes\mathrm{id}_{\mathbb{Z}_{\ell}}$ agrees with the image of $\varprojlim_{n}\overline{c}_{\ell^{n}}^{r}$. Moreover, the homomorphism $\varprojlim_{n}c_{\ell^{n}}^{r}\otimes\mathrm{id}_{\mathbb{Q}_{\ell}}$ agrees the cycle map $c_{\mathbb{Q}_{\ell}}^{r}:\mathrm{CH}^{r}(X)\otimes\mathbb{Q}_{\ell}\to\mathrm{H}_{\operatorname{et}}^{2r}(\overline{X},\mathbb{Q}_{\ell}(r))^{G_{k}}$.

In view of the previous section it is desirable being able to compute higher Brauer groups, at least for abelian varieties. The objective of this section is to extend efforts of computing transcendental Brauer groups of abelian varieties in characteristic zero to higher Brauer groups.

The main result of this section is the following theorem which for $r=1$ was shown by Skorobogatov and Zarhin [8, Corollary 2.3].

For products of elliptic curves we can prove even more precise formulae. Let $E_{0},\dots,E_{r}$ be elliptic curves and set $A=E_{0}\times E_{1}\times\dots\times E_{r}$. Our aim is to obtain a better understanding of the higher Brauer group $\mathrm{Br}^{r}(A)$. For this we need some results concerning the Chow-L group $\mathrm{CH}_{\mathrm{L}}^{r}(\overline{A})$ and the étale cohomology groups $\mathrm{H}_{\operatorname{et}}^{2r}(\overline{A},\mu_{n}^{\otimes r})$.

The map $c^{r}_{n}:\mathrm{CH}^{r}(\overline{A})\otimes\mathbb{Z}/n\mathbb{Z}\to\mathrm{H}_{\operatorname{et}}^{2r}(\overline{A},\mu_{n}^{\otimes r})$ induced by the cycle map factors through the map $\overline{c}_{n}^{r}:\mathrm{CH}_{\mathrm{L}}^{r}(\overline{A})\otimes\mathbb{Z}/n\mathbb{Z}\to\mathrm{H}_{\operatorname{et}}^{2r}(\overline{A},\mu_{n}^{\otimes r})$ which is induced by the composition of caninical homomorphisms (2).

The isomorphisms $\operatorname{Hom}(E_{i}[n],\mathbb{Z}/n\mathbb{Z})\overset{\cong}{\to}\mathrm{H}_{\operatorname{et}}^{1}(\overline{E_{i}},\mathbb{Z}/n\mathbb{Z})$ and the perfect Weil pairings furnish us with canonical isomorphisms (see [8, Section 3])

Using these isomorphisms and the Künneth formula [6, VI. §8] we obtain an isomorphism of $G_{k}$-modules

Deeper investigation of the above isomorphism shows that each of the three $\mathbb{Z}/n\mathbb{Z}$-copies is generated by a fundamental class coming from one of the elliptic curves $\overline{E_{i}}$.

In the case $r=1$ the following theorem and its applications presented below were shown by Skorobogatov and Zarhin [8].

We now consider the special case of a triple product $A=E\times E\times E$ of a elliptic curve $E$ and $r=2$. The groups $R_{n}$ and $H_{n}$ are defined as in Theorem 4.3.

For our second application let $E$ be an elliptic curve over $\mathbb{Q}$ with complex multiplication. For an odd prime $\ell$ the curve $E$ has no rational isogeny of degree $\ell$, if and only if $E[\ell]$ does not contain a Galois invariant subgroup of order $\ell$.

The author thanks Andreas Rosenschon for sugesting to work on these questions. Parts of this paper were written while the author was member of DFG Sonderforschungsbereich 701 at Bielefeld University.