# Raskind: Descent on Simply Connected Algebraic Surfaces

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

since , as you pass to the limit over , one has the exact sequence

The term is algebraic, the term transcendental.

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.

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

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

coming from the Hochschild-Serre spectral sequence. We identify

and we think of as principal homogeneous spaces under ; an element is a universal torsor if .

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

coming from the term in the spectral sequence. We expect that the group will play the role of in the above.

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

and

We have a map

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

Let be the composite of these three maps. Let .

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

where is a chosen universal gerbe. On , zero-cycles of degree 0, this is the higher -adic Abel-Jacobi map, and the image of this map is a finitely generated -module; if is a number field, one can show in some cases that that the image is finite, e.g. of Picard number over or over the CM field.

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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