Existence Of Rational Points Research Articles

Integers that are sums of two rational sixth powers

AbstractWe prove that $164\, 634\, 913$ is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$ , we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell–Weil sieve, to rule out the existence of rational points on $C_{k}$ for various k.

Invariant curves for endomorphisms of $${{\mathbb {P}}}^1\times {{\mathbb {P}}}^1$$

Let $$A_1, A_2\in {{\mathbb {C}}}(z)$$ be rational functions of degree at least two that are neither Lattès maps nor conjugate to $$z^{\pm n}$$ or $$\pm T_n.$$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of $$({{\mathbb {P}}}^1({{\mathbb {C}}}))^2$$ of the form $$(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$$ In particular, we show that if $$A\in {{\mathbb {C}}}(z)$$ is not a “generalized Lattès map”, then any (A, A)-invariant curve has genus zero and can be parametrized by rational functions commuting with A. As an application, for A defined over a subfield K of $$ {{\mathbb {C}}}$$ we give a criterion for a point of $$({{\mathbb {P}}}^1(K))^2$$ to have a Zariski dense (A, A)-orbit in terms of canonical heights, and deduce from this criterion a version of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits. We also prove a result about functional decompositions of iterates of rational functions, which implies in particular that there exist at most finitely many $$(A_1, A_2)$$ -invariant curves of any given bi-degree $$(d_1,d_2).$$

Galois criterion for torsion points of Drinfeld modules

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 [Formula: see text], but negative if the rank is [Formula: see text]. Moreover, for rank [Formula: see text] Drinfeld modules we classify those cases where the answer is positive.

Mazur's conjecture and an unexpected rational curve on Kummer surfaces and their superelliptic generalisations

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. For the proof, we investigate a highly singular rational curve on the Kummer surface $K$ associated to a product of two elliptic curves over $\mathbb{Q}$, which previously appeared in publications by Mestre, Kuwata-Wang and Satge. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves. Finally, we give a proof of Mazur's conjecture for the Kummer surface $K$ without any restrictions on the $j$-invariants of the two elliptic curves.

A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.

Hasse principle for Kummer varieties

The existence of rational points on Kummer varieties associated to 2-coverings of abelian varieties over number fields can sometimes be proved through the variation of the Selmer group in the family of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. Following Mazur and Rubin, we consider the case when the Galois action on the 2-torsion has a large image. Under mild additional hypotheses we prove the Hasse principle for the associated Kummer varieties assuming the finiteness of relevant Shafarevich-Tate groups.

CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

A conjecture of Scharaschkin and Skorobogatov states that there is a Brauer–Manin obstruction to the existence of rational points on a smooth geometrically irreducible curve over a number field. In this paper, we verify the Scharaschkin–Skorobogatov conjecture for explicit families of generalized Mordell curves. Our approach uses standard techniques from the Brauer–Manin obstruction and the arithmetic of certain threefolds.

Lois de réciprocité supérieures et points rationnels

Soit K = C ( ( x , y ) ) K=\mathbb {C}((x,y)) ou K = C ( ( x ) ) ( y ) K=\mathbb {C}((x))(y) . Soit G G un K K -groupe algébrique linéaire connexe. Il a été établi que si G G est K K -rationnel, c’est-à-dire de corps des fonctions transcendant pur sur K K , si un espace principal homogène sous G G a des points rationnels dans tous les complétés de K K par rapport aux valuations de K K , alors il a un point rationnel. Nous montrons ici qu’en général l’hypothèse de K K -rationalité ne peut être omise. Nous utilisons pour cela une obstruction d’un nouveau type, fondée sur les lois de réciprocité supérieure sur un schéma de dimension deux. Nous donnons aussi une famille d’espaces principaux homogènes pour laquelle cette obstruction raffinée à l’existence d’un point rationnel est la seule obstruction. Abstract Let K = C ( ( x , y ) ) K=\mathbb {C}((x,y)) or K = C ( ( x ) ) ( y ) K=\mathbb {C}((x))(y) . Let G G be a connected linear algebraic group over K K . Under the assumption that the K K -variety G G is K K -rational, i.e. that the function field is purely transcendental, it was proved that a principal homogeneous space of G G has a rational point over K K as soon as it has one over each completion of K K with respect to a valuation. In this paper we show that one cannot in general do without the K K -rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme. We also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points.

Versality of algebraic group actions and rational points on twisted varieties

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic varietyXX. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms ofXX(existence of rational points, existence of a dense set of rational points, etc.). We give applications of this equivalence in both directions to study versality of group actions and rational points on algebraic varieties. We obtain similar results onpp-versality for a prime integerpp. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

Existence of rational points as a homotopy limit problem

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of étale topological types under the Galois action. As our main example we show that the surjectivity statement in Grothendieck's Section Conjecture would follow from the surjectivity of the map from fixed points to continuous homotopy fixed points on the level of connected components. Along the way we define a new model for the continuous étale homotopy fixed point space of a smooth variety over a field under the Galois action.

Invariant varieties for polynomial dynamical systems

We study algebraic dynamical systems (and, more generally, s -varieties) F:A n C ?A n C given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of �clusters� from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism s:C?C those algebraic varieties X?A n C for which F(X)?X s . As a special case, we show that if f(x)?C[x] is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X?A 2 C is an irreducible curve that is invariant under the action of (x,y)?(f(x),f(y)) and projects dominantly in both directions, then X must be the graph of a polynomial that commutes with f under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA 0 , a disintegrated set defined by s(x)=f(x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of f is defined over a fixed field of a power of s , and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of f is defined over a fixed field of a power of s

Existence of rational points on smooth projective varieties

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Châtelet surfaces such that exactly one of the surfaces fails to have a k-point.

Fibre products of Kummer covers and curves with many points

We study the general fibre product of any two Kummer covers of the projective line over finite fields. Under some assumptions, we obtain an involved condition for the existence of rational points in the fibre product over a rational point of the projective line so that we determine the exact number of the rational points. Using this, we construct explicit examples of such fibre products with many rational points. In particular we obtain a record and a new entry for the table (http://www.science.uva.nl/~geer/tables-mathcomp15.ps).

Average ranks of elliptic curves: Tension between data and conjecture

Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control—thanks to ideas of Kolyvagin—the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today. We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

Elliptic and hyperelliptic curves over supersimple fields in characteristic 2

In this paper, we extend a previous result of A. Pillay and the author regarding existence of rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple fields to the even characteristic case. We give a detailed exposition of the affine models of these families of curves in characteristic 2 and the transformations between members in the same rational isomorphism class.

Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Let X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be a possibly singular hypersurface of degree d ≤ n , defined over a finite field [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓ-adic cohomology H i ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]), for ℓ ≠ char([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]), are divisible by q , without using the result on the existence of rational points by Ax and Katz.

On Moduli Spaces, Equidistribution, Estimates, and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y2 = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.

Computational experiments in arithmetic geometry over finite fields and their computer support

Some selected findings of an investigation into the following two problems of arithmetic geometry of algebraic curves over finite prime fields are presented: (a) the existence of rational points and (b) the distribution of angles of Kloosterman sums. A brief description of computer experiments supporting the investigation is presented, and some algebraic and analytical context of the problems considered is given.

Note on the existence of rational points

Note on the existence of rational points