This is joint work with V. Scharaskin.

* Surfaces of Picard Number *

Let be a field, usually finitely generated over the prime subfield ( ), a separable closure of . Let be a smooth, projective geometryicall connected, geometrically simply connected surface. ( , where .) Let , a prime number, .

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

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

**Proposition. ***[Inose-Shioda]
All surfaces over
with Picard number are defined over
, and may be realized as (double covers) of
, where are isogenous elliptic curves with CM.
*

Kummer theory says: There is an exact sequence

Tensoring with , we expect:

**Proposition. ***If is a geometrically simply connected surface, and the Tate conjecture is true, then the -primary component of
is finite.
*

**Proposition. ***Suppose as above has a good reduction modulo with the same geometric Picard number (not always true), and is a number field. If the Tate conjecture is true, then
is finite.
*

**Corollary. ***If is a of geometric Picard number , then
is finite.
*

*Rapid review of descent*

Descent by Colliot-Thélène and Sansuc. Let be a geometrically simply connected surface, and . Let be the torus whose group of characters is , and . There is an exact sequence

One has a pairing

every torsor comes with a map , and if and only if .

Now assume only geometrically simply connected (not necessarily ). has no integral structure (i.e. there is not a module such that , so we must use étale cohomology.

If is any morphism of schemes, and a sheaf on , a sheaf on , then there is a spectral sequence

Apply this general situation with the structure morphism, , . One obtains a map

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

We have a map

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

**Definition. **
A *universal -gerbe* is an element
such that
.

One can (with care and difficulty) pass to to speak of universal -adic gerbes. The set of universal -adic gerbes is either empty or a principal homogeneous space under the image of in .

One has a pairing

Gerbes | ||

which gives a partition of . This can be extended to a map

So, in these cases, have , where ranges over a finite set.

We can show if and only if there exists a universal gerbe with points everywhere locally.

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