Rational points and zero-cycles on rationally connected varieties over number fields

AbstractBased on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials.First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

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Galois criterion for torsion points of Drinfeld modules

International Journal of Number Theory ◽

10.1142/s1793042121500901 ◽

2021 ◽

pp. 1-12

Author(s):

Chien-Hua Chen

Keyword(s):

Maximal Ideal ◽

Abelian Varieties ◽

Number Fields ◽

Rational Points ◽

Drinfeld Module ◽

Torsion Point ◽

Drinfeld Modules ◽

Torsion Points ◽

Ideal Formula ◽

Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

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FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

Forum of Mathematics Sigma ◽

10.1017/fms.2020.20 ◽

2020 ◽

Vol 8 ◽

Author(s):

NATHAN CHEN ◽

DAVID STAPLETON

Keyword(s):

Minimal Degree ◽

Rational Map ◽

Rationally Connected ◽

General Complex ◽

Special Fibers ◽

Fano Index ◽

Low Dimensional ◽

The Way ◽

Rationally Connected Varieties

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

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Growth rate of ample heights and the Dynamical Mordell–Lang conjecture

International Journal of Number Theory ◽

10.1142/s1793042118501610 ◽

2018 ◽

Vol 14 (10) ◽

pp. 2673-2685

Author(s):

Kaoru Sano

Keyword(s):

Growth Rate ◽

Explicit Formula ◽

Number Fields ◽

Rational Points ◽

Positive Answer ◽

Projective Varieties ◽

Smooth Projective Varieties

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms of smooth projective varieties over number fields. As an application, we give a positive answer to a variant of the Dynamical Mordell–Lang conjecture for pairs of étale endomorphisms, which is also a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.

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Trinomials defining quintic number fields

International Journal of Number Theory ◽

10.1142/s1793042117501032 ◽

2017 ◽

Vol 13 (07) ◽

pp. 1881-1894 ◽

Cited By ~ 1

Author(s):

Jesse Patsolic ◽

Jeremy Rouse

Keyword(s):

Elliptic Curve ◽

Number Field ◽

Number Fields ◽

Rational Points

Given a quintic number field K/ℚ, we study the set of irreducible trinomials, polynomials of the form x5 + ax + b, that have a root in K. We show that there is a genus 4 curve CK whose rational points are in bijection with such trinomials. This curve CK maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on CK using elliptic curve Chabauty.

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