# Skorobogatov: Counterexamples to the Hasse Principle...

This is joint work with Laura Basile.

We work over a field with , its algebraic closure.

Definition. A bielliptic surface is a -form of a smooth projective surface of Kodaira dimension 0 that is not , neither abelian nor Enriques.

There is a complete list of such available. We have , but , for or . Over the algebraic closure, , where acts on by translations.

Proposition. There exists an abelian surface , a principal homogeneous space of , and a finite étale morphism , .

Proof. Since , then consider and normalize; the map is unramified, so . By the classification of surfaces, is an abelian surface. ( is a torsor under .)

Remark. This will not hold in higher dimension; there are just many more possibilities.

Consider , , a principal homogeneous space of , and likewise for . Now acts on so that acts on by translations, ; the action on on cannot be by translations or else itself would be a principal homogeneous space, so the action has fixed points.

Proposition. .

Proof. Take , fixed by , and write down the usual cocycle: if , .

(It arises from .)

Corollary. Let be the isogeny with kernel . Then .

We have one of the following possibilities:

Now assume , and ; we want an example where , but . We do the case .

With the notation as above:

Theorem. Assume that:

1. acts nontrivially on , which is as an abstract group;
2. ;
3.    ;
4. ;
5. , that is, for any principal homogeneous space of obtained from a class in , there exists a place where it has no point.
Then is a counterexample to the Hasse principal not explained by the Manin obstruction.

Example. If , acts by , , with acting by ; looking at the Selmer group, you look at principal homogeneous spaces of the form , so if , this has no point.

We have as Galois modules (this holds more generally if is a surface and ), and . Then (i) implies that .

The kernel of the restricted Cassels pairing     consists of elements in the image of    , where is the dual isogeny. Since must be zero because it is alternating, so lift ; then we have étale maps

to find an adelic point, find a rational point and a collection ; then has , where is the projection.

The last condition (v) says that there are no rational points on ; rational points on comes from twists of , but by assumption these have no point over a place , so they arise from .

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