Raskind: Descent on Simply Connected Algebraic Surfaces

This is joint work with V. Scharaskin.

$ K3$ Surfaces of Picard Number $ 20$

Let $ k$ be a field, usually finitely generated over the prime subfield ( $ \mathbb{Q}$), $ \overline{k}$ a separable closure of $ k$. Let $ X/k$ be a smooth, projective geometryicall connected, geometrically simply connected surface. ( $ \pi_1(\overline{X})=\{1\}$, where $ \overline{X}=X \times_k \overline{k}$.) Let $ G={\mathrm{Gal}}(\overline{k}/k)$, $ \ell$ a prime number, $ \ell \neq {\mathrm{char}}k$.

Point of the talk: It should be possible to do descent on (at least some) surfaces with nonzero geometric genus.

For example, we consider $ K3$ surfaces with geometric Picard number 20 (maximal) in characteristic zero:

Proposition. [Inose-Shioda] All $ K3$ surfaces over $ \mathbb{C}$ with Picard number $ 20$ are defined over $ \mathbb{Q}$, and may be realized as (double covers) of $ {\mathrm{Kum}}(E\times E')$, where $ E,E'$ are isogenous elliptic curves with CM.

Kummer theory says: There is an exact sequence

$\displaystyle 0 \to {\mathrm{Pic}}(\overline{X})/\ell^m {\mathrm{Pic}}(\overlin...
...erline{X},\mathbb{Z}/\ell^m(1)) \to {\mathrm{Br}}(\overline{X})[\ell^m] \to 0; $

since $ {\mathrm{Pic}}(\overline{X}) \cong NS(\overline{X})$, as you pass to the limit over $ m$, one has the exact sequence

$\displaystyle 0 \to NS(\overline{X}) \otimes \mathbb{Z}_\ell \to H^2(\overline{X},\mathbb{Z}_\ell(1)) \to T_\ell({\mathrm{Br}}(\overline{X})) \to 0. $

The term $ NS(\overline{X}) \otimes \mathbb{Z}_\ell$ is algebraic, the term $ T_\ell({\mathrm{Br}}(\overline{X}))$ transcendental.

Tensoring with $ \mathbb{Q}_\ell$, we expect:

\begin{conj}[Tate Conjecture]
The map
...(\overline{X},\mathbb{Q}_\ell(1))^{G} \end{displaymath}is surjective.

Proposition. If $ X$ is a geometrically simply connected surface, and the Tate conjecture is true, then the $ \ell$-primary component of $ {\mathrm{Br}}(X)/{\mathrm{Br}}(k)$ is finite.

Proposition. Suppose $ X$ as above has a good reduction modulo $ p$ with the same geometric Picard number (not always true), and $ k$ is a number field. If the Tate conjecture is true, then $ {\mathrm{Br}}(X)/{\mathrm{Br}}(k)$ is finite.

Corollary. If $ X$ is a $ K3$ of geometric Picard number $ 20$, then $ {\mathrm{Br}}(X)/{\mathrm{Br}}(k)$ is finite.

Proof. [Sketch of proof] Use Inose-Shioda result and Faltings-Deligne which prove Tate for $ X$, and go modulo a prime that splits in the CM-field of $ E$. $ \qedsymbol$

Rapid review of descent

Descent by Colliot-Thélène and Sansuc. Let $ X$ be a geometrically simply connected surface, and $ p_g=0$. Let $ S_k$ be the torus whose group of characters is $ {\mathrm{Pic}}(\overline{X})$, and $ S_X=S_k \times_k X$. There is an exact sequence

$\displaystyle 0 \to H^1(k,S) \to H^1(X,S) \xrightarrow{\chi} H^1(\overline{X},S)^G \to H^2(k,S) \to H^2(X,S) $

coming from the Hochschild-Serre spectral sequence. We identify

$\displaystyle H^1(\overline{X},S)^G \cong {\mathrm{Hom}}_G({\mathrm{Pic}}\overline{X},{\mathrm{Pic}}\overline{X}), $

and we think of $ H^1(X,S)$ as principal homogeneous spaces under $ S$; an element $ [\mathcal{T}] \in H^1(X,S)$ is a universal torsor if $ \chi([\mathcal{T}])={\mathrm{id}}$.

One has a pairing

$\displaystyle X(k) \times H^1(X,S)$ $\displaystyle \to H^1(k,S)$    
$\displaystyle P,[\mathcal{T}]$ $\displaystyle \mapsto \mathcal{T}_P;$    

every torsor $ \mathcal{T}$ comes with a map $ f_\mathcal{T}:\mathcal{T}\to X$, and $ \mathcal{T}_P=0$ if and only if $ P \in f_\mathcal{T}(\mathcal{T}(k))$.

Now assume $ X$ only geometrically simply connected (not necessarily $ p_g=0$). $ H^2(\overline{X},\mathbb{Z}_\ell(1))$ has no integral structure (i.e. there is not a $ \mathbb{Z}[G]$ module $ M$ such that $ M \otimes \mathbb{Z}_\ell \cong_G H^2(\overline{X},\mathbb{Z}_\ell(1))$, so we must use étale cohomology.

If $ p:X \to Y$ is any morphism of schemes, and $ \mathscr{F}$ a sheaf on $ Y{}_{\textup{\'et}}$, $ \mathscr{G}$ a sheaf on $ X{}_{\textup{\'et}}$, then there is a spectral sequence

$\displaystyle E_2^{r,s}={\mathrm{Ext}}_Y^r(\mathscr{F},R^s p_* \mathscr{G}) \Rightarrow {\mathrm{Ext}}_X^{r+s}(p^*\mathscr{F},\mathscr{G}). $

Apply this general situation with $ p:X \to {\mathrm{Spec}}(k)$ the structure morphism, $ \mathscr{F}=H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1))$, $ \mathscr{G}=\mathbb{Z}/n\mathbb{Z}(1)$. One obtains a map

$\displaystyle {\mathrm{Ext}}^2_X(p^* H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1)...
...athbb{Z}(1)) \to {\mathrm{End}}_k(H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1))) $

coming from the $ E_2^{0,2}$ term in the spectral sequence. We expect that the group $ {\mathrm{End}}_k(H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1)))$ will play the role of $ {\mathrm{Hom}}_G({\mathrm{Pic}}\overline{X},{\mathrm{Pic}}\overline{X})$ in the above.

Why is there a shift, and how does this relate to the Colliot-Thélène-Sansuc result when $ p_g=0$? Kummer theory on $ S$ gives

$\displaystyle 0 \to S[n] \to S \xrightarrow{n} S \to 0 $


$\displaystyle 0 \to H^1(X,S)/n H^1(X,S) \to H^2(X,S[n]) \to H^2(X,S)[n] \to 0. $

We have a map

  $\displaystyle H^2(X,\mathscr{H}om(p^*H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1)),\mathbb{Z}/n\mathbb{Z}(1))$    
  $\displaystyle \qquad \to {\mathrm{Ext}}^2_X(p^* H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1)),\mathbb{Z}/n\mathbb{Z}(1))$    

coming from the local-to-global spectral sequence, and we can identify

$\displaystyle H^2(X,\mathscr{H}om(p^*H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(1)...
...n\mathbb{Z}(1)))) \cong H^2(X,p^*H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(2))). $

Let $ \chi$ be the composite of these three maps. Let $ n=\ell^m$.

Definition. A universal $ n$-gerbe is an element $ \mathscr{G}\in H^2(X,p^*H^2(\overline{X},\mathbb{Z}/n\mathbb{Z}(2)))$ such that $ \chi([G])={\mathrm{id}}$.

One can (with care and difficulty) pass to $ \varinjlim_m$ to speak of universal $ \ell$-adic gerbes. The set of universal $ \ell$-adic gerbes is either empty or a principal homogeneous space under the image of $ H^2(k,H^2(\overline{X},\mathbb{Z}_\ell(2)))$ in $ H^2(X,p^* H^2(\overline{X},\mathbb{Z}_\ell(2)))$.

One has a pairing

$\displaystyle X(k) \times$   Gerbes$\displaystyle (X,k,\ell)$ $\displaystyle \to H^2(k,H^2(\overline{X},\mathbb{Z}_\ell(2)))$    
$\displaystyle P,\mathscr{G}$ $\displaystyle \mapsto \mathscr{G}_P$    

which gives a partition of $ X(k)$. This can be extended to a map

$\displaystyle \theta_\mathscr{G}:CH_0(X) \to H^2(k,H^2(\overline{X},\mathbb{Z}_\ell(2))) $

where $ \mathscr{G}$ is a chosen universal gerbe. On $ A_0(X)$, zero-cycles of degree 0, this is the higher $ \ell$-adic Abel-Jacobi map, and the image of this map is a finitely generated $ \mathbb{Z}_\ell$-module; if $ k$ is a number field, one can show in some cases that that the image is finite, e.g. $ K3$ of Picard number $ 20$ over $ \mathbb{Q}$ or over the CM field.

So, in these cases, have $ X(k) = \bigsqcup f_\alpha \mathscr{G}_\alpha(k)$, where $ \alpha$ ranges over a finite set.

We can show $ X(\mathbf A_k)^{{\mathrm{Br}}_\ell} \neq \emptyset$ if and only if there exists a universal gerbe $ \mathscr{G}$ with points everywhere locally.

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