January 01, 1970
Let \(X\) be a smooth, projective, geometrically integral variety over a global function field \(k\) of characteristic \(p\). We show that if the unipotent Brauer group of \(X\) is zero, and the Picard scheme of \(X\) is \(p\)-torsion free, then only finitely many places of \(k\) can be potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X\). On the other hand, if the unipotent Brauer group is non-zero, then almost all places of \(k\) are potentially relevant. If \(X\) is base changed from a finite field, then in both cases the finite set of exceptional places is empty.
In their recent papers, Bright and Newton [1] and Ambrosi, Newton, and Pagano [2] proved that if \(X\) is a smooth, proper, geometrically integral variety over a number field \(k\), such that \({\rm H}^0(X,\Omega^2)\neq 0\), then almost every place \(v\) of \(k\) is potentially relevant to the Brauer–Manin obstruction. By definition, this means that there is a finite field extension \(L/k\), a place \(w\) of \(L\) over \(v\), and an element \(\mathcal{A}\in {\rm{Br}}(X_L)\), such that the evaluation map \({\rm ev}_\mathcal{A}\colon X(L_w)\to {\rm{Br}}(L_w)\) is non-constant. It is a natural question as to whether or not the same is true for global fields of positive characteristic.
We therefore consider a smooth, projective, geometrically integral variety \(X\) over a global function field \(k\) of characteristic \(p>0\). We let \(\bar k\) be an algebraic closure of \(k\), and \({\boldsymbol{A}}_k\) the adèle ring of \(k\). By work of Milne, the quotient of \({\rm{Br}}(X_{\bar k})[p^\infty]\) by its divisible subgroup is the group of \(\bar k\)-points of a quasi-algebraic group \(G_X\) in the sense of Serre [3], see [4]. Thus, by [3], \(G_X\) is an extension of a finite étale group \(D_{X}\) by a smooth connected unipotent group \(U_{X}\).
Definition 1. We call \(U_X\) the unipotent Brauer group of \(X\).
The second named author asked about the role played by the unipotent Brauer group in the Brauer–Manin obstruction for \(X\). In response to this question, Valloni has shown that if \(X\) is a supersingular K3 surface, then up to replacing \(k\) by a finite inseparable extension, the Brauer–Manin set \(X({\boldsymbol{A}}_k)^{\rm{Br}}\) is ‘very small’; in particular it is not open in \(X({\boldsymbol{A}}_k)\), and every place of \(k\) is relevant to the Brauer–Manin obstruction [5].
In this paper we show that this result reflects a basic dichotomy in the behaviour of the Brauer–Manin obstruction, depending on whether \(U_X\) is zero or non-zero. Indeed, our main result is as follows.
Theorem 1 (Theorems 9 and 11). Let \(X\) be a smooth, projective, geometrically integral variety over a global function field \(k\) of characterisitc \(p\).
If \(U_X=0\), and the Picard scheme of \(X\) is \(p\)-torsion free, then only finitely many places of \(k\) are potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X\).
If \(U_X\neq 0\), then almost all places of \(k\) are potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X\).
The case of \(\ell\)-primary torsion in the Brauer group, for \(\ell\neq p\), can be handled using the approach of [6], alternatively we can easily adapt our proof of part [Ai] from the \(p\)-primary to the \(\ell\)-primary case. Combining Theorem 1 with its \(\ell\)-primary analogue we thus obtain:
Corollary 2. Let \(X\) be a smooth, projective, geometrically integral variety over a global function field \(k\), with torsion free Picard scheme and \(U_X=0\). Then only finitely many places of \(k\) are potentially relevant to the Brauer–Manin obstruction for \(X\).
The proof of Theorem 1 depends on two key ingredients: the generic representability results of Bragg–Olsson for flat cohomology [7], and, for part (ii), recent results of Krishna and Majumder [8] on Kato’s filtration in equicharacteristic \(p\). The dependence on the former in particular makes it difficult to be explicit about the precise set of places required. Indeed, if \(U_X=0\), then the set of places which are not potentially relevant comes from a smooth spreading out \(\pi\colon {\cal X}\to C\) of \(X\), where \(C\) is a smooth curve with function field \(k\), for which the fppf sheaf \({\boldsymbol{R}}^2\pi_*\mu_p\) is representable by a finite flat group scheme and \({\boldsymbol{R}}^3\pi_*\mu_p\) is representable by an affine group scheme. If \(U_X\neq 0\), then the set of potentially relevant places comes from requiring \({\boldsymbol{R}}^2\pi_*\mu_p\) to be representable by an affine group scheme of finite type. The results of Bragg–Olsson guarantee the existence of such a curve \(C\) in each case, but (to the best of our knowledge) give little control over how large we can take \(C\) to be.
In some cases, however, \(C\) can be made explicit. For example, when \(X\) is constant we can take \(C\) to be the unique smooth projective curve with function field \(k\). This leads to the following:
Theorem 3 (Corollary 14, Theorem 16). Let \(X\) be a smooth, projective, geometrically integral variety over a finite field \({\mathbb{F}}\), and let \(k/{\mathbb{F}}\) be a global function field.
If \(U_X=0\), and \(X\) has torsion-free Picard scheme, then the Brauer–Manin pairing \[X({\boldsymbol{A}}_k)\times {\rm{Br}}(X_k) \to {\mathbb{Q}}/{\mathbb{Z}}\] is identically zero.
If \(U_X\neq 0\), then every place of \(k\) is potentially relevant to the Brauer–Manin obstruction for \(X_k\).
Another case when \(C\) can be made explicit is when the geometric Brauer group of \(X\) is \(p\)-torsion free, that is, \({\rm{Br}}(X_{\bar k})[p^\infty]=0\). Then we can take \(C\) such that \(X\) has good reduction for every \(v\in |C|\), see Proposition 17. We give examples of non-isotrivial K3 surfaces with \({\rm{Br}}(X_{\bar k})[p^\infty]=0\) (Proposition 20), so this case is not subsumed by Theorem 3.
Here is a brief outline of this note. In Section 1 we put together known results from the literature to give a formula for the dimension of \(U_X\) in terms of the slopes of the crystalline cohomology groups of \(X_{\bar k}\). In Section 2 we combine the representability results of Bragg–Olsson from [7] with the properties of the evaluation filtration on the local Brauer group established by Krishna and Majumder in [8] to prove our main result, Theorem 1. In Section 3 we consider the case of constant varieties, and prove Theorem 3. Finally, in Section 4 we discuss varieties with \(p\)-torsion free geometric Brauer group.
Let \(X\) be a smooth, proper, geometrically integral variety over a field \(k\) of characteristic \(p>0\), with algebraic closure \(\bar k\). By a theorem of Grothendieck, \({\rm{Br}}(X_{\bar k})={\rm H}^2_{\rm{\acute et}}(X_{\bar k},{\mathbb{G}}_m)\) is a torsion group [9]. For primes \(\ell\neq p\), the structure of \({\rm{Br}}(X_{\bar k})[\ell^\infty]\) was determined also by Grothendieck, see [9]. We now discuss the structure of \({\rm{Br}}(X_{\bar k})[p^\infty]\).
The dimension of the unipotent Brauer group \(U_X\), as well as the corank of the divisible subgroup of \({\rm{Br}}(X_{\bar k})[p^\infty]\), can be calculated in terms of the crystalline cohomology groups of \(X_{\bar k}\) as follows. Let \(W=W(\bar k)\) be the Witt ring of \(\bar k\), with field of fractions \(K\). Write \[\rho={\rm{rk}}\, {\rm NS}(X_{\bar k}) \quad\text{and}\quad b_2={\rm{dim}}_K\big( {\rm H}^2_{\rm cris}(X_{\bar k}/W)\otimes K\big),\] for the Picard rank and the second Betti number, respectively. Write \[h={\rm{dim}}_K\big ({\rm H}^2(X,W{\cal O}_X)\otimes K\big).\] For any \(\lambda \in {\mathbb{Q}}\), we let \(\big({\rm H}^i_{\rm cris}(X_{\bar k}/W)\otimes K\big)_{[\lambda]}\) denote the slope \(\lambda\) summand of \({\rm H}^i_{\rm cris}(X_{\bar k}/W)\otimes K\), and for an interval \(I\subset {\mathbb{R}}\), we let \(\big({\rm H}^i_{\rm cris}(X_{\bar k}/W)\otimes K\big)_{I}\) denote the maximal summand of \({\rm H}^i_{\rm cris}(X_{\bar k}/W)\otimes K\) with all slopes in \(I\). Since the slope spectral sequence degenerates modulo torsion on the first page, there is, for \(i\geq 0\), an isomorphism of isocrystals \[{\rm H}^i(X,W{\cal O}_X)\otimes K\cong \big({\rm H}^i_{\rm cris}(X_{\bar k}/W)\otimes K\big)_{[0,1)},\] see [10]. We thus have \[h={\rm{dim}}_K \big({\rm H}^2_{\rm cris}(X_{\bar k}/W)\otimes K\big)_{[0,1)}\] and if \(X\) is projective, we have moreover \[\big({\rm H}^2_{\rm cris}(X_{\bar k}/W)\otimes K\big)_{[0,1)} = {\rm{dim}}_K \big({\rm H}^2_{\rm cris}(X_{\bar k}/W)\otimes K\big)_{(1,2]}\] see [10].
Lemma 1. Let \(X\) be a smooth, proper, geometrically integral variety over a field \(k\) of characteristic \(p\). The divisible subgroup of \({\rm{Br}}(X_{\bar k})[p^\infty]\) is isomorphic to \(({\mathbb{Q}}_p/{\mathbb{Z}}_p)^{b_2-2h-\rho}\).
Proof. This follows from [10]. ◻
Lemma 2. Let \(X\) be a smooth, proper, geometrically integral variety over a field \(k\) of characteristic \(p\) such that \(U_{X}=0\). Let \(\bar k\subset K\) be an extension of algebraically closed fields. Then the natural map \({\rm{Br}}(X_{\bar k})\to {\rm{Br}}(X_K)\) is an isomorphism.
Proof. For the prime-to-\(p\) torsion subgroup this is well-known, see [9], so it remains to consider the \(p\)-primary torsion subgroups.
Define \(T_p{\rm{Br}}(X_{\bar k}):=\varprojlim {\rm{Br}}(X_{\bar k})[p^n]\) and \({\rm H}^2(X_{\bar k},{\mathbb{Z}}_p(1)):=\varprojlim {\rm H}^2_{\rm fppf}(X_{\bar k},\mu_{p^n})\). Passing to the limit in the Kummer exact sequence for the fppf topology with coefficients in \(\mu_{p^n}\) we obtain a commutative diagram with exact rows (c.f.[10]): \[\xymatrix{ 0\ar[r]&{\rm NS}(X_K)\otimes{\mathbb{Z}}_p\ar[r]&{\rm H}^2(X_K,{\mathbb{Z}}_p(1))\ar[r]& T_p{\rm{Br}}(X_K)\ar[r]&0\\ 0\ar[r]&{\rm NS}(X_{\bar k})\otimes{\mathbb{Z}}_p\ar[r]\ar[u]^\cong&{\rm H}^2(X_{\bar k},{\mathbb{Z}}_p(1)) \ar[r]\ar[u]&T_p{\rm{Br}}(X_{\bar k})\ar[r]\ar[u]&0}\] The left hand vertical map is an isomorphism since the Néron–Severi scheme is étale. By [11] the map \(1-F\) is surjective on \({\rm H}^j(X,W\Omega^i_X)\), and then the exact sequence (5.5.1) of [10] gives a canonical isomorphism \[{\rm H}^2(X_{\bar k},{\mathbb{Z}}_p(1))\cong {\rm H}^1_{\rm Zar}(X_{\bar k},W\Omega^1_X)^{F=1}\] (alternatively, one can appeal to [10]). Formation of the sheaves \(W\Omega^i_X\) commutes with perfect base change [10], thus the middle vertical arrow in the diagram is an isomorphism. The diagram now implies that the right-hand vertical arrow is also an isomorphism. Hence the pullback map \({\rm{Br}}(X_{\bar k})[p^\infty]\to {\rm{Br}}(X_K)[p^\infty]\) induces an isomorphism on the divisible subgroups.
Since \(U_{X}=0\), the quotient of \({\rm{Br}}(X_{\bar k})[p^\infty]\) by its divisible subgroup is the group of \(\bar k\)-points of the finite étale group \(D_{X}\). The pullback map induces an isomorphism \(D_{X}(\bar k)\to D_{X}(K)\), and the lemma follows. ◻
Next, let us define \[h^{0,i}={\rm{dim}}_k\, {\rm H}^i(X,{\cal O}_X), \quad m^{0,i}=\sum_{\lambda\in[0,1)}(1-\lambda) {\rm{dim}}_K ({\rm H}^i(X,W{\cal O}_X)\otimes K)_{[\lambda]}.\] The numbers \(m^{0,i}\) were introduced by Ekedahl and Crew. By the above, the number \(m^{0,i}\) only depends on the slopes of \({\rm H}^i_{\rm cris}(X_{\bar k}/W)\).
Lemma 3. Let \(X\) be a smooth, projective, geometrically integral variety over a field \(k\) of characteristic \(p\). Then the Picard scheme \({\boldsymbol{P}ic}_{X/k}\) is smooth if and only if \(h^{0,1}-m^{0,1}=0\).
Proof. It is well known (see, e.g., [9]) that in this situation the group scheme \({\boldsymbol{P}ic}_{X/k}\) exists and represents the relative Picard functor; the connected component \({\boldsymbol{P}ic}^0_{X/k}\) is projective, with tangent space at the origin \({\rm H}^1(X,{\cal O}_X)\); moreover, \(A:={\boldsymbol{P}ic}^0_{X/k, {\rm red}}\) is an abelian variety over \(k\). Thus \({\boldsymbol{P}ic}^0_{X/k}\) is smooth if and only if \(h^{0,1}={\rm{dim}}\, A\). Let \(A^\vee\) be the dual abelian variety of \(A\). Thus \(A^\vee\) is the Albanese variety of \(X\), so \({\rm H}^1_{\rm cris}(X_{\bar k}/W)\cong {\rm H}^1_{\rm cris}(A^\vee_{\bar k}/W)\) is isomorphic to the contravariant Dieudonné module of \(A^\vee\), which is the same as the covariant Dieudonné module of \(A\). The dimension of the \(p\)-divisible group of \(A\) is calculated in terms of slopes of the covariant Dieudonné module of \(A\) by the same formula that defines \(m^{0,1}\), see [12] or [13]. ◻
Lemma 4. Let \(X\) be a smooth, projective, geometrically integral variety over a field \(k\) of characteristic \(p\), such that \({\boldsymbol{P}ic}_{X/k}\) is smooth. Then the following properties hold:
the formal Brauer group \(\widehat{\rm{Br}}(X)\) is representable by a formal group scheme;
if \({\rm H}^3_{\rm cris}(X/W)\) is torsion-free, then \(\widehat{\rm{Br}}(X)\) is smooth;
if \(\widehat{\rm{Br}}(X)\) is smooth, then \({\rm{dim}}(U_X)=h^{0,2}-m^{0,2}\).
Proof.
Smoothness is fpqc local on the base, so we can check the smoothness of \(\widehat{\rm{Br}}(X)\) over \(\bar k\), where the result follows from [14].
By definition, this is calculated after base changing to \(\bar k\), where the result follows from [13].
◻
Now, let \(\pi\colon X\to {\rm{Spec}}\,k\) be the structure morphism. Results of Bragg–Olsson show that if \(X\) is projective, then each fppf sheaf \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is representable by an affine group scheme of finite type over \(k\), see [7]. We then have the following relationship with the unipotent Brauer group \(U_X\):
Lemma 5. Let \(\pi\colon X\to {\rm{Spec}}\,k\) be a smooth, projective, geometrically integral variety over a field \(k\) of characteristic \(p\). Then the following are equivalent:
\(U_X= 0\);
for all \(n\geq 1\), \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is a finite (flat) group scheme over \(k\);
for some \(n\geq 1\), \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is a finite (flat) group scheme over \(k\).
Proof. We may base change to \(\bar k\), in which case \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is a product of a unipotent group and a group of multiplicative type by [15]. Since \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is \(p^n\)-torsion, it cannot have \({\mathbb{G}}_m\) as a direct factor, hence it must be a product of a unipotent group scheme \({\boldsymbol{U}}_n\) and a finite group scheme of multiplicative type.
We thus have \[{\rm H}^2(X_{\bar k},\mu_{p^n})= {\rm H}^0({\bar k},{\boldsymbol{R}}^2\pi_*\mu_{p^n}) = {\boldsymbol{U}}_n({\bar k}).\] If \(U_X=0\), we see that \(G_X=D_X\) is a finite étale group scheme, thus \[{\rm{Br}}(X_{\bar k})[p^\infty] \cong ({\mathbb{Q}}_p/{\mathbb{Z}}_p)^{b_2-2h-\rho} \oplus D_X({\bar k})\] and so \({\rm{Br}}(X_{\bar k})[p^n]\) is finite. Hence so is \({\rm H}^2(X_{\bar k},\mu_{p^n})= {\boldsymbol{U}}_n({\bar k})\), which forces \({\boldsymbol{U}}_n\) itself to be a finite unipotent group scheme. Hence [UXi] implies [UXii].
It is clear that [UXii] implies [UXiii]. Finally, if \({\rm{dim}}\, {\boldsymbol{R}}^2\pi_{*}\mu_{p^n}=0\) for some \(n\geq 1\), then \({\rm H}^0({\bar k},{\boldsymbol{R}}^2\pi_*\mu_{p^n})\) is finite, which forces \({\rm{Br}}(X_{\bar k})[p^n]\) to be finite, and hence \(U_X=0\). Thus [UXiii] implies [UXi]. ◻
We now let \(k\) be a global function field of characteristic \(p>0\), \({\boldsymbol{A}}_k\) its adèle ring, and \({\mathbb{F}}\subset k\) its field of constants. For any place \(v\) of \(k\) we let \(k_v\) denote the completion of \(k\) at \(v\), \({\cal O}_v\) its ring of integers, and \({\mathbb{F}}_v\) its residue field. We fix an algebraic closure \(\bar k\) of \(k\), let \(k^{\rm s}\subset \bar k\) the maximal separable extension of \(k\) inside \(\bar k\), and set \(\Gamma_k:={\rm Gal}(k^{\rm s}/k)\). We let \(\overline{{\mathbb{F}}}\) denote the algebraic closure of \({\mathbb{F}}\) inside \(k\).
We let \(X\) be a smooth, projective, geometrically integral variety over \(k\). For each place \(v\) of \(k\), we denote the base change of \(X\) to \(k_v\) by \(X_{k_v}\), and the image of the pullback map \({\rm{Br}}(k_v)\to {\rm{Br}}(X_{k_v})\) by \({\rm{Br}}_0(X_{k_v})\). A model for \(X\) will be a flat and projective morphism \(\pi\colon \mathcal{X}\to C\), where \(C\) is a smooth, integral, quasi-projective curve over \({\mathbb{F}}\) with function field \(k\), and \(X\) is isomorphic to the generic fibre of \(\pi\). For such a model \({\cal X}\), and a place \(v\in|C|\), we denote by \({\cal X}_{{\cal O}_v}\) the base change of \({\cal X}\) to \({\cal O}_v\).
Lemma 6. Let \(X\) be a smooth, projective, geometrically integral variety over a global function field \(k\) of characteristic \(p\), with \(p\)-torsion free Picard scheme \({\boldsymbol{P}ic}_{X/k}\). Let \(n\) be a positive integer. Then there exists a smooth projective model \({\cal X}\to C\) for \(X\), such that for any \(v\in |C|\) there is a commutative diagram with exact rows \[\label{diag:32key} \begin{align} \xymatrix@C=22pt{ 0 \ar[r] & {\rm {Pic}}({\cal X}_{{\cal O}_v})/p^n \ar[r] \ar[d]^{\cong} & {\rm H}^0({\cal O}_v,{\boldsymbol{R}}^2\pi_*\mu_{p^n}) \ar[r]\ar@{^{(}->}[d] & {\rm{Br}}({\cal X}_{{\cal O}_v})[p^n] \ar[r]\ar[d] & 0 \\ 0\ar[r] & {\rm {Pic}}(X_{k_v})/p^n \ar[r] & {\rm H}^0(k_v,{\boldsymbol{R}}^2\pi_*\mu_{p^n}) \ar[r]^{\quad\sigma} & \frac{{\rm{Br}}(X_{k_v})[p^n]}{{\rm{Br}}_0(X_{k_v})[p^n]}\ar[r] & 0 } \end{align}\qquad{(1)}\]
Remark 7. The vertical maps in ?? are the natural restriction maps. The left hand horizontal maps are the compositions \[{\rm Pic}({\cal X}_S)/p^n\to {\rm H}^2({\cal X}_S,\mu_{p^n})\to {\rm H}^0(S,{\boldsymbol{R}}^2\pi_*\mu_{p^n}),\] where \(S={\rm{Spec}}\,{\cal O}_v\) or \(S={\rm{Spec}}\,k_v\). The right hand horizontal maps are the unique maps compatible with the maps \[{\rm H}^0(S,{\boldsymbol{R}}^2\pi_*\mu_{p^n})\leftarrow{\rm H}^2({\cal X}_S,\mu_{p^n})\to {\rm{Br}}({\cal X}_S)[p^n].\]
Proof. We extend \(X/k\) to \(\pi\colon {\cal X}\to C\) by spreading out. In doing so we can ensure that \(\pi\colon {\cal X}\to C\) is a smooth projective morphism, which necessarily has geometrically integral fibres. This implies \(\pi_*\mu_{p^n}=\mu_{p^n}\).
The restriction of \({\boldsymbol{R}}^1\pi_{*}\mu_p\) to the generic point of \(C\) is \({\boldsymbol{P}ic}_{X/k}[p]=0\), hence by shrinking \(C\) we can arrange that \({\boldsymbol{R}}^1\pi_{*}\mu_p=0\). By induction on \(n\), we deduce that \({\boldsymbol{R}}^1\pi_{*}\mu_{p^n}=0\) for all \(n\geq 1\).
Applying [7] for our fixed integer \(n\), we may replace \(C\) with a dense open subscheme to ensure that \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is represented by an affine group scheme of finite type over \(C\).
We have \({\rm H}^3(k_v,\mu_{p^n})={\rm H}^3({\cal O}_v,\mu_{p^n})=0\) by [16] and [16]. We also have \({\rm H}^2({\cal O}_v,\mu_{p^n})=0\) since \({\rm {Pic}}({\cal O}_v)={\rm{Br}}({\cal O}_v)=0\). Thus the Leray spectral sequence gives rise to an isomorphism \[{\rm H}^2({\cal X}_{{\cal O}_v},\mu_{p^n})\cong {\rm H}^0({\cal O}_v,{\boldsymbol{R}}^2\pi_*\mu_{p^n}),\] as well as an exact sequence \[0\to {\rm H}^2(k_v,\mu_{p^n})\to{\rm H}^2(X_{k_v},\mu_{p^n})\to{\rm H}^0(k_v,{\boldsymbol{R}}^2\pi_*\mu_{p^n})\to 0.\] If we now combine this with the various Kummer exact sequences \[0\to {\rm {Pic}}(-)/p^n \to {\rm H}^2(-,\mu_{p^n}) \to {\rm{Br}}(-)[p^n] \to 0\] for \({\cal X}_{{\cal O}_v}\), \(k_v\) and \(X_{k_v}\), and let \(\pi^*\colon {\rm{Br}}(k_v)[p^n]\to{\rm{Br}}(X_{k_v})[p^n]\) denote the pullback map, we obtain the following commutative diagram with exact rows: \[\xymatrix@C=22pt{ & 0 \ar[r] & {\rm {Pic}}({\cal X}_{{\cal O}_v})/p^n \ar[r] \ar[d]^{\cong} & {\rm H}^0({\cal O}_v,{\boldsymbol{R}}^2\pi_*\mu_{p^n}) \ar[r]\ar[d] & {\rm{Br}}({\cal X}_{{\cal O}_v})[p^n] \ar[r]\ar[d] & 0 \\ 0\ar[r] & \ker \pi^* \ar[r] & {\rm {Pic}}(X_{k_v})/p^n \ar[r] & {\rm H}^0(k_v,{\boldsymbol{R}}^2\pi_*\mu_{p^n}) \ar[r] & {\rm coker}\, \pi^* \ar[r] & 0 }\] The left hand vertical map is an isomorphism, because \(X_{k_v}\) is the complement of a principal Cartier divisor \({\cal X}_{{\mathbb{F}}_v}\subset{\cal X}_{{\cal O}_v}\). Since \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is represented by an affine group scheme over \(C\), the middle vertical map in is injective. This implies that \(\ker \pi^*=0\), and we obtain (?? ). ◻
The curve \(C\) constructed in Lemma 6 may depend on \(n\). In the case that \(U_X=0\), we can avoid this problem as follows.
Lemma 8. With hypotheses as in Lemma 6, suppose moreover that \(U_X=0\). Then after replacing \(C\) by a dense open subscheme, the following hold:
\({\boldsymbol{R}}^2\pi_{*}\mu_{p}\) is a finite flat group scheme over \(C\);
\({\boldsymbol{R}}^3\pi_{*}\mu_{p}\) is a flat affine group scheme of finite type over \(C\);
for \(n \geq 2\), \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is a finite (but not necessarily flat) group scheme over \(C\).
Proof. It follows from [7] and Lemma 5 that \(C\) in Lemma 6 can be chosen in such a way that (i) and (ii) hold. To show that (iii) holds we argue by induction on \(n\). We have an exact sequence \[0 \to {\boldsymbol{R}}^2\pi_{*}\mu_{p}\to {\boldsymbol{R}}^2\pi_{*}\mu_{p^{n+1}}\to {\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\to {\boldsymbol{R}}^3\pi_{*}\mu_{p}.\] We know that \({\boldsymbol{R}}^3\pi_{*}\mu_{p}\) is an affine group scheme over \(C\), whilst \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\) is a finite group scheme over \(C\) by the induction hypothesis. The kernel \({\boldsymbol{K}}_n\) of the map \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\to {\boldsymbol{R}}^3\pi_{*}\mu_{p}\) is a closed subgroup of \({\boldsymbol{R}}^2\pi_{*}\mu_{p^n}\), hence it is a finite group scheme over \(C\). The exact sequence \[0\to {\boldsymbol{R}}^2\pi_{*}\mu_{p}\to {\boldsymbol{R}}^2\pi_{*}\mu_{p^{n+1}}\to{\boldsymbol{K}}_n \to 0.\] shows that \({\boldsymbol{R}}^2\pi_{*}\mu_{p^{n+1}}\) is a \({\boldsymbol{R}}^2\pi_{*}\mu_{p}\)-torsor over \({\boldsymbol{K}}_n\). It is therefore representable by a scheme which is finite and flat over \(\mathbf{K}_n\), and hence finite over \(C\). Thus (iii) holds. ◻
Theorem 9. Let \(X\) be a smooth, projective, geometrically integral variety over a global function field \(k\) of characteristic \(p\) with \(p\)-torsion free Picard scheme \({\boldsymbol{P}ic}_{X/k}\) and \(U_X=0\). Choose a spreading out \(\pi \colon {\cal X}\to C\) of \(X\) as in Lemma 8. Then for any place \(v\in |C|\), any finite extension \(L_w/k_v\) with ring of integers \({\cal O}_w\), and any \(n\geq 1\), the pullback maps induce an isomorphism \[{\rm{Br}}(L_w)[p^n] \oplus {\rm{Br}}({\cal X}_{{\cal O}_w})[p^n]\overset{\cong}{\longrightarrow}{\rm{Br}}(X_{L_w})[p^n].\] In particular, for any such \(w\), and any \(\mathcal{A} \in {\rm{Br}}(X_{L_w})[p^\infty]\), the evaluation map \[{\rm ev}_{\mathcal{A}} \colon X(L_w)\to {\mathbb{Q}}_p/{\mathbb{Z}}_p\] is constant.
Proof. We choose \(C\) as in Lemmas 6 and 8. In particular, \({\boldsymbol{R}}^2\pi_*\mu_{p^n}\) is a finite group scheme over \(C\) for \(n\geq 1\). Thus the natural map \[{\rm H}^0({\cal O}_w,{\boldsymbol{R}}^2\pi_*\mu_{p^n})\to {\rm H}^0(L_w,{\boldsymbol{R}}^2\pi_*\mu_{p^n})\] is bijective for all \(n\). Now ?? shows that \[{\rm{Br}}({\cal X}_{{\cal O}_w})[p^n] \overset{\cong}{\longrightarrow}{\rm{Br}}(X_{L_w})[p^n]/{\rm{Br}}(L_w)[p^n]\] is an isomorphism. The last statement follows from the fact that \({\rm{Br}}({\cal O}_w)=0\). ◻
For completeness, we note here that for primes \(\ell\neq p\), the same proof works without the assumption \(U_X=0\), provided that we alter the hypothesis to refer to the \(\ell\)-torsion in \({\boldsymbol{P}ic}_{X/k}\) rather than the \(p\)-torsion. The key point is that for any smooth projective model \(\pi\colon {\cal X}\to C\) of \(X\), and any \(\ell\neq p\), the sheaves \({\boldsymbol{R}}^q\pi_*\mu_{\ell^n}\) are representable by finite étale group schemes over \(C\) by standard theorems in étale cohomology. In fact, it is enough to have a smooth model locally at each place \(v\), which moreover only needs to be an algebraic space.
Theorem 10. Let \(k\) be a global function field, let \(v\) be a place of \(k\), and let \({\cal Y}\) be a smooth and proper algebraic space over \({\cal O}_v\), with geometrically integral generic fibre \(Y\). Then for any prime \(\ell\neq p\) such that \({\boldsymbol{P}ic}_{Y/k_v}\) is \(\ell\)-torsion free, and any \(\mathcal{A} \in {\rm{Br}}(Y)[\ell^\infty]\), the evaluation map \[{\rm ev}_{\mathcal{A}} \colon Y(k_v)\to {\mathbb{Q}}_\ell/{\mathbb{Z}}_\ell\] is constant. \(\Box\)
We now consider evaluation of \(p\)-primary torsion Brauer classes when \(U_X\neq 0\).
Theorem 11. Let \(X\) be a smooth, projective, geometrically integral variety over a global function field \(k\) of characteristic \(p\) with \(U_X\neq 0\). Then almost all places of \(k\) are potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X\).
Proof. Choose a smooth projective morphism \(\pi\colon {\cal X}\to C\) with geometrically integral fibres and generic fibre \(X\). Then we have \(\pi_*\mu_p=\mu_p\). Using [7] we can shrink \(C\) to arrange that \({\boldsymbol{R}}^2\pi_{*}\mu_{p}\) is represented by an affine group scheme of finite type over \(C\). We will show that any \(v\in|C|\) is potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X\).
To do so, we use the evaluation filtration on \({\rm{Br}}(X_{k_v})\), introduced in [8] following [1] in the number field case. This is an increasing filtration \({\rm Ev}_m{\rm{Br}}(X_{k_v})\) on \({\rm{Br}}(X_{k_v})\), defined for \(m\geq -2\), with the following properties:
\(\mathcal{A}\in {\rm Ev}_{-1}{\rm{Br}}(X_{k_v})\) if and only if for every finite extension \(L_w/k_v\) the evaluation map \({\rm ev}_{\mathcal{A}}\colon X(L_w)\to {\mathbb{Q}}/{\mathbb{Z}}\) is constant;
there is a residue map \(\partial \colon {\rm Ev}_{-1}{\rm{Br}}(X_{k_v}) \to {\rm H}^1({\mathbb{F}}_v,{\mathbb{Q}}/{\mathbb{Z}})\) such that \[\mathcal{A}\in {\rm Ev}_{-2}{\rm{Br}}(X_{k_v})\iff\partial(\mathcal{A})=0\iff \mathcal{A} \in {\rm{Br}}({\cal X}_{{\cal O}_v}).\]
These properties are [8] (in the definition of \(\partial\) we used that the closed fibre \({\cal X}_{{\mathbb{F}}_v}\) is geometrically integral, hence \({\rm H}^1({\mathbb{F}}_v,{\mathbb{Q}}/{\mathbb{Z}})\) injects into \({\rm H}^1({\cal X}_{{\mathbb{F}}_v},{\mathbb{Q}}/{\mathbb{Z}})\)). We deduce that \[{\rm Ev}_{-1}{\rm{Br}}(X_{k_v}) = {\rm{Br}}({\cal X}_{{\cal O}_v}) \oplus {\rm{Br}}_0(X_{k_v}).\] For a finite field extension \(L/k\) and a place \(w\) of \(L\) above \(v\), we consider the natural map \[\xi_{L,w}\colon {\rm H}^0(L,{\boldsymbol{R}}^2\pi_*\mu_{p})\to{\rm H}^0(L_w,{\boldsymbol{R}}^2\pi_*\mu_{p})/{\rm H}^0({\cal O}_w,{\boldsymbol{R}}^2\pi_*\mu_{p}),\] where \({\cal O}_w\) is the ring of integers of \(L_w\).
Lemma 12. Up to replacing \(k\) by a finite field extension, and replacing \(C\) by its normalisation in this extension, there exists \(a\in {\rm H}^0(k,{\boldsymbol{R}}^2\pi_*\mu_{p})\) such that for any finite field extension \(L/k\), and any prime \(w\) of \(L\) above \(v\), we have \(\xi_{L,w}(a)\neq 0\).
Proof. It follows from Lemma 5 that the generic fibre of \({\boldsymbol{R}}^2\pi_*\mu_{p}\) has positive dimension. Replacing \(k\) by a finite field extension, we can assume that the reduced subscheme of \({\boldsymbol{R}}^2\pi_*\mu_{p}\) is a subgroup. After another extension, using [15], we ensure that \({\boldsymbol{R}}^2\pi_*\mu_{p}\) contains a smooth connected unipotent group scheme of positive dimension. Passing to yet another finite extension, we can ensure that it is split and so contains a copy of \({\mathbb{G}}_{a,k}\) [15]. We rename this field \(k\), and replace \(C\) by its normalisation in this field.
Let \(G\) be the scheme theoretic closure of \({\mathbb{G}}_{a,k}\subset {\boldsymbol{R}}^2\pi_*\mu_{p,k}\) in \({\boldsymbol{R}}^2\pi_*\mu_p\). Thus \(G\) is an affine group scheme of finite type over \(C\), with generic fibre \(G_k={\mathbb{G}}_{a,k}\). Denote by \(f_v\) the natural map \(G({\cal O}_v)\to G(k_v)={\mathbb{G}}_{a,k}(k_v)=k_v\). For any pair \((L,w)\) as in the statement of the lemma, the map \(f_v\) naturally extends to a map \(f_w\colon G({\cal O}_w)\to L_w\). Since \(G\) is of finite type, there is a non-zero element \(c_v\in k_v\), depending only on \(G\), such that for every finite field extension \(L_w/k_v\) we have \(f_w(G({\cal O}_w))\subset c_v{\cal O}_w\). If \(a\in G(k)=k\) is such that \(a/c_v\notin {\cal O}_v\), then the image of \(a\) in \(G(k_v)=k_v\) is not in \(G({\cal O}_v)\), so that \(\xi_{k,v}(a)\neq 0\). In this case we have \(a/c_v\notin{\cal O}_w\), so the image of \(a\) in \(G(L)=L\) maps to an element of \(G(L_w)\) which is not in \(G({\cal O}_w)\), thus \(\xi_{L,w}(a)\neq 0\). ◻
To finish the proof of Theorem 11 we take \(k\) and \(C\) as in Lemma 12. Let \(v\in |C|\). Write \(\widetilde{\rm H}^2(X,\mu_p)\) for the cokernel of the natural map \({\rm H}^2(k,\mu_p)\to {\rm H}^2(X,\mu_p)\), and use a similar notation for \(X_{k_v}\). The Kummer exact sequences for \({\cal X}_{{\cal O}_v}\), \(X_{k_v}\), and \(X\) give rise to the following commutative diagram with exact rows \[\label{11} \begin{align} \xymatrix@C=22pt{ 0 \ar[r] & {\rm {Pic}}({\cal X}_{{\cal O}_v})/p \ar[r] \ar[d]^{\cong} & {\rm H}^0({\cal X}_{{\cal O}_v},\mu_p) \ar[r]\ar@{^{(}->}[d] & {\rm{Br}}({\cal X}_{{\cal O}_v})[p] \ar[r]\ar@{^{(}->}[d] & 0 \\ 0\ar[r] & {\rm {Pic}}(X_{k_v})/p \ar[r] & \widetilde{\rm H}^2(X_{k_v},\mu_p) \ar[r] & \frac{{\rm{Br}}(X_{k_v})[p^n]}{{\rm{Br}}_0(X_{k_v})[p]}\ar[r] & 0\\ 0\ar[r] & {\rm {Pic}}(X)/p \ar[r]\ar[u] & \widetilde{\rm H}^2(X,\mu_p) \ar[r]^{\;\sigma}\ar[u] & \frac{{\rm{Br}}(X)[p^n]}{{\rm{Br}}_0(X)[p]}\ar[r]\ar[u] & 0 } \end{align}\tag{1}\] We note that \({\rm H}^3(k,\mu_p)=0\): this follows from the exact sequence \[{\rm{Br}}(k)\xrightarrow{p}{\rm{Br}}(k)\to{\rm H}^3(k,\mu_p)\to {\rm H}^3(k,{\mathbb{G}}_m)\xrightarrow{p}{\rm H}^3(k,{\mathbb{G}}_m),\] since \({\rm{Br}}(k)\) is \(p\)-divisible [9] and \({\rm H}^3(k,{\mathbb{G}}_m)=0\) [16]. Thus the Leray spectral sequence gives an exact sequence \[\widetilde{\rm H}^2(X,\mu_p)\to{\rm H}^0(k, {\boldsymbol{R}}^2\pi_*\mu_p)\xrightarrow{d} {\rm H}^2(k, {\boldsymbol{R}}^1\pi_*\mu_p).\] We have a similar exact sequence for every finite field extension \(L/k\).
By Lemma 12 we can find an element \(x\in {\rm H}^0(k,{\boldsymbol{R}}^2\pi_*\mu_p)\) such that the image of \(x\) in \({\rm H}^0(L_w, {\boldsymbol{R}}^2\pi_*\mu_p)\) is not contained in \({\rm H}^0({\cal O}_w, {\boldsymbol{R}}^2\pi_*\mu_p)\), for any finite field extension \(L/k\) and any place \(w\) of \(L\) above \(v\). By [17], any class in \({\rm H}^2(k, {\boldsymbol{R}}^1\pi_*\mu_p)\) is killed by a finite field extension of \(k\), so we can choose \(L\) such that \(d(x)\) goes to zero in \({\rm H}^2(L, {\boldsymbol{R}}^1\pi_*\mu_p)\). Then the image of \(x\) in \({\rm H}^0(L,{\boldsymbol{R}}^2\pi_*\mu_{p^n})\) lifts to an element \(y\in\widetilde{\rm H}^2(X_L,\mu_p)\). We now rename \(L\) by \(k\) and \(L_w\) by \(k_v\).
Let \(\mathcal{A}\in{\rm{Br}}(X)[p]\) be a lifting of \(\sigma(y)\). An inspection of diagram (1 ) shows that the image of \(\mathcal{A}\) in \({\rm{Br}}(X_{k_v})\) is not contained in \({\rm{Br}}({\cal X}_{{\cal O}_v})\oplus{\rm{Br}}_0(k_v)\), otherwise the image of \(x\) in \({\rm H}^0(k_v, {\boldsymbol{R}}^2\pi_*\mu_p)\) would be in \({\rm H}^0({\cal O}_v, {\boldsymbol{R}}^2\pi_*\mu_p)\), contrary to the choice of \(x\). Thus \(\mathcal{A}\notin {\rm Ev}_{-1}{\rm{Br}}(X_{k_v})\), so that \(v\) is potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X\). ◻
The smooth curve \(C\) required seems rather difficult to determine explicitly, but in particular cases we can say more. We discuss these cases in the next two sections.
In this section, we consider constant varieties over \(k\). We will therefore keep the same notation as in the previous section, except that now \(X\) will be defined over the constant field \({\mathbb{F}}\) of \(k\). In this case, the sheaves \({\boldsymbol{R}}^q\pi_{*}\mu_{p^n}\) are obtained via base change along \(C\to{\rm{Spec}}\, {\mathbb{F}}\), hence in Lemma 8 we may take \(C\) to be the unique complete smooth curve with function field \(k\). Thus Theorem 9 becomes:
Theorem 13. Let \(X\) be a smooth, projective, geometrically integral variety over a finite field \({\mathbb{F}}\) with \(p\)-torsion free Picard scheme and \(U_X=0\). Then for any global function field \(k/{\mathbb{F}}\), any place \(v\) of \(k\), and any \(\mathcal{A} \in {\rm{Br}}(X_{k_v})[p^\infty]\), the evaluation map \[{\rm ev}_{\mathcal{A}} \colon X(k_v)\to {\mathbb{Q}}_p/{\mathbb{Z}}_p\] is constant.
Corollary 14. Let \(X\) be a smooth, projective and geometrically integral variety over a finite field \({\mathbb{F}}\), with torsion free Picard scheme and \(U_X=0\). Then, for any global function field \(k/{\mathbb{F}}\), the Brauer–Manin pairing \[X({\boldsymbol{A}}_k)\times {\rm{Br}}(X_k) \to {\mathbb{Q}}/{\mathbb{Z}}\] is identically zero.
Proof. By the Lang–Weil estimates, \(X\) admits a \(0\)-cycle of degee \(1\), and hence so does \(X_k\). Write this \(0\)-cycle as \(\sum_i n_i Q_i\) and let \(k_i=k(Q_i)\). Thus \(\sum_i n_i[k_{i}:k]=1\).
Let \((P_v)_v\in \prod_v X(k_v)\) and let \(\mathcal{A}\in {\rm{Br}}(X)\). For any place \(w\) of \(k_i\) over \(v\) we have \[{\rm inv}_w\,\mathcal{A}(Q_i)={\rm inv}_w({\rm res}_{k_{i,w}/k_v}\,\mathcal{A}(P_v))=[k_{i,w}:k_v]\cdot{\rm inv}_v\,\mathcal{A}(P_v),\] where the first equality is from local constancy of the evaluation map (Theorems 13 and 10), and the second is [9]. Global reciprocity now gives \(\sum_w {\rm inv}_w\,\mathcal{A}(Q_i)=0\) for all \(i\), and hence \[\begin{align} 0 &=\sum_i n_i\sum_{w} {\rm inv}_w\,\mathcal{A}(Q_i)=\sum_i n_i\sum_v\sum_{w\mid v} [k_{i,w}:k_v]\cdot{\rm inv}_v\,\mathcal{A}(P_v) \\ &= \left(\sum_i n_i[k_i:k]\right)\left(\sum_v{\rm inv}_v\,\mathcal{A}(P_v)\right) = \sum_v{\rm inv}_v\,\mathcal{A}(P_v), \end{align}\] as required. ◻
Remark 15. This corollary can be proved more directly. By Lemma 2, when \(U_X=0\) and \(K/{\mathbb{F}}\) is algebraically closed, the group \({{\rm{Br}}}(X_{K})\) is independent of \(K\). One can use this, together with the Hochschild–Serre spectral sequence, to show that the natural map \[{\rm{Br}}(X)\oplus {\rm{Br}}(k) \to {\rm{Br}}(X_k)\] is an isomorphism. Any element coming from \({\rm{Br}}(k)\) pairs trivially with \(X({\boldsymbol{A}}_k)\). Using the valuative criterion of properness, together with the fact that \({\rm{Br}}({\cal O}_v)=0\) for all \(v\), one shows that the same is true for any element of \({\rm{Br}}(X)\).
In the case that \(U_X\neq 0\), we can, in Theorem 11, again take \(C\) to be the unique complete smooth curve with function field \(k\). We then obtain the following result.
Theorem 16. Let \(X\) be a smooth, projective, geometrically integral variety over a finite field \({\mathbb{F}}\) with \(U_X\neq 0\). Then for any global function field \(k/{\mathbb{F}}\), every place of \(v\) is potentially relevant to the \(p\)-primary Brauer–Manin obstruction for \(X_k\).
For varieties over a global function field with torsion free Picard scheme and \(p\)-torsion free geometric Brauer group, we can give an explicit sufficient condition for a prime not to be potentially relevant to the Brauer–Manin obstruction. We say that \(X\) has good reduction at \(v\) if there exists a smooth projective scheme over \({\cal O}_v\) with geometrically integral fibres and generic fibre \(X_{k_v}\).
Proposition 17. Let \(X\) be a smooth, projective and geometrically integral variety over a global function field \(k\) of characteristic \(p\), with torsion-free Picard scheme, and such that \({\rm{Br}}(X_{\bar k})[p^\infty]=0\). Any place of \(k\) where \(X\) has good reduction is not potentially relevant to the Brauer–Manin obstruction.
Proof. Since \({\boldsymbol{P}ic}_{X/k}\) is smooth, the natural map \({\rm{Br}}(X_{k^{\rm s}})\to{\rm{Br}}(X_{\bar k})\) is injective, see [9], [18], so we have \({\rm{Br}}(X_{k^{\rm s}})[p^\infty]=0\). We conclude that \({\rm{Br}}(X)={\rm{Br}}_1(X)+{\rm{Br}}(X)(p')\), where \({\rm{Br}}_1(X)\) is the kernel of the natural map \({\rm{Br}}(X)\to{\rm{Br}}(X_{k^{\rm s}})\) and \({\rm{Br}}(X)(p')\) is the subgroup of elements of order prime to \(p\).
Let \(v\) be a place of \(k\) where \(X\) has good reduction. For any \(\mathcal{A}\in {\rm{Br}}_1(X)\) the evaluation map \({\rm ev}_{\mathcal{A}}\colon X(k_v)\to{\mathbb{Q}}/{\mathbb{Z}}\) is constant by [6] or [9]. (These results are stated in characteristic zero, but the same proofs work in any characteristic.) The same holds for \(\mathcal{A}\in{\rm{Br}}(X)(p')\) by Theorem 10. These arguments can be applied after any finite extension of \(k\), thus proving our statement. ◻
For surfaces, there is the following explicit criterion for the triviality of the geometric Brauer group.
Proposition 18. Let \(X\) be a smooth, projective and geometrically integral surface, such that \({\boldsymbol{P}ic}_{X/k}\) is smooth and \(U_X=0\). If \(b_2=\rho+2h\) and \({\rm NS}(X_{\bar k})[p^\infty]=0\), then \({\rm{Br}}(X_{\bar k})[p^\infty]=0\).
Proof. By Lemma 1 the condition \(b_2=\rho+2h\) implies that the divisible subgroup of \({\rm{Br}}(X_{\bar k})[p^\infty]\) is trivial. Since \(X\) is a surface, \({\rm NS}(X_{\bar k})[p^\infty]=0\) implies that \(D_X=0\) by [19]. But \(U_X=0\), so we have \({\rm{Br}}(X_{\bar k})[p^\infty]=0\). ◻
Let us now discuss the case of K3 surfaces \(X/k\). In this case, the Picard scheme \({\boldsymbol{P}ic}_{X/k}\) is torsion-free, and we have \(U_X=0\) if and only if \(h>0\), which is equivalent to \(X\) being non-supersingular. In this case \({\rm{Br}}(X_{\bar k})\) is divisible [10], so to conclude that \({\rm{Br}}(X_{\bar k})[p^\infty]=0\) we only need to ensure that \(\rho+2h=22\) and \(h>0\).
Example 19.
Kazuhiro Ito proved that for \(p\geq 5\) there exist K3 surfaces over \(\overline{\mathbb{F}}_p\) with arbitrary positive integer values of \(\rho\) and \(h\) subject to the conditions that \(\rho\) is even and \(\rho+2h\leq 22\), see [20]. In particular, for any \(h\) between 1 and 10 there are K3 surfaces over \({\mathbb{F}}_{p^n}\), for some \(n\geq 1\), with \(\rho=22-2h\).
Let \(E\) be an ordinary elliptic curve over \(k={\mathbb{F}}_p\), where \(p\) is a prime (including \(p=2\)). The Kummer surface \(X={\rm Kum}(E\times E)\) is defined as the minimal desingularisation of the quotient of the abelian surface \(A=E\times E\) by the involution \(x\mapsto -x\). It is a K3 surface, so \(b_2=22\). (This also holds when \(p=2\) since \(A\) is ordinary, see [21]). We have \({\rm {End}}(E_{\bar k})\cong{\mathbb{Z}}^{\oplus 2}\), hence the rank of \({\rm NS}(A_{\bar k})\) is 4. The Picard rank of \(X_{\bar k}\) is \(\rho=16+4=20\) and \(h_X=h_A=1\), see [22].
Let \(k\) be any field of positive characteristic \(p\) (including \(p=2\)). Let \(E\) and \(E'\) be elliptic curves over \(k\) such that \(E\) is ordinary and \(E'\) is supersingular. Since \({\rm {Hom}}(E_{\bar k},E'_{\bar k})=0\), the rank of \({\rm NS}(A_{\bar k})\) is 2. The Kummer surface \(X={\rm Kum}(E\times E')\) is defined in the same way as in (ii). The Picard rank of \(X_{\bar k}\) is \(\rho=16+2=18\). We have \(h_X=h_A=2\) again by [22].
Recall that a variety \(X\) over a global function field \(k\) is called isotrivial if there exist a finite extension \(L/k\), a finite field \({\mathbb{F}}\subset L\), and a variety \(X_0/{\mathbb{F}}\) such that \(X_L \cong X_{0,L}\). Given the results of §3, Proposition 17 is only interesting in the non-isotrivial case.
If \(k\) is a global function field, with fixed separable closure \(k^{\rm s}\), we let \(\overline{{\mathbb{F}}}\) denote the algebraic closure of the field of constants of \(k\) inside \(k^{\rm s}\), and define the geometric Galois group of \(k\) to be \({\rm{Gal}}(k^{\rm s}/\overline{{\mathbb{F}}}k)\). If \(X/k\) is isotrivial, then the action of the geometric Galois group of \(k\) on any étale cohomology group of \(X_{k^{\rm s}}\) has to be through a finite quotient. The following proposition shows that there exist non-isotrivial varieties with torsion free Picard scheme and \(p\)-torsion free geometric Brauer group.
Proposition 20. Is the situation of Example 19[iii], assume that \(E\) has non-constant \(j\)-invariant. Then \(X\) is non-isotrivial.
Proof. For any prime \(\ell\neq p\), the cohomology group \({\rm H}^2_{\rm{\acute et}}(X_{k^{\rm s}},{\mathbb{Q}}_\ell)\) contains a direct summand isomorphic to \({\rm H}^1_{\rm{\acute et}}(E_{k^{\rm s}},{\mathbb{Q}}_\ell)\otimes {\rm H}^1_{\rm{\acute et}}(E'_{k^{\rm s}},{\mathbb{Q}}_\ell)\). Since \(E'\) is supersingular, it is defined over a finite field, and so the geometric Galois group of \(k\) acts on \({\rm H}^1_{\rm{\acute et}}(E'_{k^{\rm s}},{\mathbb{Q}}_\ell)\) through a finite quotient. It is therefore enough to show that the geometric Galois group of \(k\) does not act on \({\rm H}^1_{\rm{\acute et}}(E_{k^{\rm s}},{\mathbb{Q}}_\ell)\) through a finite quotient.
To see this, the fact that the \(j\)-invariant of \(E\) is non-constant means there is a place \(v\) of \(k\) at which it is non-integral, and hence at which \(E\) does not have potential good reduction. In particular, the inertia group at \(v\), which is contained within the geometric Galois group of \(k\), cannot act on \({\rm H}^1_{\rm{\acute et}}(E_{k^{\rm s}},{\mathbb{Q}}_\ell)\) through a finite quotient, so we are done. ◻
A.S.is grateful to Subhadip Majumder and Margherita Pagano for helpful discussions. He thanks the Lodha Mathematical Sciences Institute in Mumbai and the Max Planck Institute for Mathematics in Bonn for excellent working conditions and support. C.L.would like to thank the Max Planck Institute for Mathematics in Bonn for hospitality.
Department of Mathematics, Harrison Building, University of Exeter, EX4 4QF, United Kingdom
c.d.lazda@exeter.ac.uk
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