Rational points on curves

Michael Stoll

Displaying similar documents to “Rational points on curves”

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

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For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

Rational points on $X_{0}^{+} (p^{r})$

Yuri Bilu, Pierre Parent, Marusia Rebolledo (2013)

Annales de l’institut Fourier

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Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of $X_{0}^{+} (p^{r}) (ℚ)$ , for $r > 1$ and $p$ a prime number exceeding $2 \cdot 10^{11}$ . This includes the case of the curves $X_{split} (p)$ . We then prove, with the help of computer calculations, that the same holds true for $p$ in the range $11 \leq p \leq 10^{14}$ , $p \neq 13$ . The combination of those results completes the qualitative study of rational points on $X_{0}^{+} (p^{r})$ undertook in our previous work, with the only exception of $p^{r} = 13^{2}$ .

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(- K_{X})$ -degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$ ) of all rational curves through $x$ are at least $dim X + 1$ , then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(- K_{X})$ -degrees of all rational curves through $x$ are at least $dim X$ . Our study uses the projective variety $𝒞_{x} \subset ℙ T_{x} (X)$ , called the VMRT at $x$ , defined as the union of tangent directions to the...

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves such that $W (ℝ)$ is orientable. J. Kollár proved that a connected component $N$ of $W (ℝ)$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$ , our...

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

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We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $\{\begin{matrix} n = a_{1} + a_{2} + \dots + a_{s - 1}, \\ a_{1} a_{2} \dots a_{s - 1} (a_{1} + a_{2} + \dots + a_{s - 1}) = b^{s} \end{matrix}$ has positive integer or rational solutions $n$ , $b$ , $a_{i}$ , $i = 1, 2, \dots, s - 1$ , $s \geq 3 .$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s = 3$ , but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s \geq 4$ . As a corollary, for $s \geq 4$ and any positive integer $n$ , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions...

An $a b c d$ theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

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We provide a lower bound for the number of distinct zeros of a sum $1 + u + v$ for two rational functions $u, v$ , in term of the degree of $u, v$ , which is sharp whenever $u, v$ have few distinct zeros and poles compared to their degree. This sharpens the “ $a b c d$ -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^{a} + y^{a} + z^{c} = 1$ contains only finitely many rational or elliptic...

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

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Consider the families of curves $C^{n, A} : y ² = x ⁿ + A x$ and $C_{n, A} : y ² = x ⁿ + A$ where A is a nonzero rational. Let $J^{n, A}$ and $J_{n, A}$ denote their respective Jacobian varieties. The torsion points of $C^{3, A} (ℚ)$ and $C_{3, A} (ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7, A} (ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7, A} (ℚ) [2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3, A}$ (A ≠ 4) and $J^{5, A}$ . We...

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

A local-global principle for rational isogenies of prime degree

Andrew V. Sutherland (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a number field. We consider a local-global principle for elliptic curves $E / K$ that admit (or do not admit) a rational isogeny of prime degree $ℓ$ . For suitable $K$ (including $K = ℚ$ ), we prove that this principle holds for all $ℓ \equiv 1 mod 4$ , and for $ℓ < 7$ , but find a counterexample when $ℓ = 7$ for an elliptic curve with $j$ -invariant $2268945 / 128$ . For $K = ℚ$ we show that, up to isomorphism, this is the only counterexample.

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

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Consider a recurrence sequence ${(x_{k})}_{k \in ℤ}$ of integers satisfying $x_{k + n} = a_{n - 1} x_{k + n - 1} + . . . + a ₁ x_{k + 1} + a ₀ x_{k}$ , where $a ₀, a ₁, . . ., a_{n - 1} \in ℤ$ are fixed and a₀ ∈ -1,1. Assume that $x_{k} > 0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k ₀} < 0$ then for each negative integer -D there exist infinitely many rational primes q such that $q | x_{k}$ for some k ∈ ℕ and (-D/q) = -1.

Nodal curves in $\mathbb{P}_{3}(\mathbb{C})$

Edoardo Ballico, Paolo Oliverio (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Siano $d$ , $g$ , $t$ interi con $0\leq t\leq g$ ; se esiste in $\mathbb{P}_{3}(\mathbb{C})$ una curva connessa, non singolare di grado $d$ e genere $g$ , allora esiste in $\mathbb{P}_{3}(\mathbb{C})$ una curva irriducibile di grado $d$ , genere aritmetico $g$ e $t$ nodi.

On covering and quasi-unsplit families of curves

Laurent Bonavero, Cinzia Casagrande, Stéphane Druel (2007)

Journal of the European Mathematical Society

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Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$ , we find conditions allowing one to construct a geometric quotient $q : X \to Y$ , with $q$ regular on the whole of $X$ , such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$ . Among other results, we show that on a normal and $ℚ$ -factorial projective variety $X$ with canonical singularities and $dim X \leq 4$ , every covering and quasi-unsplit family $V$ of rational curves generates a geometric...