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
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
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
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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