¹

²

Introduction↩︎

For an algebraic variety \(X\) over a field \(k\) with separable closure \(k_{\rm s}\) and a prime \(\ell\) not divisible by \({\rm char}(k)\), the action of the absolute Galois group \(\Gamma={\rm{Gal}}(k_{{\rm s}}/k)\) on the étale cohomology groups \({\rm H}^i_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}_{\ell})\) contains arithmetic information about the variety \(X\), such as the number of rational points in the case when \(k\) is a finite field.

The action of the Galois group \(\Gamma\) on individual étale cohomology groups comes from its action on the étale cohomology complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}_{\ell})\). The phenomenon that we study in this paper is that the quasi-isomorphism class of this complex in general contains more information than individual cohomology Galois modules.

Remarkably, for a smooth projective variety \(X\) over \(k\), by Deligne’s decomposition theorem [1], the rational étale cohomology complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Q}}_{\ell})\) is quasi-isomorphic to the direct sum \(\bigoplus\limits_{i\geq 0}{\rm H}^i_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Q}}_{\ell})[-i]\) of individual cohomology groups as an object of the derived category of \({\mathbb{Q}}_{\ell}\)-vector spaces equipped with an action of \(\Gamma\).

This is no longer true for cohomology with integral coefficients, and is witnessed by the fact that the Hochschild–Serre spectral sequence \[E_{2}^{p,q}= {\rm H}^p(k,{\rm H}_{\rm{\acute et}}^q(X_{k_{\rm s}},{\mathbb{Z}}_\ell))\Rightarrow{\rm H}_{\rm{\acute et}}^{p+q}(X,{\mathbb{Z}}_\ell) \label{ss32ell}\tag{1}\] in general has non-zero differentials. This is the case when \(\ell=2\), \(k={\mathbb{R}}\), \(X({\mathbb{R}})=\emptyset\), as follows from work of Artin–Verdier [2], see also [3]. If \(X\) is a smooth projective real curve with no real points and \(\ell=2\), then a differential on the 2nd (respectively, 3rd) page is non-zero if the genus of \(X\) is odd (respectively, even), see Examples [real32curves32odd32genus] and [real32curves32even32genus].

Our first main result is that (1 ) may fail to degenerate even for abelian varieties. We denote by \(D(\Gamma,{\mathbb{Z}}/n)\) the derived category of the abelian category of discrete \({\mathbb{Z}}/n\)-modules with a continuous action of \(\Gamma={\rm{Gal}}(k_{{\rm s}}/k)\).

By Theorem 3 below, the Hochschild–Serre spectral sequence degenerates for any abelian variety isomorphic to a direct factor of a product of Jacobians of curves each with a rational divisor class of degree 1. In particular, the abelian surface \(A\) from Theorem 1 is an example of an abelian variety not of this type.

Let us sketch the proof of Theorem 1. Let \(C\) be a smooth proper curve of genus \(g\) over \(k\) and let \(J\) be the Jacobian of \(C\). We identify \(J\) with the dual abelian variety using the canonical principal polarisation. Using a general description of \(\delta_2^{0,2}\) given in Theorem 2 below, we show that the mod 2 first Chern class of the canonical principal polarisation is an element of \({\rm H}^2_{\rm{\acute et}}(J_{k_{\rm s}},{\mathbb{Z}}/2)^\Gamma\) which is sent by \(\delta_2^{0,2}\) to the image of the class of the torsor of theta-characteristics under the Bockstein map \({\rm H}^1(k,J[2])\to {\rm H}^2(k,J[2])\). Thus \(\delta_2^{0,2}\neq 0\) whenever \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\in {\rm H}^1(k,J)\) is not divisible by 2. A systematic method to construct such curves over number fields was found by B. Creutz [4]. The idea is to exploit the fine arithmetic structure available in this case, namely, that \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\) is automatically a 2-torsion element of the Tate–Shafarevich group \(\textcyr{Sh}(J)\). The class \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\) is divisible by 2 in \({\rm H}^1(k,J)\) if and only if its image in \(\textcyr{Sh}^2(k,J[2])\) under the Bockstein map is zero, which is equivalent to being orthogonal to \(\textcyr{Sh}^1(k,J[2])\) under the non-degenerate Poitou–Tate duality pairing. By compatibility of pairings, this happens if and only if \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\) is orthogonal to the image of \(\textcyr{Sh}^1(k,J[2])\) in \(\textcyr{Sh}(J)\) under the Cassels–Tate pairing.

We have two strategies to construct an element of \(\textcyr{Sh}^1(k,J[2])\) whose image in \(\textcyr{Sh}(J)\) pairs non-trivially with \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\).

The first strategy, following Manin, interprets the Cassels–Tate pairing in terms of the Brauer–Manin obstruction on the everywhere locally soluble variety \(X={\boldsymbol{P}ic}^{g-1}_{C/k}\): any \(t\in\textcyr{Sh}(J)\) gives rise to a Brauer class in \({\rm{Br}}(X)\) whose Brauer–Manin pairing with an arbitrary adelic point of \(X\) equals the Cassels–Tate pairing of \(t\) and \([X]\). When \(g=2\), the variety \(X\) is the Albanese torsor of \(C\), so \(C\) is a closed subvariety of \(X\). If \(C\) itself is everywhere locally soluble, we can take an adelic point of \(X\) contained in \(C\) and then we only need to know the restriction of our Brauer class to \(C\). The easiest way to arrange for this is to consider hyperelliptic curves \(C\) given by the equation \(y^2=f(x)\), where the polynomial \(f(x)\in k[x]\) of degree 6 is itself everywhere locally soluble. In this setting, one can construct elements of \({\rm{Br}}(C)\) coming from \(\textcyr{Sh}^1(k,J[2])\subset{\rm H}^1(k,J[2])\) by an explicit formula, which allows one to compute the Brauer–Manin pairing with the adelic points of \(f(x)=0\). For a worked-out example borrowed from Creutz [4] and Creutz–Viray [5] see Theorem 22 (a).

The second strategy is based on the approach of Poonen and Stoll [6]. The class \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\) comes from the torsor of theta-characteristics in \({\rm H}^1(k, J[2])\), so if this torsor is everywhere locally trivial and the curve \(C\) is odd, which means that the Cassels–Tate pairing of \([{\boldsymbol{P}ic}^{g-1}_{C/k}]\) with itself is non-zero, then we are done. For a worked-out example see Theorem 22 (b).

Let us now give a general description of the first potentially non-trivial extension in the étale cohomology complex of a semiabelian variety over an arbitrary field \(k\) (\({\rm char}(k)\neq 2\)). It is easy to see that all differentials on the \(i\)-th page of (1 ) are zero for \(\ell>i\), see Corollary 3. For the study of the differentials on the second page we can thus assume that \(\ell=2\). For a \({\mathbb{Z}}_2\)-module \(M\) we denote by \(Q(M)\) the module of quadratic functions on \(M^{\vee}={\rm {Hom}}_{{\mathbb{Z}}_2}(M,{\mathbb{Z}}_2)\) with values in \({\mathbb{Z}}_2\). It fits into the exact sequence \[\label{intro:32quadratic32ext2} 0\to M\to Q(M)\to M^{\otimes 2}\to \wedge^2 M\to 0,\tag{2}\] where the middle map sends a quadratic function \(f:M^{\vee}\to{\mathbb{Z}}_2\) to the bilinear form \[f(x+y)-f(x)-f(y)\in{\rm {Hom}}(M^{\vee}\otimes M^{\vee},{\mathbb{Z}}_2)\cong M^{\otimes 2}.\] The sequence (2 ) is natural in \(M\), so if \(M\) is equipped with an action of \(\Gamma\) then (2 ) becomes an exact sequence of \(\Gamma\)-modules.

In the case of abelian varieties this was proved in [7], but here we give a new shorter proof exhibiting an explicit quasi-isomorphism from the 2-term complex \(Q(M)\to M^{\otimes 2}\) to the truncated bar complex of the Tate module \(T_2(A)\). Theorem 2, in particular, gives a description of the differentials \[{\rm H}^i(k, {\rm H}^2_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2))\to {\rm H}^{i+2}(k, {\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2))\] of (1 ), as well as the analogous differentials in the Hochschild–Serre spectral sequence with \({\mathbb{Z}}/2^n\)-coefficients, for any \(n\), see also Theorem 13.

Theorem 1 shows that the class described in Theorem 2 is not always zero for abelian varieties, but at the moment we do not know if it is ever non-zero for tori, cf. Remark [tori:32nonvanishing32remark].

In contrast, for the Jacobians of curves with a rational divisor class of degree \(1\) the étale cohomology complex decomposes in all degrees. We give a proof of this result, which may be well-known to the experts, but does not seem to be available in the literature.

Another class of examples of non-zero differentials in the Hochschild–Serre spectral sequence comes from varieties without a rational point. If a geometrically connected \(k\)-variety \(X\) has a \(k\)-point, then \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}_{\ell})\) is quasi-isomorphic to \({\mathbb{Z}}_{\ell}\oplus\tau^{\geq 1}\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}_{\ell})\) in the derived category of \(\Gamma\)-modules. This is no longer true for varieties without a rational point, and we calculate the exact obstruction to splitting off \({\rm H}^0(X_{k_{{\rm s}}},{\mathbb{Z}}_{\ell})\) from \(\tau^{\leq 1}\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}_{\ell})\) when \(X\) is a torsor for a semiabelian variety. We denote by \(\mathcal{A}(\Gamma,{\mathbb{Z}}/n)\) the abelian category of discrete \({\mathbb{Z}}/n\)-modules equipped with a continuous action of \(\Gamma\).

As we point out in Remark [section32remark], in the setting of Theorem 4, the obstruction to the existence of a rational \(0\)-cycle of degree \(1\) coming from \(\tau^{[0,1]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},\widehat{{\mathbb{Z}}})\) being non-split coincides with the obstruction coming from the failure of the abelianised fundamental exact sequence for the arithmetic fundamental group to have a section.

We give several complements of the above results, summarised below.

In Theorem 23 we describe a differential on the \(3\)rd page of the Hochschild–Serre spectral sequence with \({\mathbb{G}}_m\)-coefficients for a torus, which allows one to compute the Brauer group of any algebraic torus, answering a question raised in [8]. When the first version of this paper was completed, Julian Demeio told us about his result that for a torus \(T\) over a field \(k\) of characteristic zero the natural map \({\rm{Br}}(T)\to{\rm{Br}}(T_{k_{\rm s}})^\Gamma\) is surjective when \(T\) is quasi-trivial or when \(k\) is a number field [9]. This also follows from our Theorem 23, see Corollary 25. Demeio’s proof is based on an elaborate analysis of differentials of the Hochschild–Serre spectral sequence by L.S. Charlap, A.T. Vasques, and C.H. Sah (see the references in [9]). The proof of Theorem 23 in this paper is self-contained: it reduces the computation of the differential with \({\mathbb{G}}_m\)-coefficients to the case of finite coefficients which is handled in our explicit proof of Theorem 2.

In Proposition 30 we combine Theorem 4 with Poincaré duality to describe all differentials on the second page of the Hochschild–Serre spectral sequence with coefficients in \(\mu_n\) for a smooth proper curve. When all differentials on the second page vanish, in Proposition 34 we describe the differentials on the \(3\)rd page of this spectral sequence, generalising a result of Suslin in the case of genus \(0\).

Notation. For an object \(C\) of the derived category \(D(\mathcal{A})\) of an abelian category \(\mathcal{A}\) its truncation \(\tau^{[i,i+1]}C\) in any two consecutive degrees fits into a distinguished triangle \[{\rm H}^i(C)[-i]\to \tau^{[i,i+1]}C\to {\rm H}^{i+1}(C)[-i-1]\xrightarrow{\delta_i}{\rm H}^i(C)[-i+1]\] We refer to the map \[\delta_i\in {\rm {Hom}}_{D(\mathcal{A})}({\rm H}^{i+1}(C)[-i-1],{\rm H}^i(C)[-i+1])\cong {\rm Ext}^2_{\mathcal{A}}({\rm H}^{i+1}(C), {\rm H}^i(C))\] as the extension class of \(\tau^{[i,i+1]}C\).

For a profinite group \(G\) and an integer \(n\) we denote by \(\mathcal{A}(G,{\mathbb{Z}}/n)\) the abelian category of discrete \({\mathbb{Z}}/n\)-modules equipped with a continuous action of \(G\), cf. [10]. We write \(D(G,{\mathbb{Z}}/n)\) for the derived category of \(\mathcal{A}(G,{\mathbb{Z}}/n)\). For a prime \(\ell\) we denote by \(D(G,{\mathbb{Z}}_{\ell})\) the derived category of sheaves of \({\mathbb{Z}}_{\ell}\)-modules on the pro-étale site \(BG_{\mathrm{proet}}\), where \({\mathbb{Z}}_{\ell}\) is the sheaf of rings obtained from the topological ring \({\mathbb{Z}}_{\ell}\) with a trivial action of \(G\), cf. [11]. Recall that an \(\ell\)-adically complete \({\mathbb{Z}}_{\ell}\)-module \(M\) equipped with a continuous action of \(G\) gives rise to an object of the triangulated category \(D(G,{\mathbb{Z}}_{\ell})\), and \({\rm {Hom}}_{D(G,{\mathbb{Z}}_{\ell})}({\mathbb{Z}}_{\ell},M[i])\) is isomorphic to \(i\)-th continuous cohomology group \({\rm H}^i_{\mathrm{cont}}(G,M)\) defined using the complex of continuous cochains on \(G\).

When \(\Gamma={\rm{Gal}}(k_{{\rm s}}/k)\) is the absolute Galois group of a field \(k\), we denote \({\rm H}^i_{\mathrm{cont}}(\Gamma, M)\) by \({\rm H}^i(k, M)\) for any continuous \(\Gamma\)-module \(M\).

Acknowledgements. The first named author was supported by the Clay Research Fellowship and was a member of the Institute for Advanced Study in Princeton during the preparation of this paper. Work on this paper benefited from visits of the second named author to the Institute for Advanced Study and the Max Planck Institute for Mathematics in Bonn. We thank Jean-Louis Colliot-Thélène and Tamás Szamuely for help with references, and Vadim Vologodsky for explaining to us the result obtained in Theorem 3.

1 Semiabelian varieties↩︎

Let \(A\) be a semiabelian variety over \(k\), that is, an extension of an abelian variety by a torus. Let \(\ell\) be a prime number, \(\ell\neq{\rm char}(k)\). For all \(i\geq 0\), \(n\geq 1\), cup product induces an isomorphism \({\rm H}^i_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/\ell^n)\simeq\wedge^i{\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/\ell^n)\) compatible with the natural action of the Galois group \(\Gamma={\rm{Gal}}(k_{{\rm s}}/k)\). (By [12] this follows from the same statement with \(k_{\rm s}\) replaced by an algebraically closed field extension, which is proved in [13].)

We would like to study the differentials \(\delta_i^{p,q}\colon E_{i}^{p,q}\to E_i^{p+i,q-i+1}\) of the Hochschild–Serre spectral sequence \[\label{Leray1} E_{2}^{p,q}= {\rm H}^p(k,{\rm H}_{\rm{\acute et}}^q(A_{k_{\rm s}},{\mathbb{Z}}/\ell^n))\Rightarrow{\rm H}_{\rm{\acute et}}^{p+q}(A,{\mathbb{Z}}/\ell^n).\tag{3}\]

Let us make some simple remarks about these differentials.

Proof. The spectral sequence (3 ) is functorial in \(A\), so it is compatible with multiplication by \(m\) map \([m]\colon A\to A\), for any integer \(m\). The induced map on \({\rm H}^r_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/\ell^n)\cong\wedge^r{\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/\ell^n)\) is multiplication by \(m^r\). Each \(E_i^{p,q}\) for \(i\geq 2\) is a subquotient of \(E_{2}^{p,q}\), hence \([m]^*\) acts on it via multiplication by \(m^q\). Since \([m]^*\) commutes with all differentials on the \(i\)-th page, we have \(m^{q-i+1}\delta_{i}^{p,q}=\delta_{i}^{p,q}m^q\) for all \(p,q\geq 0\). Thus we have \[(m^q-m^{q-i+1})\delta^{p,q}_i=m^{q-i+1}(m^{i-1}-1)\delta^{p,q}_i=0.\] Taking \(m=\ell\) we see that \(\ell^{q-i+1}\) annihilates \(\delta^{p,q}_i\). From now on we assume that \((m,\ell)=1\). Then we have \((m^{i-1}-1)\delta^{p,q}_i=0\).

Suppose \(\ell\neq 2\). If \(\ell-1\) does not divide \(i-1\), then we can find an integer \(m\) such that \((m,\ell)=1\) and \(m^{i-1}-1\) is non-zero modulo \(\ell\). Then \(\delta^{p,q}_i=0\).

If \(\ell-1\) divides \(i-1\), we claim that the lowest value of \({\rm val}_\ell(m^{i-1}-1)\), where \((m,\ell)=1\), is \(n+1\). If \((m,\ell)=1\), then \(m^{r\ell^n(\ell-1)}-1\) is clearly divisible by \(\ell^{n+1}\), so it is enough to check that \({\rm val}_\ell((1+\ell)^{r\ell^n(\ell-1)}-1)=n+1\) when \((r,\ell)=1\). This is immediate for \(n=0\), and the general case follows by induction in \(n\). This proves (a).

Suppose \(\ell=2\). If \(i\) is even, then taking \(m=-1\) we prove the first statement of (b). If \(i\) is odd, so that \(n\geq 1\), we claim that the smallest value of \({\rm val}_2(m^{r 2^n}-1)\), where \(r\) and \(m\) are odd, is \(n+2\). We have \[m^{r2^n}-1=(m^{2^n}-1)(m^{(r-1)2^n}+\ldots+1).\] Since \(r\) and \(m\) are odd, the second factor in the right hand side is odd. Thus we can assume that \(r=1\). For \(n=1\) the statement is obvious, and the general case immediately follows by induction. This proves (b). \(\Box\)

Note that the statement of Remark 1 is a particular case of Lemma 2.

Thus the differentials on the second page can be non-zero only for \(\ell=2\), and the first non-trivial case is the differential \(\delta^{p,2}_2\), where \(p\geq 0\). Let us start with the spectral sequence \[\label{Leray2} E_{2}^{p,q}= {\rm H}^p(k,{\rm H}_{\rm{\acute et}}^q(A_{k_{\rm s}},{\mathbb{Z}}_2))\Rightarrow{\rm H}_{\rm{\acute et}}^{p+q}(A,{\mathbb{Z}}_2).\tag{4}\]

Let \(M\) be a free, finitely generated \({\mathbb{Z}}_2\)-module. Recall that \(S^2(M)\) is the quotient of \(M^{\otimes 2}\) by the \({\mathbb{Z}}_2\)-submodule generated by \(x\otimes y-y\otimes x\) for \(x,y\in M\), and \(\wedge^2(M)\) is the quotient of \(M^{\otimes 2}\) by the \({\mathbb{Z}}_2\)-submodule generated by \(x\otimes x\) for \(x\in M\). We write \(M^\vee={\rm {Hom}}_{{\mathbb{Z}}_2}(M,{\mathbb{Z}}_2)\).

For example, \(Q({\mathbb{Z}}_2)\) is a free rank \(2\) module consisting of functions of the form \(a\cdot x+b\cdot \frac{x^2-x}{2}\) with \(a,b\in{\mathbb{Z}}_2\).

There is a natural injection \(M\to Q(M)\) sending an element of \(M\) to the linear function on \(M^\vee\) that it defines. The cokernel of this map is \({\rm {Hom}}_{{\mathbb{Z}}_2}(S^2(M^\vee),{\mathbb{Z}}_2)\). One immediately checks (for example, by choosing a basis of \(M\) and the dual basis of \(M^\vee\)) that under the natural pairing \[M^{\otimes 2}\times (M^\vee)^{\otimes 2}\to{\mathbb{Z}}_2\] the \({\mathbb{Z}}_2\)-submodules \((M\otimes M)^{S_2}=\langle m\otimes m|m\in M\rangle\) and \(\langle a\otimes b-b\otimes a|a,b\in M^{\vee}\rangle\) are exact annihilators of each other. Thus there is a canonical isomorphism \({\rm {Hom}}_{{\mathbb{Z}}_2}(S^2(M^\vee),{\mathbb{Z}}_2)\cong(M\otimes M)^{S_2}\). We obtain a canonical exact sequence \[0\to M\to Q(M)\to M^{\otimes 2}\to \wedge^2 M\to 0. \label{genius}\tag{5}\] The \({\mathbb{Z}}_2\)-module \(Q(M)\) contains the submodule of quadratic forms \(M^\vee\to{\mathbb{Z}}_2\). The map \(Q(M)\to M^{\otimes 2}\) sends a quadratic form to the associated bilinear form.

The extension defined by (5 ) is equivalent to an extension with smaller terms. Denote by \((M^{\otimes 2})_{S_2,\mathrm{sgn}}=(M^{\otimes 2})/\langle m_1\otimes m_2+m_2\otimes m_1|m_1,m_2\in M\rangle\) the coinvariants of the involution \(m_1\otimes m_2\mapsto -m_2\otimes m_1\) acting on \(M^{\otimes 2}\). This module fits into the following exact sequence: \[0\to M\stackrel{2}\longrightarrow M\to (M^{\otimes 2})_{S_2,{\rm sgn}}\to \wedge^2 M\to 0, \label{genius1}\tag{6}\] where the map \(M\to (M^{\otimes 2})_{S_2,\mathrm{sgn}}\) sends \(m\) to \(m\otimes m\), and the rightmost map sends \(m_1\otimes m_2\in (M^{\otimes 2})_{S_2,\mathrm{sgn}}\) to \(m_1\wedge m_2\in \wedge^2 M\). A direct computation shows:

The extension (6 ) is the Yoneda product of \[0\to M\stackrel{2}\longrightarrow M\to M/2\to 0 \label{Bock}\tag{7}\] and \[0\to M/2\to (M^{\otimes 2})_{S_2,{\rm sgn}}\to \wedge^2 M\to 0. \label{alpha42}\tag{8}\] All of the above constructions and exact sequences are compatible with the action of the automorphisms of the \({\mathbb{Z}}_2\)-module \(M\). In particular, if \(M\) is a \({\mathbb{Z}}_2\)-module equipped with the \(2\)-adic topology and a continuous action of \(\Gamma\), all the sequences above are naturally sequences of \(\Gamma\)-modules.

The following statement is the particular case \(p=2\) of [7]. Here we give a short elementary proof of this result.

Proof. We begin by recasting étale cohomology of \(A_{k_{{\rm s}}}\) in terms of group cohomology of the profinite abelian group \(M^{\vee}\), using that semiabelian varieties are étale \(K(\pi,1)\)-spaces:

Given Lemma 7, Theorem 6 follows from the following description of the extension class in \(\tau^{[1,2]}\mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}_2)\):

Proof of Lemma 7. For any connected scheme \(X\) with a geometric base point \(\bar{x}\) and any integer \(m\geq 1\) there is a natural map \[\label{pi1coh32to32etalecoh32map} \mathrm{R}\Gamma(\pi_1^{{\rm{\acute et}}}(X,\bar{x}),{\mathbb{Z}}/m)\to \mathrm{R}\Gamma_{{\rm{\acute et}}}(X,{\mathbb{Z}}/m)\tag{9}\] induced by the pullback along the map of sites \(X_{{\rm{\acute et}}}\to X_{\mathrm{f\acute et}}\), where the target site consists of schemes finite étale over \(X\), cf. [14]. Moreover, if \(X=Y_{k_{{\rm s}}}\), where \(Y\) is a scheme over \(k\), and \(\bar{x}\) is above a \(k\)-point of \(Y\), this map naturally lifts to a map in \(D(\Gamma,{\mathbb{Z}}/m)\), where the action of \(\Gamma\) on the continuous cochain complex \(\mathrm{R}\Gamma(\pi_1^{{\rm{\acute et}}}(X,\bar{x}),{\mathbb{Z}}/m)\) is induced by the Galois action on the fundamental group \(\pi_1^{{\rm{\acute et}}}(X,\bar{x})\).

The map (9 ) induces a natural map \[{\rm H}^1_{\rm cont}(\pi_1^{{\rm{\acute et}}}(X,\bar{x}),{\mathbb{Z}}/m)\to {\rm H}^1(X,{\mathbb{Z}}/m). \label{25a}\tag{10}\] Since étale \(X\)-torsors for \({\mathbb{Z}}/m\) are the same as Galois covers of \(X\) with Galois group \({\mathbb{Z}}/m\), this map is an isomorphism [15]. Consider the composition \[\label{pi1coh32to32etalecoh32map32prime32to32p} \mathrm{R}\Gamma(\pi_1^{{\rm{\acute et}}}(X,\bar{x})^{(p')},{\mathbb{Z}}/m)\to \mathrm{R}\Gamma(\pi_1^{{\rm{\acute et}}}(X,\bar{x}),{\mathbb{Z}}/m)\to \mathrm{R}\Gamma_{{\rm{\acute et}}}(X,{\mathbb{Z}}/m),\tag{11}\] where the first map is induced by the surjection \(\pi_1^{{\rm{\acute et}}}(X,\bar{x})\twoheadrightarrow\pi_1^{{\rm{\acute et}}}(X,\bar{x})^{(p')}\) onto the prime-to-\(p\) completion of the fundamental group where \(p={\rm char}(k)\); if \({\rm char}(k)=0\) we set \(\pi_1^{{\rm{\acute et}}}(X,\bar{x})^{(p')}:=\pi_1^{{\rm{\acute et}}}(X,\bar{x})\). The map (11 ) induces an isomorphism on \({\rm H}^1\) whenever \(m\) is not divisible by \({\rm char}(k)\).

Let us now specialise to the case \(X=A_{k_{{\rm s}}}\), where \(A\) is a semiabelian variety over \(k\). We take \(\bar x\) to be the origin of the group law on \(A\) and omit it from the notation.

We know that the cohomology ring of the target of (11 ) is the free exterior algebra on \({\rm H}^1(A_{k_{\rm s}},{\mathbb{Z}}/m)\). We claim that the same is true for the source of (11 ). Indeed, [13] implies that \(\pi_1^{{\rm{\acute et}}}(A_{k_{{\rm s}}})^{(p')}\) is a finite product of copies of \(\widehat{{\mathbb{Z}}}^{(p')}\). The cohomological dimension of \(\widehat{{\mathbb{Z}}}^{(p')}\) is 1 and the cohomology groups \({\rm H}^0(\widehat{{\mathbb{Z}}}^{(p')},{\mathbb{Z}}/m)\simeq {\rm H}^1(\widehat{{\mathbb{Z}}}^{(p')},{\mathbb{Z}}/m)\simeq{\mathbb{Z}}/m\) are free \({\mathbb{Z}}/m\)-modules, hence the claim follows from the Künneth formula. Thus for \(X=A_{k_{{\rm s}}}\) the map (11 ) is a quasi-isomorphism in \(D(\Gamma,{\mathbb{Z}}/m)\).

Next, \(\pi_1^{{\rm{\acute et}}}(A_{k_{{\rm s}}})^{(p')}\) is the product of the \(2\)-adic Tate module \(T_2(A_{k_{{\rm s}}})\cong M^{\vee}\) and a 2-divisible profinite group. Passing to the limit over \(m=2^n\) we obtain a quasi-isomorphism \(\mathrm{R}\Gamma(\pi_1^{{\rm{\acute et}}}(A_{k_{{\rm s}}})^{(p')},{\mathbb{Z}}_2)\simeq \mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}_2)\) in \(D(\Gamma,{\mathbb{Z}}_2)\), hence also a quasi-isomorphism \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2)\simeq \mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}_2)\). \(\Box\)

Proof of Proposition 9. We can calculate \(\mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}_2)\) using the standard bar complex of \({\mathbb{Z}}_2\)-modules for computing continuous cohomology of the group \(M^{\vee}\): \[0\longrightarrow{\mathbb{Z}}_2\stackrel{d_0=0}\longrightarrow{\rm Func}(M^\vee,{\mathbb{Z}}_2)\stackrel{d_1}\longrightarrow {\rm Func}(M^\vee\times M^\vee,{\mathbb{Z}}_2)\stackrel{d_2}\longrightarrow\ldots,\] where \({\rm Func}\) denotes the \({\mathbb{Z}}_2\)-module of continuous functions. Considering this complex as a complex of \(\Gamma\)-modules for the natural action of the Galois group \(\Gamma\) induced by its action on \(M\), we get a complex in \(D(\Gamma,{\mathbb{Z}}_2)\) that calculates the cohomology of the semi-direct product \(M^\vee\rtimes\Gamma\).

The differential \(d_1\) sends a function \(f\colon M^\vee\to{\mathbb{Z}}_2\) to the function \[(x,y)\mapsto f(x+y)-f(x)-f(y).\] The inclusion \(Q(M)\subset {\rm Func}(M^\vee,{\mathbb{Z}}_2)\) gives rise to a commutative diagram \[\begin{align} \label{main32thm32proof32commdiag}\xymatrix{Q(M)\ar[r]\ar[d]&M^{\otimes 2}\ar[d]\\ {\rm Func}(M^\vee,{\mathbb{Z}}_2)\ar[r]^{d_1 \;\;\;\;}&{\rm Func}(M^\vee\times M^\vee,{\mathbb{Z}}_2)} \end{align}\tag{12}\] The right-hand vertical map is the inclusion of bilinear functions on \(M^\vee\times M^\vee\) into all continuous functions. Commutativity of the diagram is immediate from the definitions of the maps. The submodule \(M\subset Q(M)\) of linear functions maps isomorphically to \({\rm H}^1_{\mathrm{cont}}(M^{\vee},{\mathbb{Z}}_2)\). Moreover, the right vertical map lands in the kernel of the next differential \(d_2\), and the induced map \(M^{\otimes 2}\to {\rm H}^2_{\mathrm{cont}}(M^{\vee},{\mathbb{Z}}_2)\) coincides with the cup-product map \({\rm H}^1_{\mathrm{cont}}(M^{\vee},{\mathbb{Z}}_2)^{\otimes 2}\to {\rm H}^2_{\mathrm{cont}}(M^{\vee},{\mathbb{Z}}_2)\).

Exactness of (5 ) then implies that (12 ) gives a quasi-isomorphism in \(D(\Gamma,{\mathbb{Z}}_2)\) between the two-term complex \(Q(M)\to M^{\otimes 2}\) and the truncation \(\tau^{[1,2]}\mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}_2)\), as desired. The description of the analogous truncation \(\tau^{[1,2]}\mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}/2^n)\) now follows formally because cohomology groups \({\rm H}^i_{\mathrm{cont}}(M^{\vee},{\mathbb{Z}}_2)\) as well as terms of the extension (5 ) are torsion-free. \(\Box\)

Theorem 6 is now proved because, by Lemma 7, the differentials under consideration in spectral sequence (4 ) (resp. (3 )) are induced by the \({\rm Ext}^2\) class of \(\tau^{[1,2]}\mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}_2)\) (resp. \(\tau^{[1,2]}\mathrm{R}\Gamma(M^{\vee},{\mathbb{Z}}/2^n)\)).

The rest of this section is mainly devoted to the proof of Theorem 13 below, which gives a more computable description of the differential \(\delta_2^{p,2}\) of the spectral sequence \((\ref{Leray1})\) with coefficients \({\mathbb{Z}}/\ell^m\).

When \(N\) is a free, finitely generated \({\mathbb{Z}}_2\)-module with continuous action of \(\Gamma\), we denote by Bock the morphism \(N/2^m\to N[1]\) defined by the exact sequence \[0\to N\stackrel{2^m}\longrightarrow N\to N/2^m\to 0.\] For \(i\geq 1\), by a slight abuse of notation, we also denote by Bock the composition \(N/2^m\to N[1]\to (N/2^i)[1]\). This map is defined by the exact sequence \[0\to N/2^i\to N/2^{m+i}\to N/2^m\to 0,\] which is the push-out of the previous exact sequence by the map \(N\to N/2^i\).

Let us also introduce the following notation. For an \({\mathbb{F}}_2\)-vector space \(V\), equipped with an action of \(\Gamma\) continuous with respect to the discrete topology on \(V\), consider the short exact sequence \[\label{alpha} 0\to V\xrightarrow{v\mapsto v\cdot v} S^2 V\xrightarrow{v_1\cdot v_2\mapsto v_1\wedge v_2} \wedge^2 V\to 0.\tag{13}\] Its extension class is a map \(\wedge^2 V\to V[1]\) in \(D(\Gamma,{\mathbb{F}}_2)\) which we denote by \(\alpha(V)\). Note that the extension (8 ) is obtained from (13 ) for \(V=M/2\) by pulling back along \(\wedge^2 M\to \wedge^2(M/2)\). We have \(S^2V=(V^{\otimes 2})_{S_2}\cong (V^{\otimes 2})_{S_2,{\rm sgn}}\). The following lemma shows that when the dimension of \(V\) is small, the connecting map \(\alpha(V)\) defined by (13 ) is zero.

Proof. (i) Let \(u,v,w\) be the three non-zero elements of \(V\). Then the unique non-zero element in \(\wedge^2 V\) lifts to \(uv+vw+wu\in S^2V\). This lifting is \({\rm {GL}}(2,{\mathbb{F}}_2)\)-equivariant.

(ii) Assume that there is a non-zero element \(x\in (\wedge^2 V)^\Gamma\). We have a perfect bilinear pairing of \({\mathbb{F}}_2\)-vector spaces \(V\times \wedge^2V\to {\mathbb{F}}_2\). The elements of \(V\) that pair trivially with \(x\) form an \({\mathbb{F}}_2\)-subspace \(N\subset V\) of dimension 2 which is stable under the action of \(\Gamma\). Moreover, \(x\in \wedge^2N\). We have a commutative diagram of \(\Gamma\)-modules with exact rows \[\xymatrix{0\ar[r]&N\ar[r]\ar[d]&S^2 N\ar[r]\ar[d]&\wedge^2 N\ar[r]\ar[d]&0\\ 0\ar[r]& V\ar[r]& S^2V\ar[r]& \wedge^2 V\ar[r]& 0}\] By part (i), the top sequence is split. Thus \(x\) lifts to an element of \((S^2V)^\Gamma\). \(\Box\)

Proof. (i) Let \(e_1,\ldots,e_n\) be a \(\Gamma\)-stable basis of \(V\). The union of sets \(\{e_i^2\}\) and \(\{e_ie_j|i<j\}\) is a \(\Gamma\)-stable basis of \(S^2V\), which gives a \(\Gamma\)-equivariant splitting of \((\ref{alpha})\).

(ii) By (i) it is enough to show that any representation of \({\mathbb{Z}}/2\) in \(V\) is permutational, but this is well-known. Indeed, endow \(V\) with the structure of an \({\mathbb{F}}_2[x]\)-module by letting \(x\) act as the generator of \({\mathbb{Z}}/2\). By the classification of finitely generated torsion modules over a PID, the \({\mathbb{F}}_2[x]\)-module \(V\) is a direct sum of cyclic submodules isomorphic to \({\mathbb{F}}_2={\mathbb{F}}_2[x]/(x-1)\) or \({\mathbb{F}}_2[x]/(x^2-1)\), both of which are permutational. \(\Box\)

The proof of Theorem 13 relies on the following general computation.

Proof. The mod \(p^m\) reduction of the map \(Y\to X[2]\) is the Yoneda class of the complex \((X\xrightarrow{\psi} \widetilde{E})\otimes^L_{{\mathbb{Z}}_p}{\mathbb{Z}}/p^m\) concentrated in degrees \(-1\) and \(0\). It can be computed as the totalisation of the bicomplex \[\label{modp32bicomplex} \begin{tikzcd} X\arrow[r,"{\psi}"] & \widetilde{E}\\ X\arrow[r,"{\psi}"]\arrow[u, "{p^m}"] & \widetilde{E}\arrow[u, "{p^m}"] \end{tikzcd}\tag{14}\] The compositions of (?? ) and (?? ) are represented, respectively, by the following Yoneda extensions \[\label{deltabock32yoneda} 0\to X/p^m\xrightarrow{a} R_m\xrightarrow{b} Y/p^{m+1}\xrightarrow{c} Y/p^m\to 0\tag{15}\] \[\label{bockdelta32yoneda} 0\to X/p^m\xrightarrow{a'} X/p^{m+1}\xrightarrow{b'} T_m\xrightarrow{c'} Y/p^m\to 0,\tag{16}\] where \(R_m\) (respectively, \(T_m\)) is the following pushout (respectively, pullback) \[\begin{tikzcd} X/p\arrow[r]\arrow[d] & E\arrow[d] & & T_m \arrow[d]\arrow[r] & Y/p^m\arrow[d] \\ X/p^{m}\arrow[r] & R_m & & E \arrow[r] & Y/p \end{tikzcd}\] and the map \(b:R_m\to Y/p^{m+1}\) is the composition \(R_m\to Y/p\hookrightarrow Y/p^{m+1}\), while \(b':X/p^{m+1}\to T_m\) is the composition \(X/p^{m+1}\twoheadrightarrow X/p\to T_m\). The result of subtracting the map \(Y/p^m\to X/p^m[2]\) corresponding to (15 ) from that corresponding to (16 ) is then represented by the extension \[\label{difference32yoneda} X/p^m\xrightarrow{(a,0)} \frac{R_m\oplus X/p^{m+1}}{(a(x),-a'(x))|x\in X/p^m}\xrightarrow{b\oplus b'} \ker(Y/p^{m+1}\oplus T_m\xrightarrow{c-c'} Y/p^m)\xrightarrow{c} Y/p^m\tag{17}\] We can now write down a chain level map from the totalisation of (14 ) to the two-term complex from (17 ): \[\label{bock32main32map} \begin{tikzcd} X\arrow[r,"{(\psi,-p^m)}"] & \widetilde{E}\oplus X \arrow[r,"{(p^m,\psi)}"]\arrow[d,"{f_1}"] & \widetilde{E}\arrow[d, "{f_2}"] \\ & \frac{R_m\oplus X/p^{m+1}}{(a(x),-a'(x))|x\in X/p^m}\arrow[r,"{b\oplus b'}"] & \ker(Y/p^{m+1}\oplus T_m\xrightarrow{c-c'} Y/p^m) \end{tikzcd}\tag{18}\] Let \(f_1\) be the map induced by the direct sum of maps \(\widetilde{E}\to E\to R_m\) and \(X\twoheadrightarrow X/p^{m+1}\), and \(f_2\) is the sum of compositions \(\widetilde{E}\to Y\to Y/p^{m+1}\) and \(\widetilde{E}\twoheadrightarrow T_m\). The map \(f_2\) indeed lands in the kernel of \(c-c'\), the square in (18 ) commutes, and the composition \(f_1\circ (\psi,-p^m)\) is zero.

The complexes given by rows of (18 ) both have two non-zero cohomology groups given by \(X/p^m\) and \(Y/p^m\). The map of complexes induced by \(f_1,f_2\) induces the identity map on cohomology, giving the desired identification between the two classes in \({\rm Ext}^2_{\mathcal{A}(\Gamma,{\mathbb{Z}}_p)}(Y/p^m,X/p^m)\). \(\Box\)

We now specialise the above computation to our case of interest. As before, for our semiabelian variety \(A\) we denote the continuous \(\Gamma\)-module \({\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2)\) by \(M\). Recall that we write \(\alpha(M/2)\colon \wedge^2 (M/2)\to (M/2)[1]\) for the morphism in \(D(\Gamma,{\mathbb{F}}_2)\) (as well as the corresponding morphism between the same objects in \(D(\Gamma,{\mathbb{Z}}_2)\)) defined by (13 ) with \(V=M/2\).

The next statement is a more general variant of the particular case \(p=2\) of [7]. We give an elementary proof of this result using Lemma 12.

Proof. By Theorem 6 the degree \(2\) extension class corresponding to \(\tau^{[1,2]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}_2)\) is equal to the composition \[\wedge^2M\to\wedge^2(M/2)\xrightarrow{\alpha(M/2)}M/2[1]\xrightarrow{\mathrm{Bock}_M}M[2]\] To calculate the induced map on mod \(2^m\) reductions, we apply Lemma 12 to the case \(p=2\) with \(X=M\), \(Y=\wedge^2 M\), and the extension \(E\) given by \[0\to M/2\to S^2(M/2)\to \wedge^2(M/2)\to 0.\] We obtain that the extension class of \(\tau^{[1,2]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/2^m)\) is equal to the difference of the following compositions: \[\wedge^2(M/2^m)\longrightarrow\wedge^2(M/2)\stackrel{\alpha(M/2)}\longrightarrow(M/2)[1] \stackrel{\rm Bock}\longrightarrow(M/2^m)[2], \label{genius2}\tag{19}\] \[\wedge^2(M/2^m)\stackrel{\rm Bock}\longrightarrow\wedge^2(M/2)[1]\stackrel{\alpha(M/2)[1]}\longrightarrow(M/2)[2] \longrightarrow(M/2^m)[2]. \label{genius3}\tag{20}\] The differential \(\delta_2^{p,2}\) is equal to the map induced by this difference on the degree \(p\) derived functor of \(\Gamma\)-invariants, which gives the desired result. \(\Box\)

2 Jacobians↩︎

Let \(C\) be a smooth, projective, geometrically integral curve over a field \(k\). Let \({\boldsymbol{P}ic}_{C/k}\) be the Picard scheme of \(C\). Denote by \(J:={\boldsymbol{P}ic}^0_{C/k}\) the Jacobian of \(C\). The connected component \({\boldsymbol{P}ic}_{C/k}^{1}\subset{\boldsymbol{P}ic}_{C/k}\) parametrising divisors of degree 1 is a torsor for \(J\), usually called the Albanese torsor. There is a canonical map \(C\hookrightarrow {\boldsymbol{P}ic}_{C/k}^{1}\). After choosing a point \(x_0\in C(k_{\rm s})\), this map is identified with the usual Abel–Jacobi map \(C_{k_{\rm s}}\to J_{k_{\rm s}}\) sending \(x\) to the class of the divisor \(x-x_0\).

It is well known that \({\boldsymbol{P}ic}_{C/k}(k)\) is canonically isomorphic to \({\rm {Pic}}(C_{k_{\rm s}})^\Gamma\), see, e.g., [8]. Thus the class of the torsor \({\boldsymbol{P}ic}^1_{C/k}(k)\) in \({\rm H}^1(k,J)\) is zero if and only if \(C\) has a \(k\)-rational divisor class of degree 1.

Let \(n\) be an integer not divisible by \({\rm char}(k)\). As before, we denote by \(D(\Gamma,{\mathbb{Z}}/n)\) the derived category of the abelian category of discrete \({\mathbb{Z}}/n\)-modules equipped with a continuous action of \(\Gamma={\rm{Gal}}(k_{\rm s}/k)\).

Suppose that \(A\) and \(B\) are abelian varieties over \(k\) such that there are quasi-isomorphisms \[\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\simeq \bigoplus\limits_{i\geq 0} {\rm H}^i_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)[-i], \quad \mathrm{R}\Gamma_{{\rm{\acute et}}}(B_{k_{\rm s}},{\mathbb{Z}}/n)\simeq \bigoplus\limits_{j\geq 0} {\rm H}^j_{{\rm{\acute et}}}(B_{k_{\rm s}},{\mathbb{Z}}/n)[-j]\] in \(D(\Gamma,{\mathbb{Z}}/n)\). The Künneth formula gives a quasi-isomorphism \[\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{\rm s}}\times B_{k_{\rm s}},{\mathbb{Z}}/n)\simeq \mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\otimes^L_{{\mathbb{Z}}/n}\mathrm{R}\Gamma_{{\rm{\acute et}}}(B_{k_{\rm s}},{\mathbb{Z}}/n).\] The groups \({\rm H}^i_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\) and \({\rm H}^j_{{\rm{\acute et}}}(B_{k_{\rm s}},{\mathbb{Z}}/n)\) are free \({\mathbb{Z}}/n\)-modules for all \(i\) and \(j\), thus for each \(m\geq 0\) we have isomorphisms \[{\rm H}^m_{\rm{\acute et}}(A_{k_{\rm s}}\times B_{k_{\rm s}})\cong\bigoplus_{i+j=m} {\rm H}^i_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\otimes_{{\mathbb{Z}}/n}{\rm H}^j_{{\rm{\acute et}}}(B_{k_{\rm s}},{\mathbb{Z}}/n).\] We deduce a quasi-isomorphism \[\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{\rm s}}\times B_{k_{\rm s}},{\mathbb{Z}}/n)\simeq \bigoplus\limits_{m\geq 0} {\rm H}^m_{{\rm{\acute et}}}(A_{k_{\rm s}}\times B_{k_{\rm s}},{\mathbb{Z}}/n)[-m].\] Conversely, given such a quasi-isomorphism, we can produce a quasi-isomorphism \[\begin{gather} \mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/n)\xrightarrow{p_1^*}\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{\rm s}}\times B_{k_{\rm s}},{\mathbb{Z}}/n)\simeq \bigoplus\limits_{m\geq 0} {\rm H}^m_{{\rm{\acute et}}}(A_{k_{\rm s}}\times B_{k_{\rm s}},{\mathbb{Z}}/n)[-m]\\ \xrightarrow{({\rm id}_{A_{k_{{\rm s}}}}\times e_B)^*}\bigoplus\limits_{m\geq 0}{\rm H}^m_{\rm{\acute et}}(A_{k_{{\rm s}}},{\mathbb{Z}}/n)[-m]. \end{gather}\] Hence direct products of abelian varieties with decomposable étale cohomology complex, as well as direct factors of such varieties, also have decomposable étale cohomology complexes. Therefore Theorem 3 of the introduction is a consequence of the following

Proof. We will deduce this decomposition from the fact that \(J\) admits integral Künneth projectors defined over the field \(k\), as proved in [16].

Composing the canonical map \(C\to {\boldsymbol{P}ic}_{C/k}^{1}\) with an isomorphism of torsors \({\boldsymbol{P}ic}_{C/k}^{1}\cong J\) provided by the given degree \(1\) divisor class, we get a map \(\alpha:C\to J\). Denote by \(\alpha_n:\mathrm{Sym}^nC\to J\) the maps from the symmetric powers of \(C\) induced by \(\alpha\) using the group structure on \(J\). For each \(1\leq n\leq g\), denote by \(w^{[n]}\in Z^n(J)\) the codimension \(n\) cycle in the Jacobian variety obtained by pushing forward the fundamental cycle \([\mathrm{Sym}^{g-n} C]\) along \(\alpha_{g-n}\). We also denote by \(w^{[0]}\in Z^0(J)\) the fundamental cycle of \(J\) itself.

Following [16], for each \(0\leq i\leq 2g\) we define a codimension \(g\) cycle on \(J\times J\) by the formula \[\pi_i:=(-1)^i\displaystyle\sum_{\substack{2a+b=2g-i \\ b+2c=i}}p_1^*w^{[a]}\cdot p_2^*w^{[c]}\cdot \sum\limits_{d+e+f=b}(-1)^{d+f}p_1^*w^{[d]}\cdot \mu^*w^{[e]}\cdot p_2^*w^{[f]}\] where \(p_1,p_2,\mu:J\times J\to J\) are the two projections and the multiplication map, respectively.

We will use the following key property of the cycles \(\pi_i\), established in [16]:

Proof. This is proved in [16] in the case \(k_{\rm s}=\mathbb{C}\). The proof goes through over an arbitrary separably closed field: its only geometric input is Poincaré’s formula \(r!\cdot w^{[r]}=(w^{[1]})^{r}\) in cohomology of \(J\), see [17] for a proof of this over arbitrary fields. \(\Box\)

Let us now upgrade the action of \(\pi_i\) from individual cohomology groups to the complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\), as an object of \(D(\Gamma,{\mathbb{Z}}/n)\).

Proof. The two projections \(p_1,p_2:Y\times Y\to Y\) define maps in \(D(\Gamma,{\mathbb{Z}}/n)\): \[p_1^*:\mathrm{R}\Gamma_{{\rm{\acute et}}}(Y_{k_{\rm s}},{\mathbb{Z}}/n)\to \mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)\] and \[p_{2*}:\mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)\to \mathrm{R}\Gamma_{{\rm{\acute et}}}(Y_{k_{{\rm s}}},{\mathbb{Z}}/n)(-d)[-2d],\] where \((-d)\) refers to the twist by the \((-d)\)-th power of the cyclotomic character \(\Gamma\to {\rm Aut}_{{\mathbb{Z}}/n}(\mu_n)\).

There are cycle classes \({\rm cl}(c)\in {\rm H}^{2i}_{{\rm{\acute et}}}(Y\times Y,({\mathbb{Z}}/n)(i))\) in absolute étale cohomology of \(Y\times Y\), as defined in [18]. The absolute étale cohomology complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(Y\times Y,({\mathbb{Z}}/n)(i))\) can be identified with the continuous group cohomology complex \(\mathrm{R}\Gamma_{\mathrm{cont}}(\Gamma, \mathrm{R}\Gamma_{\rm{\acute et}}((Y\times Y)_{k_{\rm s}},({\mathbb{Z}}/n)(i)))\) with coefficients in geometric étale cohomology, hence \({\rm cl}(c)\) corresponds to a map \({\rm cl}(c):({\mathbb{Z}}/n)[-2i]\to \mathrm{R}\Gamma_{\rm{\acute et}}((Y\times Y)_{k_{\rm s}},({\mathbb{Z}}/n)(i))\) in the derived category \(D(\Gamma,{\mathbb{Z}}/n)\).

We can now define the endomorphism of the complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(Y_{k_{\rm s}},{\mathbb{Z}}/n)\) induced by \(c\) as the composition \[\begin{gather} \mathrm{R}\Gamma_{{\rm{\acute et}}}(Y_{k_{\rm s}},{\mathbb{Z}}/n)\xrightarrow{p_1^*} \mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)\xrightarrow{\cup {\rm cl}(c)}\\ \mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)(i)[2i]\xrightarrow{p_{2*}}\mathrm{R}\Gamma_{{\rm{\acute et}}}(Y_{k_{\rm s}},{\mathbb{Z}}/n)(i-d)[2i-2d] \end{gather}\] Here the middle map \(\cup {\rm cl}(c)\) denotes the composition \[\begin{gather} \mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)\xrightarrow{{\rm id}\otimes {\rm cl}(c)}\mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)^{\otimes 2}(i)[2i]\xrightarrow{}\\ \mathrm{R}\Gamma_{{\rm{\acute et}}}((Y\times Y)_{k_{\rm s}},{\mathbb{Z}}/n)(i)[2i] \end{gather}\] where the second map is the cup-product on the level of cohomology complexes. \(\Box\)

We can now construct the desired quasi-isomorphism (?? ). It suffices to produce, for each \(i\geq 0\), a map \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\to {\rm H}^i_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)[-i]\) in \(D(\Gamma,{\mathbb{Z}}/n)\) that induces the identity map on the \(i\)-th cohomology. By Lemma 16, the cycle \(\pi_i\) defines a map \([\pi_i]:\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\to \mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\) in \(D(\Gamma,{\mathbb{Z}}/n)\) inducing \(0\) on \({\rm H}^j\) for \(j\neq i\), and the identity map on \({\rm H}^i\). The endomorphism \([\pi_i]\) fits into the following map of distinguished triangles \[\begin{tikzcd} \tau^{\leq i}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\arrow[r]\arrow[d,"{\tau^{\leq i}[\pi_i]}"] & \mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n) \arrow[d,"{[\pi_i]}"]\arrow[r] & \tau^{\geq i+1}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\arrow[d,"{\tau^{\geq i+1}[\pi_i]}"] \\ \tau^{\leq i}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\arrow[r] & \mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n) \arrow[r] & \tau^{\geq i+1}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n) \end{tikzcd}\] The map \(\tau^{\geq i+1}[\pi_i]\) induces the zero map on all cohomology groups, but a priori it may be a non-zero map in \(D(\Gamma,{\mathbb{Z}}/n)\). Since the complex \(\tau^{\geq i+1}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\) has non-zero cohomology only in the range \([i+1,2g]\), the \((2g-i)\)-th power of the endomorphism \(\tau^{\geq i+1}[\pi_i]\) is nonetheless equal to zero in \(D(\Gamma,{\mathbb{Z}}/n)\), see Lemma 17 below.

Therefore \([\pi_i]^{2g-i}\) gives rise to a map \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\to \tau^{\leq i}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\) inducing the identity map on \(i\)-th cohomology. Composing it with the natural map \[\tau^{\leq i}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\to \tau^{[i,i]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\cong {\rm H}^i_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)[-i]\] we obtain the desired map \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)\to {\rm H}^i_{{\rm{\acute et}}}(J_{k_{\rm s}},{\mathbb{Z}}/n)[-i]\) in \(D(\Gamma,{\mathbb{Z}}/n)\) that induces the identity map on \({\rm H}^i\). Summing up these maps over all \(i\in [0,2g]\) we obtain the quasi-isomorphism (?? ).

Applying the derived functor of \(\Gamma\)-invariants to (?? ) gives the isomorphisms (?? ).\(\Box\)

Proof. Without loss of generality we can assume \(a=0\). Proceed by induction on \(n\). If \(n=0\), then \(C\) is isomorphic in \(D^b(A)\) to an object represented by the one-term complex \({\rm H}^0(C)\) in degree \(0\). Under this isomorphism, \(f\) corresponds to an endomorphism induced by \({\rm H}^0(f)\in{\rm {Hom}}_A({\rm H}^0(C),{\rm H}^0(C))\), but \({\rm H}^0(f)=0\).

Now suppose that \(f^n\) is zero on \(\tau^{<n}C\). The composition \(C\xrightarrow{f} C \to {\rm H}^n(C)[-n]\) is zero, because it is equal to the composition \(C\to {\rm H}^n(C)[-n]\to {\rm H}^n(C)[-n]\), where the last map is \({\rm H}^n(f)=0\). Given the distinguished triangle \[\tau^{<n}C\to C\to{\rm H}^n(C)[-n],\] this implies that \(f\colon C\to C\) factors as \(C\xrightarrow{g}\tau^{<n}C\to C\) for some map \(g\). Then \(f^{n+1}\) is the composition \(C\xrightarrow{g}\tau^{<n}C\to C\xrightarrow{f^{n}}C\). The truncation \(\tau^{<n}C\) is functorial in \(C\), so the composition of the last two maps is equal to \(\tau^{<n}C\xrightarrow{f^{n}}\tau^{<n}C\to C\). By the induction assumption \(f^n\) is zero on \(\tau^{<n}C\), so the iterate \(f^{n+1}\) is zero on \(C\). \(\Box\)

3 Abelian surfaces over \({\mathbb{Q}}\) with \(\delta_2^{0,2}\neq 0\)↩︎

In this section we give examples of abelian varieties \(A\) over \({\mathbb{Q}}\) such that the differential \(\delta_2^{0,2}\) of the spectral sequence \[E_{2}^{p,q}= {\rm H}^p(k,{\rm H}_{\rm{\acute et}}^q(A_{k_{\rm s}},{\mathbb{Z}}_2(1)))\Rightarrow{\rm H}_{\rm{\acute et}}^{p+q}(A,{\mathbb{Z}}_2(1)), \label{ss32twist}\tag{21}\] and the same differential of the spectral sequence with coefficients \({\mathbb{Z}}/2\), are both non-zero.

Let \(A\) be an abelian variety over a field \(k\). We write \(A^\vee\) for the dual abelian variety. We denote by \(\beta_2: {\rm H}^1(k, A)\to {\rm H}^2(k, A[2])\) the connecting homomorphism induced by the short exact sequence of \(\Gamma\)-modules \[0\to A[2](k_{{\rm s}})\to A(k_{{\rm s}})\xrightarrow{2}A(k_{{\rm s}})\to 0.\]

We use the following crucial proposition.

Proof. The antipodal involution \([-1]\colon A\to A\) induces an action of \({\mathbb{Z}}/2\) on \({\rm {Pic}}(A_{k_{\rm s}})\) which turns (?? ) into an exact sequence of abelian groups with an action of \({\mathbb{Z}}/2\). The induced action on \({\rm NS\,}(A_{k_{\rm s}})\) is trivial, cf. [19]. The involution \([-1]_A\) induces the involution \([-1]_{A^\vee}\) on \(A^\vee\). Recall that for an abelian group \(M\) with an action of \({\mathbb{Z}}/2=\langle\sigma\rangle\) we have \({\rm H}^1({\mathbb{Z}}/2,M)\cong M^{\sigma=-1}/(1-\sigma)M\). Since \(A^\vee(k_{\rm s})\) is 2-divisible, the endomorphism \(1-[-1]_{A^{\vee}}=[2]_{A^{\vee}}\) is surjective, and we obtain \({\rm H}^1({\mathbb{Z}}/2,A^\vee(k_{\rm s}))=0\). Thus the long exact sequence of cohomology for the group \({\mathbb{Z}}/2\) gives an exact sequence of \(\Gamma\)-modules \[0 \longrightarrow A^\vee[2] \longrightarrow{\rm {Pic}}(A_{k_{\rm s}})^{[-1]^*} \longrightarrow{\rm NS\,}(A_{k_{\rm s}}) \longrightarrow 0. \label{g2}\tag{22}\] Let \(c_\lambda\in {\rm H}^1(k,A^\vee[2])\) be the image of \(\lambda\) under the connecting map of \((\ref{g2})\). It is clear that \(c'_\lambda\) is the image of \(c_\lambda\) in \({\rm H}^1(k,A^\vee)\).

Let \(T_2(A^\vee)\) be the 2-adic Tate module, and let \(e_2\colon A[2]\times A^\vee[2]\to{\mathbb{Z}}/2\) be the Weil pairing. The short exact sequences (22 ) and (13 ) are compatible, so that there is a commutative diagram (see diagram (16) in [20]): \[\xymatrix{ 0 \ar[r]& A^\vee[2] \ar[r]\ar[d]^{\rm id}& {\rm {Pic}}(A_{k_{\rm s}})^{[-1]^*} \ar[r]\ar[d]& {\rm NS\,}(A_{k_{\rm s}}) \ar[r]\ar[d]& 0\\ 0 \ar[r]& A^\vee[2] \ar[r]&S^2(A^\vee[2])\ar[r]&\wedge^2(A^\vee[2])\ar[r]&0}\] Furthermore, the right-hand vertical map sends \(\lambda\) to the element of \(\wedge^2(A^\vee[2])\) corresponding to \[e_2(x,\lambda(y))\in {\rm {Hom}}(\wedge^2A[2],{\mathbb{Z}}/2).\] This map factors as the first Chern class map \({\rm c}_1\) \[{\rm NS\,}(A_{k_{\rm s}})\cong{\rm {Hom}}(A_{k_{\rm s}},A^\vee_{k_{\rm s}})^{\rm sym}\stackrel{{\rm c}_1}\longrightarrow{\rm {Hom}}(T_2(A),T_2(A^\vee))^{\rm sym}\cong\wedge^2 T_2(A^\vee)(-1) \cong{\rm H}^2_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}_2(1))\] followed by the reduction modulo 2 map \({\rm H}^2_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}_2(1))\to{\rm H}^2_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/2)\), see [21]. Theorem 6 gives \[\label{main32formula32on32clambda} \delta_2^{0,2}({\rm c}_1(\lambda))={\rm Bock}_{T_2(A^\vee)}(c_\lambda),\tag{23}\] where \({\rm Bock}_{T_2(A^\vee)}\) is the connecting map attached to the exact sequence \[0\to T_2(A^\vee)\xrightarrow{2} T_2(A^\vee)\to A^\vee[2]\to 0.\] This sequence fits into the following commutative diagram with exact rows: \[\xymatrix{0\ar[r]&A^\vee[2]\ar[r]& A^\vee\ar[r]^{2} &A^\vee\ar[r]& 0\\ 0\ar[r]& A^\vee[2]\ar[r]\ar[u]^\cong& A^\vee[4]\ar[r]\ar[u] &A^\vee[2]\ar[r]\ar[u]& 0\\ 0\ar[r]& T_2(A^\vee)\ar[r]^{2}\ar[u]& T_2(A^\vee)\ar[r]\ar[u]& A^\vee[2]\ar[r]\ar[u]_\cong& 0}\] This implies a relation between connecting homomorphisms induced by the first and last rows: the image of \(\mathrm{Bock}_{T_2(A^{\vee})}(c_{\lambda})\in {\rm H}^2(k, T_2A^{\vee})\) in \({\rm H}^2(k, A^{\vee}[2])\) equals \(\beta_2(c'_{\lambda})\), where \(\beta_2:{\rm H}^1(k, A^{\vee})\to {\rm H}^2(k, A^{\vee}[2])\) is the connecting homomorphism induced by the top exact row, as defined above. Combining this with (23 ) gives the desired equality.

To prove the final assertion of the proposition, note that the short exact sequence of Galois modules \[0\to {\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2(1))\xrightarrow{2}{\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2(1))\to {\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/2)\to 0\] shows that \(\delta^{0,2}_2(c_1(\lambda))\in {\rm H}^2(k,{\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}_2(1)))\) is divisible by \(2\) if and only if its image in \({\rm H}^2(k, {\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/2))\cong {\rm H}^2(k, A^{\vee}[2])\) is zero. Similarly, the short exact sequence of Galois modules \[0\to A^{\vee}[2]\to A^{\vee}(k_{{\rm s}})\xrightarrow{2}A^{\vee}(k_{{\rm s}})\to 0\] gives that \(c_{\lambda}'\in {\rm H}^1(k,A^{\vee})\) is divisible by \(2\) if and only if its image \(\beta_2(c'_{\lambda})\in {\rm H}^2(k,A^{\vee}[2])\) under the connecting homomorphism \(\beta_2\) is zero. This implies the equivalence of \(2\)-divisibility of \(\delta_{2}^{0,2}(c_1(\lambda))\) and \(c'_{\lambda}\). \(\Box\)

Proposition 18 demonstrates that the Hochschild–Serre spectral sequence for \(A\) (both with \({\mathbb{Z}}_2\) and \({\mathbb{Z}}/2\) coefficients) does not degenerate if the class \(c'_\lambda\in {\rm H}^1(k,A^{\vee})\) is not divisible by \(2\). In the course of the above proof we also introduced the class \(c_{\lambda}\in {\rm H}^1(k,A^{\vee}[2])\) lifting \(c_{\lambda}'\). Classes \(c_{\lambda}\) and \(c_{\lambda}'\) have an explicit interpretation when \(A\) is the Jacobian of a curve, which is particularly convenient when \(C\) is hyperelliptic.

Let \(C\) be a smooth, projective, geometrically integral curve of genus \(g\) over a field \(k\) of characteristic not equal to 2. Let \(J:={\boldsymbol{P}ic}^0_{C/k}\) be the Jacobian of \(C\) with its canonical principal polarisation \(\lambda\). By [20], \(c_\lambda\) is the class of the torsor of theta characteristics, which is the closed subvariety of \({\boldsymbol{P}ic}_{C/k}^{g-1}\) given by \(2x=K_C\), where \(K_C\) is the canonical class. Thus \(c'_\lambda\) is the class of \({\boldsymbol{P}ic}_{C/k}^{g-1}\) in \({\rm H}^1(k,J)\).

If \(C\) is a hyperelliptic curve of odd genus or with a rational Weierstrass point, then \(c_\lambda=0\), see [20]. If \(C\) is a hyperelliptic curve of even genus, then we have an isomorphism \({\boldsymbol{P}ic}_{C/k}^{g-1}\cong{\boldsymbol{P}ic}_{C/k}^{1}\) given by subtracting \(\frac{g-2}{2} H_C\), where \(H_C\) is the hyperelliptic divisor class. Then \(c_\lambda\) is the class of the torsor for \(J[2]\) in \({\boldsymbol{P}ic}_{C/k}^{1}\) given by \(2x=H_C\), and \(c'_\lambda=[{\boldsymbol{P}ic}_{C/k}^{1}]\). Explicitly, the \(k_{\rm s}\)-points of the torsor of theta characteristics are Galois-equivariantly identified with the subsets of the set of Weierstrass points of \(C\) of odd cardinality, after identifying each subset with its complement.

Now let \(k\) be a number field. In this case \(c_\lambda\) lies in the 2-Selmer group \({\rm Sel}_2(J)\), see [6]. Thus \(c'_\lambda\) lies in the Tate–Shafarevich group \(\textcyr{Sh}(J)\).

We have the Cassels–Tate pairing \(\textcyr{Sh}(J)\times \textcyr{Sh}(J^\vee)\to{\mathbb{Q}}/{\mathbb{Z}}\), see [22]. Identifying \(J\) with its dual abelian variety \(J^\vee\) via \(\lambda\), we obtain a pairing \[\langle x,y\rangle \colon \textcyr{Sh}(J)\times \textcyr{Sh}(J)\to{\mathbb{Q}}/{\mathbb{Z}},\] which we shall also call the Cassels–Tate pairing. By [6] we have \(\langle x,x+c'_\lambda\rangle=0\) for any \(x\in \textcyr{Sh}(J)\).

For a finite \(\Gamma\)-module \(M\) one defines \(\textcyr{Sh}^1(k,M)\) as the subgroup of \({\rm H}^1(k,M)\) consisting of the classes that restrict to zero in \({\rm H}^1(k_v,M)\), for all places \(v\) of \(k\). It is clear that \(\textcyr{Sh}^1(k,J[2])\subset{\rm Sel}_2(J)\).

Over a number field \(k\), we can test the divisibility of \(c'_\lambda\) in \({\rm H}^1(k,J)\) in terms of the Cassels–Tate pairing. Let us denote by \(i\) the natural map \({\rm H}^1(k,J[2])\to{\rm H}^1(k,J)\).

Recall that a curve \(C\) over a number field \(k\) is called odd if \(\langle c'_\lambda,c'_\lambda\rangle\neq 0\).

Proof. This is a particular case of [4], which is based on the non-degeneracy of the Poitou–Tate pairing [22] \[\textcyr{Sh}^1(k,J[2])\times \textcyr{Sh}^2(k,J^\vee[2])\to{\mathbb{Z}}/2\] and its compatibility with the Cassels–Tate pairing \(\textcyr{Sh}(J)\times \textcyr{Sh}(J^\vee)\to{\mathbb{Q}}/{\mathbb{Z}}\). For the second claim we consider the pairing of \(c'_\lambda=i(c_\lambda)\) with itself. \(\Box\)

Manin pointed out that the Cassels–Tate pairing can be interpreted as a particular case of what is now called the Brauer–Manin pairing. Let us recall this interpretation. Let \({\rm{Br}}(C)\) be the Brauer group of \(C\) and let \({\rm{Br}}_0(C)={\rm {Im}}[{\rm{Br}}(k)\to{\rm{Br}}(C)]\). Since \({\rm{Br}}(C_{k_{\rm s}})=0\), the spectral sequence \({\rm H}^p(k,{\rm H}^q_{\rm{\acute et}}(C_{k_{\rm s}},{\mathbb{G}}_m))\Rightarrow{\rm H}^{p+q}_{\rm{\acute et}}(C,{\mathbb{G}}_m)\) gives an exact sequence \[0\to {\rm{Br}}_0(C)\to{\rm{Br}}(C)\to{\rm H}^1(k,{\rm {Pic}}(C_{k_{\rm s}}))\to 0, \label{night}\tag{24}\] where we used that \({\rm H}^3(k,k_{\rm s}^\times)=0\) which holds since \(k\) is a number field. We have an exact sequence of \(\Gamma\)-modules \[0\to J(k_{\rm s})\to {\rm {Pic}}(C_{k_{\rm s}})\xrightarrow{{\rm{deg}}}{\mathbb{Z}}\to 0.\] It induces an isomorphism of \({\rm H}^1(k,{\rm {Pic}}(C_{k_{\rm s}}))\) with the quotient of \({\rm H}^1(k,J)\) by the cyclic subgroup generated by \([{\boldsymbol{P}ic}_{C/k}^{1}]\). For an element \(x\in{\rm H}^1(k,J)\) we say that a Brauer class \({\mathcal{A}}\in {\rm{Br}}(C)\) is associated to \(x\) if the images of \(x\) and \({\mathcal{A}}\) in \({\rm H}^1(k,{\rm {Pic}}(C_{k_{\rm s}}))\) are equal. If \(\mathcal{A}\) is associated to \(x\), then there is a natural choice of an extension of \(\mathcal{A}\) to a Brauer class on \({\boldsymbol{P}ic}_{C/k}^{1}\).

Proof. Since \(C\) is hyperelliptic of even genus, we have \({\boldsymbol{P}ic}_{C/k}^{g-1}\cong {\boldsymbol{P}ic}_{C/k}^{1}\). For each place of \(k\) choose a local point \(P_v\in C(k_v)\subset {\boldsymbol{P}ic}_{C/k}^{1}(k_v)\). By a theorem of Manin [23], we have \[\sum_v{\rm inv}_v({\mathcal{A}}(P_v))=\langle x,c'_\lambda\rangle\in\frac{1}{2}{\mathbb{Z}}/{\mathbb{Z}},\] for any \(x\in\textcyr{Sh}(J)\) and \({\mathcal{A}}\in {\rm{Br}}(C)\) associated to \(x\), using that \(c'_\lambda=[{\boldsymbol{P}ic}_{C/k}^{1}]\in\textcyr{Sh}(J)\). Taking \(x\in i(\textcyr{Sh}^1(k,J[2]))\) we conclude by Lemma 19. \(\Box\)

Proof. (a) By Creutz [4] and Creutz and Viray [5], this curve satisfies the assumption of Lemma 21. Here is a sketch of this calculation.

Let \(C\) be the hyperelliptic curve over a field \(k\) of characteristic different from 2 given by \(y^2=cf(x)\) where \(c\in k^\times\) and \(f(x)\in k[x]\) is a separable monic polynomial of even degree. Let \(J\) be the Jacobian of \(C\). Let \(L=k[x]/(f(x))\) and let \(\theta\in L\) be the image of \(x\). The well-known identification of the group \(k\)-scheme \(J[2]\) with the quotient of the kernel of the norm map \(R_{L/k}(\mu_2)\to \mu_2\) by the image of \(\mu_2\) gives a natural inclusion \[(L^\times/k^\times L^{\times 2})_1\subset{\rm H}^1(k,J[2])/\langle c_\lambda\rangle,\] where the subscript \(1\) denotes the subgroup of elements with norm \(1\in k^\times/k^{\times 2}\). To \(l\in L^\times\) we associate the class of the quaternion algebra \((l,x-\theta)\in{\rm{Br}}(L(x))\), which only depends on the image of \(l\) in \(L^\times/L^{\times 2}\). The corestriction \[{\rm cores}_{L(x)/k(x)}\big((l,x-\theta)\big)\in {\rm{Br}}(k({\mathbb{P}}^1_k))\] is unramified away from the ramification locus of \(C\to {\mathbb{P}}^1_k\) and the point at infinity. If the norm of \(l\) is a square in \(k^\times\), then it is unramified at infinity. It follows that the pullback of this element to \({\rm{Br}}(k(C))\) is unramified, and so belongs to the subgroup \({\rm{Br}}(C)\subset {\rm{Br}}(k(C))\). We denote the resulting element by \({\mathcal{A}}_l\in{\rm{Br}}(C)\). Finally, multiplying \(l\) by \(s\in k^\times\) gives \({\mathcal{A}}_{ls}={\mathcal{A}}_l+(s,c)\), so this does not change the image of \({\mathcal{A}}_l\) in \({\rm{Br}}(C)/{\rm{Br}}_0(C)\). An important property of this construction is that the map sending \(l\) to \({\mathcal{A}}_l\) coincides with the composition \[{\rm H}^1(k,J[2])/\langle c_\lambda\rangle\stackrel{i}\longrightarrow{\rm H}^1(k,J)/\langle c'_\lambda\rangle\longrightarrow{\rm{Br}}(C)/{\rm{Br}}_0(C),\] where the second arrow comes from (24 ).

Now let \(k={\mathbb{Q}}\) and \(L={\mathbb{Q}}(\sqrt{-1})\oplus{\mathbb{Q}}(\sqrt{-17})\oplus{\mathbb{Q}}(\sqrt{17})\). The 0-dimensional scheme \({\rm{Spec}}(L)\) over \({\mathbb{Q}}\) is everywhere locally soluble, hence the curve \(C\) given by \[y^2=c(x^2+1)(x^2+17)(x^2-17)\] is everywhere locally soluble for any \(c\in {\mathbb{Q}}^\times\) and we have \(c_\lambda\in\textcyr{Sh}^1({\mathbb{Q}},J[2])\). Take \(l=(1,1,-1)\in L\). One checks that \(l\) gives rise to an everywhere locally trivial element of \((L^\times/{\mathbb{Q}}^\times L^{\times 2})_1\), and thus to an element of \(\textcyr{Sh}^1({\mathbb{Q}},J[2])/\langle c_\lambda\rangle\). We have \({\mathcal{A}}_l=(-1,x^2-17)\). A local computation [5] shows that if the number of odd prime factors \(p\) of \(c>0\) such that neither \(17\) nor \(-1\) is a square modulo \(p\), is odd, then \(\sum_v{\rm inv}_v({\mathcal{A}}_l(P_v))=1/2\). For example, one can take \(c=3\). Thus Lemma 21 can be applied, so that \(c'_\lambda\) is not divisible by \(2\) in \({\rm H}^1({\mathbb{Q}},J)\), hence \(\delta^{0,2}_2\) is non-zero by Proposition 18.

(b) This curve was also communicated to us by Creutz (personal communication). It is the quadratic twist by \(-1\) of the curve in [20]. The arguments in loc. cit. show that for the primes \(p\) where \(C\) has bad reduction, and also for \(p=2\), the polynomial \(f(x)\) has a root in \({\mathbb{Q}}_p\). In view of Lemma 19 and Remark [poonenrains] it remains to show that \(C\) is odd. By [6] this happens if and only if the number of places \(v\) such that \(C\) does not have a \({\mathbb{Q}}_v\)-rational divisor of degree 1, is odd. Such places are necessarily real or places of bad reduction, so we only need to consider the real place. We check that \(C({\mathbb{R}})=\emptyset\), so \(C\) has no real divisor of degree 1, and hence is odd. \(\Box\)

4 The Brauer group of a torus↩︎

Let \(T\) be a torus over a field \(k\) of characteristic exponent \(p\). Let \(\widehat{T}={\rm {Hom}}_{k_{\rm s}-{\rm gps}}(T_{k_{\rm s}},{\mathbb{G}}_m)\) be the \(\Gamma\)-module of characters of \(T\). The Brauer group \({\rm{Br}}(T)={\rm H}^2_{\rm{\acute et}}(T,{\mathbb{G}}_m)\) is computed by the spectral sequence \[E_{2}^{p,q} = {\rm H}^p(k, {\rm H}^q_{{\rm{\acute et}}}(T_{k_{\rm s}}, {\mathbb{G}}_{m})) \Rightarrow {\rm H}^{p+q}_{{\rm{\acute et}}}(T, {\mathbb{G}}_{m}). \label{Leray-torus}\tag{25}\] We denote by \(d^{p,q}_r\) the differential on the \(r\)th page emanating from the \((p,q)\)-entry.

Since \(T_{k_{\rm s}}\) is a dense open subscheme of \({\mathbb{A}}^n_{k_{\rm s}}\), we have \({\rm H}^1(T_{k_{\rm s}},{\mathbb{G}}_{m})\cong{\rm {Pic}}(T_{k_{\rm s}})=0\). Thus \(d^{0,2}_2=0\), and the first interesting differential is \(d^{\,0,2}_3\colon{\rm{Br}}(T_{k_{\rm s}})^\Gamma\to {\rm H}^3(k,k_{\rm s}[T]^\times)\).

The origin \(e\in T(k)\) of the group law on \(T\) gives a section of the structure morphism \(T\to{\rm{Spec}}(k)\), hence \({\rm H}^r(k,{\mathbb{G}}_m)\to{\rm H}^r_{\rm{\acute et}}(T,{\mathbb{G}}_m)\) is injective for all \(r\geq 0\). Likewise, the natural map \({\rm{Br}}(k)\to{\rm{Br}}(T)\) is injective. Let \({\rm{Br}}_e(T) ={\rm{Ker}}[{\rm{Br}}(T) \to {\rm{Br}}(k)]\) be the kernel of specialisation at \(e\).

We have an isomorphism of \(\Gamma\)-modules \({\rm H}^0_{\rm{\acute et}}(T_{k_{\rm s}},{\mathbb{G}}_{m})=k_{\rm s}[T]^\times\cong k_{\rm s}^\times \oplus \widehat{T}\) (cf. [24]). The spectral sequence (25 ) thus gives rise to an exact sequence \[0 \to {\rm H}^2(k,\widehat{T}) \to {\rm{Br}}_e(T) \to {\rm{Br}}(T_{k_{\rm s}})^\Gamma\xrightarrow{\bar d_3^{\,0,2}} {\rm H}^3(k, \widehat{T}). \label{ias1}\tag{26}\] To compute \({\rm{Br}}(T)\) one needs to describe the map \(\bar d_3^{\, 0,2}\colon {\rm{Br}}(T_{k_{\rm s}})^\Gamma\to {\rm H}^3(k, \widehat{T})\), which is the composition of the differential \(d_3^{0,2}\) with the map induced by the projection \(k_{{\rm s}}[T]^{\times}\to \widehat{T}\).

Let \(\ell\) be a prime number not equal to \(p\). For an abelian group \(A\) we write \(A(p')\) for the subgroup of \(A\) consisting of the elements of finite order not divisible by \(p\).

Since \({\rm {Pic}}(T_{k_{\rm s}})=0\), the Kummer sequence gives rise to an isomorphism \[\kappa\colon{\rm H}^2_{\rm{\acute et}}(T_{k_{\rm s}},\mu_{\ell^n})\tilde{\longrightarrow{\rm{Br}}}(T_{k_{\rm s}})[\ell^n].\] Using the isomorphism of \(\Gamma\)-modules \({\rm H}^0_{\rm{\acute et}}(T_{k_{\rm s}},{\mathbb{G}}_{m})\cong k_{\rm s}^\times \oplus \widehat{T}\), we also deduce from the Kummer sequence a natural isomorphism \[\widehat T/\ell^n\tilde{\longrightarrow{\rm H}}^1_{\rm{\acute et}}(T_{k_{\rm s}},\mu_{\ell^n}).\] We note that \({\rm H}^2_{\rm{\acute et}}(T_{k_{\rm s}},{\mathbb{Z}}/\ell^n)\cong\wedge^2{\rm H}^1_{\rm{\acute et}}(T_{k_{\rm s}},{\mathbb{Z}}/\ell^n)\) and thus the multiplication by \(m\) map \([m]\colon T\to T\) acts on \({\rm H}^2_{\rm{\acute et}}(T_{k_{\rm s}},\mu_{\ell^n})\), and hence on \({\rm{Br}}(T_{k_{\rm s}})(p')\), as \(m^2\). On the other hand, \([m]\) acts on \(\widehat{T}\) as \(m\). Taking \(m=-1\) we see that \(2\bar d_3^{\, 0,2}=0\), so that \(\bar d_3^{\, 0,2}\) sends the elements of \({\rm{Br}}(T_{k_{\rm s}})(p')\) of odd order to zero. Then it follows from (26 ) that every element of odd order in \({\rm{Br}}(T_{k_{\rm s}})(p')^\Gamma\) lifts to \({\rm{Br}}(T)\).

The question of an explicit description of the map \(\bar d_3^{\, 0,2}\) on the 2-primary torsion subgroup of \({\rm{Br}}(T_{k_{\rm s}})^\Gamma\) was asked at the top of [8]. We now give such a description.

Proof. We will show that the restriction of \(d_3^{0,2}\) to the \(2^n\)-torsion subgroup can be read off from the Hochschild–Serre spectral sequence for the cohomology of \(T\) with coefficients in \(\mu_{2^n}\). The natural map \(\tau^{\leq 2}\mathrm{R}\Gamma_{{\rm{\acute et}}}(T_{k_{{\rm s}}},\mu_{2^n})\to \tau^{\leq 2}\mathrm{R}\Gamma_{{\rm{\acute et}}}(T_{k_{{\rm s}}},{\mathbb{G}}_m)\) of truncations in degrees \(\leq 2\) induces a commutative diagram relating the extensions between \({\rm H}^2\) and \(\tau^{\leq 1}\) in these complexes: \[\label{torus:32kummer32diagram} \begin{tikzcd} {\rm H}^2_{{\rm{\acute et}}}(T_{k_{{\rm s}}},\mu_{2^n})\arrow[r]\arrow[d] & \tau^{\leq 1}\mathrm{R}\Gamma(T_{k_{{\rm s}}},\mu_{2^n})[3]\arrow[d] \\ {\rm H}^2_{{\rm{\acute et}}}(T_{k_{{\rm s}}},{\mathbb{G}}_m)\arrow[r] & {\rm H}^0_{\rm{\acute et}}(T_{k_{{\rm s}}},{\mathbb{G}}_m)[3] \end{tikzcd}\tag{27}\] where we used that \({\rm H}^1_{\rm{\acute et}}(T_{k_{{\rm s}}},{\mathbb{G}}_m)={\rm {Pic}}(T_{k_{{\rm s}}})\) vanishes. The differential \(d_{3}^{0,2}\) is obtained by applying the functor \({\rm H}^0(k,-)\) to the bottom horizontal map in the diagram.

Applying the functor of Galois invariants to the left vertical map we get the inclusion of the \(2^n\)-torsion \(\wedge^2(\widehat{T}/2^n)(-1)^{\Gamma}\cong {\rm H}^2_{{\rm{\acute et}}}(T_{k_{{\rm s}}},\mu_{2^n})\) into the Galois invariants in the Brauer group of \(T_{k_{{\rm s}}}\), so we are looking to compute the result of applying \({\rm H}^0(k,-)\) to the counter-clockwise composition in (27 ), composed with the projection \({\rm H}^3(k,{\rm H}^0_{\rm{\acute et}}(T_{k_{{\rm s}}},{\mathbb{G}}_m))\to {\rm H}^3(k,\widehat{T})\).

From the Kummer sequence, we see that \(\tau^{\leq 1}\mathrm{R}\Gamma_{{\rm{\acute et}}}(T_{k_{{\rm s}}},\mu_{2^n})\) can be represented by the two-term complex \({\rm H}^0_{\rm{\acute et}}(T_{k_{{\rm s}}},{\mathbb{G}}_m)\xrightarrow{2^n}{\rm H}^0_{\rm{\acute et}}(T_{k_{{\rm s}}},{\mathbb{G}}_m)\) with the right vertical arrow in (27 ) given by shift by \([3]\) of the projection onto the \(0\)th term of this complex. This map followed by the projection to \(\widehat{T}\) sends \({\rm H}^0_{\rm{\acute et}}(T_{k_{\rm s}},\mu_{2^n})\) to zero, so the composition factors through \(\tau^{[1,1]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(T_{k_{{\rm s}}},\mu_{2^n})\). Applying \({\rm H}^0(k,-)\) to this composition, we obtain a map that factors as follows: \[{\rm H}^0(k, \tau^{\leq 1}\mathrm{R}\Gamma_{\rm{\acute et}}(T_{k_{{\rm s}}},\mu_{2^n})[3])\to {\rm H}^2(k, {\rm H}^1_{\rm{\acute et}}(T_{k_{{\rm s}}},\mu_{2^n}))\cong {\rm H}^2(k,\widehat{T}/2^n)\xrightarrow{-\mathrm{Bock}_{\widehat{T}}}{\rm H}^3(k, \widehat{T}).\] Therefore the clockwise composition in (27 ) evaluated on \({\rm H}^0(k,-)\), composed with the projection onto \({\rm H}^3(k,\widehat{T})\), is given by \[{\rm H}^2_{\rm{\acute et}}(T_{k_{{\rm s}}},\mu_{2^n})^{\Gamma}\xrightarrow{\delta_{2}^{0,2}} {\rm H}^2(k, {\rm H}^1_{\rm{\acute et}}(T_{k_{{\rm s}}},\mu_{2^n}))\xrightarrow{-\mathrm{Bock}_{\widehat{T}}}{\rm H}^3(k,\widehat{T}),\] where \(\delta^{0,2}_2\) is a differential of the spectral sequence \[E^{p,q}_2={\rm H}^p(k,{\rm H}^q_{\rm{\acute et}}(T_{k_{\rm s}},\mu_{2^n}))\Rightarrow{\rm H}^{p+q}_{\rm{\acute et}}(T,\mu_{2^n}).\] By Theorem 13, \(\delta^{0,2}_2\) is obtained by applying \(-\otimes_{{\mathbb{Z}}_2}^L\mu_{2^n}\) to the terms of the 2-extension given by the difference of the maps (19 ) and (20 ) with \(M={\rm H}^1_{\rm{\acute et}}(T_{k_{\rm s}},{\mathbb{Z}}_2)\). The composition of two Bockstein maps is zero, so only (20 ) contributes to \(\bar d_3^{\,0,2}\), thus proving that \(\bar d_3^{\,0,2}\) is the composition of the four maps in the theorem. \(\Box\)

When the first version of this paper was completed, the authors became aware of the following result of Julian Demeio [9]. Recall that a torus is called quasi-trivial if \(\widehat T\) is a permutational \(\Gamma\)-module, that is, \(\widehat T\) has a \(\Gamma\)-stable \({\mathbb{Z}}\)-basis.

Proof. Let us show that \(\bar d^{\,0,2}_3=0\). The case of quasi-trivial torus is immediate from Theorem 23 and Lemma 11 (i). If \(k\) is a \(p\)-adic field, then \({\rm H}^3(k,\widehat T)=0\) since \(k\) has strict cohomological dimension 2 [25], so there is nothing to prove. The case \(k_v={\mathbb{R}}\) follows from Theorem 23 and Lemma 11 (ii). If \(k\) is a number field, we have an isomorphism \({\rm H}^3(k,\widehat T)\tilde{\longrightarrow}\prod {\rm H}^3(k_v,\widehat T)\), where the product is over the real places of \(k\), see [25], so this case follows from the case of local fields. \(\Box\)

5 sec:Torsors↩︎

Let \(X\) be a \(k\)-torsor for a semiabelian variety \(A\). In this section we address the problem of computing the étale cohomology groups \({\rm H}^i_{\rm{\acute et}}(X,{\mathbb{Z}}/n)\), where \(n\) is not divisible by \({\rm char}(k)\).

Translations by elements of \(A(k_{\rm s})\) act trivially on the groups \({\rm H}^i_{\rm{\acute et}}(X_{k_{\rm s}},{\mathbb{Z}}/n)\), for \(i\geq 0\). Let us explain this. The choice of a \(k_{\rm s}\)-point in \(X\) gives an isomorphism \(X_{k_{\rm s}}\simeq A_{k_{\rm s}}\). By the canonical isomorphism \({\rm H}^i_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\cong\wedge^i {\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\) it suffices to prove the triviality of action on \({\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\). Let \(p_1,p_2,\mu:A\times A\to A\) be the two projections and the multiplication map, respectively. We need to show that for any \(a\in A(k_{\rm s})\) the composition \[{\rm H}^1_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\xrightarrow{\mu^*} {\rm H}^1_{{\rm{\acute et}}}(A_{k_{\rm s}}\times A_{k_{\rm s}},{\mathbb{Z}}/n)\xrightarrow{(a\times{\rm Id}_{A})^*}{\rm H}^1_{{\rm{\acute et}}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\] is the identity map. Using that \({\rm H}^1(k_{\rm s},{\mathbb{Z}}/n)=0\), we see that it suffices to prove that \(\mu^*=p_1^*+p_2^*\), but this follows from the canonical isomorphism \[{\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\cong{\rm {Hom}}(\pi_1^{\rm{\acute et}}(A_{k_{\rm s}}),{\mathbb{Z}}/n)\cong{\rm {Hom}}(A[n],{\mathbb{Z}}/n), \label{isomo}\tag{28}\] see (10 ).

Therefore, any choice of a trivialisation \(X_{k_{{\rm s}}}\simeq A_{k_{{\rm s}}}\) of the torsor \(X\) over \(k_{{\rm s}}\) gives rise to the same \(\Gamma\)-equivariant isomorphism \[{\rm H}^i_{\rm{\acute et}}(X_{k_{\rm s}},{\mathbb{Z}}/n)\cong{\rm H}^i_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n), \quad i\geq 0.\] But the complexes \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\) and \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/n)\) need not be isomorphic as objects of \(D(\Gamma,{\mathbb{Z}}/n)\): the truncation of the latter in degrees \([0,1]\) is the direct sum of its cohomology groups, but the truncation \(\tau^{[0,1]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\) need not be split, as we demonstrate in this section.

The Hochschild–Serre spectral sequence for \(X\) has the form \[E^{p,q}_2={\rm H}^p(k,{\rm H}^q_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n))\cong{\rm H}^p(k,{\rm H}^q_{\rm{\acute et}}(X_{k_{\rm s}},{\mathbb{Z}}/n))\Rightarrow{\rm H}^{p+q}_{\rm{\acute et}}(X,{\mathbb{Z}}/n). \label{ssX}\tag{29}\] The question we address is the explicit form of the differentials \[\delta^{p,1}_2\colon{\rm H}^p(k,{\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)) \to {\rm H}^{p+2}(k,{\mathbb{Z}}/n),\] where \(p\geq 0\). Each of these differentials is induced by the map in the derived category of discrete \({\mathbb{Z}}/n\)-modules with a continuous action of \(\Gamma\) \[\delta_X:{\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/n)\cong {\rm H}^1_{\rm{\acute et}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\to {\mathbb{Z}}/n[2]\] arising from the complex \(\tau^{[0,1]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\in D(\Gamma,{\mathbb{Z}}/n)\).

Recall that \(\mathcal{A}(\Gamma,{\mathbb{Z}}/n)\) is the abelian category of discrete \({\mathbb{Z}}/n\)-modules equipped with a continuous action of \(\Gamma\). Writing \(M={\rm H}^1_{{\rm{\acute et}}}(A_{k_{{\rm s}}},{\mathbb{Z}}/n)\), we can think of \(\delta_X\) as an element of \({\rm Ext}^2_{\mathcal{A}(\Gamma,{\mathbb{Z}}/n)}(M,{\mathbb{Z}}/n)\). Since \({\rm {Hom}}_{\mathcal{A}(\Gamma,{\mathbb{Z}}/n)}(M,{\mathbb{Z}}/n)={\rm {Hom}}_{{\mathbb{Z}}/n}(M,{\mathbb{Z}}/n)^\Gamma\), we have a spectral sequence \[E^{p,q}_2={\rm H}^p(k,{\rm Ext}^q_{{\mathbb{Z}}/n}(M,{\mathbb{Z}}/n))\Rightarrow{\rm Ext}^{p+q}_{\mathcal{A}(\Gamma,{\mathbb{Z}}/n)}(M,{\mathbb{Z}}/n). \label{ss95ext}\tag{30}\] In our case, \(M\) is a free, hence projective \({\mathbb{Z}}/n\)-module, thus \({\rm Ext}^q_{{\mathbb{Z}}/n}(M,{\mathbb{Z}}/n)=0\) for \(q>0\). Now (30 ) gives a natural isomorphism \[{\rm Ext}^2_{\mathcal{A}(\Gamma,{\mathbb{Z}}/n)}(M,{\mathbb{Z}}/n)\cong {\rm H}^2(k,{\rm {Hom}}_{{\mathbb{Z}}/n}(M,{\mathbb{Z}}/n))\cong {\rm H}^2(k,A[n]), \label{iso95delta}\tag{31}\] where the second isomorphism is the \({\mathbb{Z}}/n\)-linear dual of a natural isomorphism (28 ) of \(\Gamma\)-modules \({\rm {Hom}}_{{\mathbb{Z}}/n}(A[n],{\mathbb{Z}}/n)\cong{\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\) induced by the map (9 ). Using isomorphisms (31 ), we identify \(\delta_X\) with an element of \({\rm H}^2(k,A[n])\).

We will express the class \(\delta_X\) in terms of the class of the torsor \(X\). The exact sequence of \(\Gamma\)-modules \[0\to A[n]\to A\to A\to 0 \label{torsor}\tag{32}\] gives rise to the homomorphism of Galois cohomology groups \[\label{torsor32bockstein} \beta_n\colon {\rm H}^1(k,A)\to{\rm H}^2(k,A[n]).\tag{33}\]

Proof. Our goal is to compute the morphism \[\delta_X:{\rm {Hom}}_{{\mathbb{Z}}/n}(A[n],{\mathbb{Z}}/n)\cong {\rm H}^1_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\to {\rm H}^0_{\rm{\acute et}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)[2]\cong{\mathbb{Z}}/n[2]\] in \(D(\Gamma, {\mathbb{Z}}/n)\) corresponding to the complex \(\tau^{[0,1]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\). The \({\mathbb{Z}}/n\)-linear dual of the map \(\delta_X\) shifted by \(2\) is a map \(\delta_X^{\vee}[2]:{\mathbb{Z}}/n\to (A[n])[2]\) which is the following composition \[{\mathbb{Z}}/n\xrightarrow{\Delta}{\rm {End}}_{{\mathbb{Z}}/n}(A[n])={\rm {Hom}}_{{\mathbb{Z}}/n}(A[n],{\mathbb{Z}}/n)\otimes^L_{{\mathbb{Z}}/n}A[n]\xrightarrow{\delta_X\otimes{\rm id}}{\mathbb{Z}}/n[2]\otimes_{{\mathbb{Z}}/n}^L A[n],\] where \(\Delta\) sends \(1\in{\mathbb{Z}}/n\) to \({\rm id}_{A[n]}\in{\rm {End}}_{{\mathbb{Z}}/n}(A[n])\). After applying the derived functor of the functor of \(\Gamma\)-invariants, we see that \({\rm id}_{A[n]}\in{\rm {End}}_{{\mathbb{Z}}/n}(A[n])^\Gamma\) goes to \(\delta_X\in {\rm H}^2(k,A[n])\).

The map \(\delta_X\otimes{\rm id}_{A[n]}\) can be identified with the connecting map of the complex \(\tau^{[0,1]}\mathrm{R}\Gamma_{\rm{\acute et}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\otimes^L_{{\mathbb{Z}}/n}A[n]\simeq \tau^{[0,1]}\mathrm{R}\Gamma_{\rm{\acute et}}(X_{k_{{\rm s}}},A[n])\). Thus the differential \[d\colon{\rm H}^1_{\rm{\acute et}}(X_{k_{\rm s}},A[n])^\Gamma\to{\rm H}^2_{\rm{\acute et}}(k,A[n])\] of the spectral sequence \[E_2^{p,q}={\rm H}^p(k, {\rm H}^q_{\rm{\acute et}}(X_{k_{\rm s}},A[n]))\to{\rm H}^{p+q}_{\rm{\acute et}}(X,A[n])\] is induced by \(\delta_X\otimes {\rm id}_{A[n]}\). There are natural isomorphisms of \(\Gamma\)-modules \[{\rm H}^1_{\rm{\acute et}}(X_{k_{\rm s}},A[n])\cong {\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},A[n])\cong{\rm H}^1_{\rm{\acute et}}(A_{k_{\rm s}},{\mathbb{Z}}/n)\otimes_{{\mathbb{Z}}/n}A[n]\cong {\rm {End}}_{{\mathbb{Z}}/n}(A[n]).\] In the notation of [23], under these isomorphisms, \({\rm id}_{A[n]}\in {\rm {End}}_{{\mathbb{Z}}/n}(A[n])\) corresponds to the class \(\tau\in {\rm H}^1_{\rm{\acute et}}(X_{k_{\rm s}},A[n])\) of the \(X_{k_{\rm s}}\)-torsor for \(A[n]\) given by (32 ) after an identification of \(X_{k_{\rm s}}\) with \(A_{k_{\rm s}}\). (Note that \(\tau\) does not depend on the choice of a \(k_{\rm s}\)-point of \(X\), thus \(\tau\in {\rm H}^1_{\rm{\acute et}}(X_{k_{\rm s}},A[n])^\Gamma\).) This gives \(d(\tau)=\delta_X\).

On the other hand, by [23], which is a restatement of [26], we have \(d(\tau)=\beta_n([X])\). \(\Box\)

Among the \(k\)-torsors \(X\) for a semiabelian variety \(A\) whose class in \({\rm H}^1(k,A)\) is not divisible by \(n\), we have the following well-known examples in dimension 1.

6 Curves↩︎

Let \(C\) be a smooth, projective, geometrically integral curve over a field \(k\). Let \(J:={\boldsymbol{P}ic}_{C/k}^0\) be the Jacobian of \(C\) and let \(X:={\boldsymbol{P}ic}_{C/k}^1\) be the Albanese torsor of \(C\). The canonical map \(C\to X\) gives isomorphisms of \(\Gamma\)-modules \[{\rm H}^1_{\rm{\acute et}}(C_{k_{\rm s}},{\mathbb{Z}}/n)\cong {\rm H}^1_{{\rm{\acute et}}}(X_{k_{{\rm s}}},{\mathbb{Z}}/n)\cong {\rm H}^1_{\rm{\acute et}}(J_{k_{\rm s}},{\mathbb{Z}}/n).\] We use this identification in the following proposition.

Proof. The canonical map \(C\to X\) induces a map \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(X_{k_{{\rm s}}},\mu_n^{\otimes i})\to \mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n^{\otimes i})\) in \(D(\Gamma,{\mathbb{Z}}/n)\), which induces isomorphisms of cohomology groups in degrees 0 and 1. This gives (a).

Poincaré duality states that in \(D(\Gamma,{\mathbb{Z}}/n)\) the objects \({\rm {Hom}}_{{\mathbb{Z}}/n}(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{Z}}/n),\mu_n)\) and \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{Z}}/n)[2]\) are canonically isomorphic. Since each cohomology group of \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{Z}}/n)\) is a free \({\mathbb{Z}}/n\)-module, this implies a duality on truncations: the objects \({\rm {Hom}}_{{\mathbb{Z}}/n}(\tau^{[0,1]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{Z}}/n),\mu_n)\) and \(\tau^{[1,2]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{Z}}/n)[2]\) are canonically isomorphic. Therefore the differential \(\delta^{0,2}_{2,C}\) of the spectral sequence with coefficients \({\mathbb{Z}}/n\) is given by the class of a 2-extension in \({\rm Ext}^2_k({\mathbb{Z}}/n,{\rm H}^1_{\rm{\acute et}}(J^\vee_{k_{\rm s}},\mu_n))\) that is dual to the extension defining \(\delta^{0,1}_{2,C}\). It follows that \(\delta^{0,2}_{2,C}\) sends the generator of \({\rm H}^2_{\rm{\acute et}}(C_{k_{\rm s}},\mu_n)^\Gamma\cong{\mathbb{Z}}/n\) to the image of \([X]\) in \({\rm H}^2(k,J[n])\cong{\rm H}^2(k,{\rm H}^1_{\rm{\acute et}}(J^\vee_{k_{\rm s}},\mu_n))\). \(\Box\)

One can describe the differentials \(\delta_3^{p,2}\) in the spectral sequence (?? ) in general, assuming that all differentials on the \(2\)nd page vanish. It follows from Proposition 30 that the term \(E^{0,2}_2={\rm H}^2_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n)^{\Gamma}\simeq {\mathbb{Z}}/n\) remains intact on the \(3\)rd page if and only if the class of Albanese torsor \([{\boldsymbol{P}ic}^1_{C/k}]\in {\rm H}^1(k, J)\) is divisible by \(n\). Let us assume that this is the case and choose a \(k\)-torsor for \(J\), which we denote by \({\boldsymbol{P}ic}^{1/n}_{C/k}\), together with an isomorphism \([n]_*{\boldsymbol{P}ic}^{1/n}_{C/k}\simeq {\boldsymbol{P}ic}^1_{C/k}\). Equivalently, we have a pushout diagram of extensions of group schemes

\[\label{curves:32div32by32n32torsor32diagram} \begin{tikzcd} J\arrow[r] & A\arrow[r,"{\pi}"] & {\mathbb{Z}}\\ J\arrow[r]\arrow[u, equal] & {\boldsymbol{P}ic}_{C/k}\arrow[u,"g"]\arrow[r] & {\mathbb{Z}}\arrow[u,"{n}"] \end{tikzcd}\tag{34}\] with \({\boldsymbol{P}ic}^{1/n}_{C/k}\) isomorphic to the fibre of \(\pi\) over \(1\in{\mathbb{Z}}\).

The Picard stack of \(C\) is a \({\mathbb{G}}_m\)-gerbe over the Picard scheme \({\boldsymbol{P}ic}_{C/k}\) and it has the structure of a group stack compatible with the group structure on \({\boldsymbol{P}ic}_{C/k}\). In particular, it defines a degree \(2\) extension of Galois modules \[\delta:{\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\to k_{{\rm s}}^{\times}[2].\] By definition, the map \(g\) in the diagram (34 ) defines a short exact sequence of Galois modules \[\label{curves:32div32by32n32exact32sequence} 0\to {\boldsymbol{P}ic}_C(k_{{\rm s}})\xrightarrow{g} A(k_{{\rm s}})\to{\mathbb{Z}}/n\to 0\tag{35}\] and we define a degree \(3\) extension as the composition \[\label{curves:32deg32332extension32over32Z} {\mathbb{Z}}/n\to {\boldsymbol{P}ic}_C(k_{{\rm s}})[1]\xrightarrow{\delta[1]} k_{{\rm s}}^{\times}[3]\tag{36}\] with the first map being the connecting morphism in \(D(\Gamma,{\mathbb{Z}})\) induced by (35 ).

Recall that for an object \(M\in D(\Gamma,{\mathbb{Z}})\) there are natural isomorphisms for all \(i\): \[\label{curves:32mod32n32adjunction} {\rm H}^i(k,M\otimes^{L}_{{\mathbb{Z}}}{\mathbb{Z}}/n)\cong {\rm Ext}^{i+1}_{\Gamma}({\mathbb{Z}}/n, M)\tag{37}\] Indeed, the right-hand side can be calculated as \({\rm H}^{i+1}(k,\mathrm{RHom}_{{\mathbb{Z}}}({\mathbb{Z}}/n,M))\) and \(\mathrm{RHom}_{{\mathbb{Z}}}({\mathbb{Z}}/n,M)\in D(\Gamma,{\mathbb{Z}})\) is quasi-isomorphic to \(M\otimes^{L}_{{\mathbb{Z}}}{\mathbb{Z}}/n[-1]\).

For a Galois module \(M\), which is \(n\)-divisible as an abelian group, there is a natural isomorphism \({\rm Ext}^i_{\Gamma}({\mathbb{Z}}/n, M)\cong {\rm Ext}^i_{\Gamma}({\mathbb{Z}},M[n])={\rm H}^i(k,M[n])\). Let \(c_{1/n}\in {\rm H}^3(k,\mu_n)\) be the image of the composition of maps in (36 ) under this isomorphism for \(M=k_{{\rm s}}^{\times}\). As a consequence of the following proposition, the class \(c_{1/n}\) does not depend on the choice of the torsor \({\boldsymbol{P}ic}^{1/n}_{C/k}\).

Proof. Under the assumption that \([{\boldsymbol{P}ic}_{C/k}^1]\) is divisible by \(n\), all differentials on the second page vanish by Proposition 30. To access the differentials on the next page, we relate the extensions between the cohomology modules of the complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n)\) to those of \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{G}}_m)\).

The complex \(\tau^{\leq 2}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{G}}_m)\) has non-zero cohomology modules only in degrees \(0\) and \(1\), isomorphic to \(k_{{\rm s}}^{\times}\) and \({\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\), respectively, cf. [8]. The degree \(2\) extension between them is exactly the class \(\delta\). According to the Kummer sequence, applying the functor \(-\otimes_{{\mathbb{Z}}}^L{\mathbb{Z}}/n:D(\Gamma,{\mathbb{Z}})\to D(\Gamma,{\mathbb{Z}}/n)\) to the object \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{G}}_m)\) gives \(\mathrm{R}\Gamma(C_{k_{{\rm s}}},\mu_n)[1]\), and the extension \[{\rm H}^0_{\rm{\acute et}}(C_{k_{{\rm s}}},{\mathbb{G}}_m)\to \tau^{\leq 2}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},{\mathbb{G}}_m)\to {\rm H}^1_{\rm{\acute et}}(C_{k_{{\rm s}}},{\mathbb{G}}_m)[-1]\] is carried to \[{\rm H}^0_{\rm{\acute et}}(C_{k_{{\rm s}}},\mu_n)[1]\to \mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n)[1]\to (\tau^{[1,2]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n))[1].\] Hence the extension \[\label{curves:32ext32to32compute} \tau^{[1,2]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n)[2]\to \mu_n[3]\tag{38}\] arising from the complex \(\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n)\) is the result of applying the functor \(-\otimes_{{\mathbb{Z}}}^L{\mathbb{Z}}/n\) to the map \(\delta:{\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\to k_{{\rm s}}^{\times}[2]\).

Under the assumption that \([{\boldsymbol{P}ic}_{C/k}^1]\) is divisible by \(n\), the natural map \[\label{curves:32mod32n32map32to32split} \tau^{[1,2]}\mathrm{R}\Gamma_{{\rm{\acute et}}}(C_{k_{{\rm s}}},\mu_n)[2]\simeq {\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\otimes^L_{{\mathbb{Z}}}{\mathbb{Z}}/n\to {\mathbb{Z}}/n\tag{39}\] has a section in \(D(\Gamma,{\mathbb{Z}}/n)\) and the task of computing the \(3\)rd differential is equivalent to computing the composition of this section with (38 ).

Recall from (37 ) that for any Galois module \(M\) there is a natural isomorphism \({\rm Ext}^i_{\Gamma}({\mathbb{Z}}/n, M[1])\cong {\rm H}^i(k, M\otimes^L_{{\mathbb{Z}}}{\mathbb{Z}}/n)\). The morphism \(h:{\mathbb{Z}}/n\to {\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})[1]\) corresponding to the extension \(0\to {\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\to A(k_{{\rm s}})\to {\mathbb{Z}}/n\to 0\) is carried to a class in \({\rm H}^0(k, {\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\otimes_{{\mathbb{Z}}}^L{\mathbb{Z}}/n)\) which defines a splitting of (39 ) because the composition \[{\mathbb{Z}}/n\xrightarrow{h}{\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})[1]\xrightarrow{{\rm{deg}}[1]}{\mathbb{Z}}[1]\] is the Bockstein map.

The composition of this splitting with (38 ) is then the map \({\mathbb{Z}}/n\to \mu_n[3]\) whose class in \({\rm H}^3(k,\mu_n)\) corresponds to the composition \({\mathbb{Z}}/n\xrightarrow{h} {\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})[1]\xrightarrow{\delta[1]}k_{{\rm s}}^{\times}[3]\), and we arrive at the definition of the class \(c_{1/n}\), as desired. \(\Box\).

If the torsion order of the Albanese torsor \([{\boldsymbol{P}ic}^1_{C/k}]\in {\rm H}^1(k,J)\) is coprime to \(n\), then we can make the above formula for the \(3\)rd differential more explicit. The map \(\delta\) arising from the Picard stack of \(C\) induces a map \({\boldsymbol{P}ic}_{C/k}(k)\to {\rm{Br}}(k)\) sending a rational point to the obstruction to lifting it to an actual line bundle on \(C\). We denote by \(\mathrm{Bock}_{{\mathbb{G}}_m}:{\rm{Br}}(k)={\rm H}^2(k,{\mathbb{G}}_m)\to {\rm H}^3(k,\mu_n)\) the connecting homomorphism of the Kummer sequence.

Proof. Since \(n\) is invertible modulo \(m\), the class \([{\boldsymbol{P}ic}^1_{C/k}]\) is divisible by \(n\), so we are in the setup of Proposition 34. Specifically, choose \(m'\) to be any integer such that \(m'\cdot n\equiv 1\bmod m\) and let \({\boldsymbol{P}ic}_{C/k}^{1/n}:=[m']_*{\boldsymbol{P}ic}^1_{C/k}\). The point \(x\) defines a map of Galois modules \(f_x:{\mathbb{Z}}\to {\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\) sending \(1\in {\mathbb{Z}}\) to \(x\). The composition \[{\mathbb{Z}}\xrightarrow{f_x}{\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})\xrightarrow{\delta}k_{{\rm s}}^{\times}[2]\xrightarrow{\mathrm{Bock}_{{\mathbb{G}}_m}}\mu_n[3]\] corresponds under the isomorphism \({\rm H}^3(k,\mu_n)\simeq {\rm Ext}^3_{\Gamma}({\mathbb{Z}}/n,k_{{\rm s}}^{\times})\) to the composition \[\label{curves:32n32divisible32albanese32deg32332ext32formula} {\mathbb{Z}}/n\to{\mathbb{Z}}[1]\xrightarrow{f_x[1]}{\boldsymbol{P}ic}_{C/k}(k_{{\rm s}})[1]\xrightarrow{\delta[1]}k_{{\rm s}}^{\times}[3].\tag{40}\] The composition of the first two maps in (40 ) equals \[{\mathbb{Z}}/n\xrightarrow{f_x\otimes^L_{{\mathbb{Z}}}{\mathbb{Z}}/n}{\boldsymbol{P}ic}_C(k_{{\rm s}})\otimes^L_{{\mathbb{Z}}}{\mathbb{Z}}/n\xrightarrow{{\rm id}\otimes\mathrm{Bock}}{\boldsymbol{P}ic}_C(k_{{\rm s}})[1].\] Since \(f_x\otimes^L_{{\mathbb{Z}}}{\mathbb{Z}}/n\) is a section of the natural map \({\boldsymbol{P}ic}_C(k_{{\rm s}})\otimes^L_{{\mathbb{Z}}}{\mathbb{Z}}/n\to {\mathbb{Z}}/n\), the above composition (40 ) is the map inducing the differentials on the \(3\)rd page of our spectral sequence, as in the proof of Proposition 34. \(\Box\).

We finally explain why Corollary 35 implies Suslin’s lemma [30] discussed in Example [suslin32lemma32remark].

Proof. The spectral sequence \({\rm H}^p(k,{\rm Ext}_{\mathbb{Z}}^q({\mathbb{Z}}/2,{\mathbb{Z}}/2))\Rightarrow{\rm Ext}^{p+q}_{\Gamma,{\mathbb{Z}}}({\mathbb{Z}}/2,{\mathbb{Z}}/2)\) gives an exact sequence \[0\to {\rm H}^1(k,{\mathbb{Z}}/2)\to{\rm Ext}^1_{\Gamma,{\mathbb{Z}}}({\mathbb{Z}}/2,{\mathbb{Z}}/2)\to {\rm Ext}^1_{\mathbb{Z}}({\mathbb{Z}}/2,{\mathbb{Z}}/2).\] As pointed out in [30], the difference between the image of \([-1]\in {\rm H}^1(k,{\mathbb{Z}}/2)\) in \({\rm Ext}^1_{\Gamma,{\mathbb{Z}}}({\mathbb{Z}}/2,{\mathbb{Z}}/2)\) and the class of (?? ) is the class of the extension where \({\mathbb{Z}}/4\) is equipped with the trivial Galois action \[0\to{\mathbb{Z}}/2\to{\mathbb{Z}}/4\to{\mathbb{Z}}/2\to 0. \label{Z474}\tag{41}\] By the Milnor–Bloch–Kato conjecture, the natural map \[{\rm H}^n(k,\mu_4^{\otimes n})\to{\rm H}^n(k,\mu_2^{\otimes n})={\rm H}^n(k,{\mathbb{Z}}/2)\] is surjective. For even \(n\) we have \(\mu_4^{\otimes n}\cong {\mathbb{Z}}/4\), so the connecting map of (41 ) is zero on \({\rm H}^n(k,{\mathbb{Z}}/2)\). \(\Box\)

If \(C\) is a conic, then \(\delta\) sends \(1\in {\mathbb{Z}}\cong{\rm {Pic}}(C_{k_{\rm s}})^\Gamma\cong{\boldsymbol{P}ic}_{C/k}(k)\) to the class of the associated quaternion algebra in \({\rm{Br}}(k)\) (see [8]), that is, to the image of the symbol \((a,b)\) under the natural map \({\rm H}^2(k,{\mathbb{Z}}/2)\to {\rm H}^2(k,{\mathbb{G}}_m)\). By Lemma 38 the image of \((a,b)\) under the Bockstein map \({\rm H}^2(k,{\mathbb{G}}_m)\to{\rm H}^3(k,{\mathbb{Z}}/2)\) is \((a,b,-1)\). We of course do not need to invoke the Milnor–Bloch–Kato conjecture for this particular computation, because the Kummer classes of \(a\) and \(b\) in \({\rm H}^1(k,{\mathbb{Z}}/2)\) lift to \({\rm H}^1(k,\mu_4)\), so \((a,b)\in {\rm H}^2(k,{\mathbb{Z}}/2)\) lifts to \({\rm H}^2(k, \mu_4^{\otimes 2})\cong{\rm H}^2(k,{\mathbb{Z}}/4)\).

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

alexander.petrov.57@gmail.com

Department of Mathematics, South Kensington Campus, Imperial College London SW7 2AZ United Kingdom and Institute for the Information Transmission Problems, Russian Academy of Sciences, Moscow 127994 Russia

a.skorobogatov@imperial.ac.uk

References↩︎

---

1. Mathematics Subject Classification (2020): 18G40, 11G10↩︎

2. Keywords: spectral sequences, abelian varieties, torsors, tori, algebraic curves↩︎