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
, .
*

**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. ***
.
*

(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:
*

- acts nontrivially on , which is as an abstract group;
- ;
- ;
- ;
- , that is, for any principal homogeneous space of obtained from a class in , there exists a place where it has no point.

**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

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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