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

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

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

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

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:

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

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

coming from the Hochschild-Serre spectral sequence. We identify

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

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

$$X(k) \times H^1(X,S) \to H^1(k,S)$$

$$P,[\mathcal{T}] \mapsto \mathcal{T}_P;$$

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

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

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

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

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

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

and

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

We have a map

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

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

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

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

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

$$X(k) \times (X,k,\ell) \to H^2(k,H^2(\overline{X},\mathbb{Z}_\ell(2)))$$

$$P,\mathscr{G} \mapsto \mathscr{G}_P$$

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

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

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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