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.
(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:
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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