This is joint work with Graber, Harris, and Starr.
Throughout, we let be any finite field and a field of transcendence degree over . We have two classical results, due to Chevalley-Warning and Tsen, for and , respectively: a hypersurface of low degree has a rational point; a hypersurface of low degree is one with .
Inspired by these theorems, one (i.e. Artin) defines a field to be quasi-algebraically closed if every hypersurface of low degree over has a -rational point. In view of resent results due to Kollár, Kollár-Miyaoka-Mori, and Graber-Harris-Starr, we ask similar questions not for hypersurfaces but for certain other classes of varieties.
We generalize the notion of hypersurfaces of low degree to rationally connected varieties over which are projective and smooth: a variety is rationally connected if for any two points , there exists a rational curve with . Rationally connected varieties are closed under birational transformation, products, domination (if is dominant, and is rationally connected, then is rationally connected), and specialization. This is a much better class of varieties than, say, rational varieties (consider the difficulty in determining which cubic -folds are rational).
There is another candidate for a generalization of hypersurfaces of low degree: a variety over is -acyclic if
The only rationally connected curves are rational curves; the only rationally connected surfaces are rational surfaces. However, there are -acyclic surfaces which are not rational, e.g. Enriques surfaces.
Theorem. If is a smooth hypersurface over , then it is equivalent for to be of low degree, rationally connected, and -acyclic.
Theorem. [Generalized Chevalley-Warning; Katz] Any -acyclic variety over has a -rational point.
Over , and given an endomorphism , one defines the Lefschetz number ; this complex number measures the fixed point locus of . Over a finite field, the Lefschetz number counts the number of fixed points, at least modulo ; one computes that the Lefschetz number is .
For the generalization to rationally connected varieties over , a variety over can be thought of as a family of rationally connected varieties over a curve . By family we always mean that although the base might not be proper or smooth, the morphism is proper and generically smooth.
Theorem. [Generalized Tsen; Graber, Harris, Starr] Any rationally connected variety over has a -rational point.
Given a family , a section is a triangle
We can rephrase the GHS theorem in this language as follows: If is a family with a pseudo-section, then its restriction to every smooth curve has a section.
Theorem. [Weak converse to GHB] If is a family such that every restriction for every smooth curve has a section, then has a pseudo-section.
This theorem is related to Lefschetz's theorem about as can be seen by restricting attention to finite étale covers.
This theorem has application to finding varieties with no rational point; in particular:
Corollary. There exists an Enriques surface over some with no -rational point.
This is completely ineffective; an open question is to find the genus of the curve given by this counterexample.
Every Enriques surfaces over has a point over ; this is not the case over , so we have distinguished finite fields from function fields of transcendence degree over . We ask: in the Artin-Lang philosophy, what kinds of varieties are cut out by -acyclic varieties?
We have another corollary:
Corollary. A family of curves of genus over a base has a section if and only if it has a section over every curve .
The corollary is clear: a family of curves of genus has no room for a pseudo-section.
Number Theoretic Applications
Now we consider a family defined over a number field; we say is arithmetically surjective if and only if is surjective for all finite extensions .
If is a family of curves of genus , is it the case that arithmetic surjectivity is equivalent to the existence of a section over ? This question is unapproachable in full generality.
Instead, let us take a very small fragment of it: let be a nonempty open subset of over , a family of genus curves, we say it belongs to its Jacobian . We consider quadratic twist elliptic pencils; given any , we have the pencil . We have the problem: For all belonging to , is is true that arithmetic surjectivity holds if and only if a section exists? Work of Skinner-Ono can be used to establish this for all elliptic curves .
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