Greatest common divisors of $u-1$, $v-1$ in positive characteristic and rational points on curves over finite fields

Pietro Corvaja; Umberto Zannier

Displaying similar documents to “Greatest common divisors of $u - 1$ , $v - 1$ in positive characteristic and rational points on curves over finite fields”

Solutions to xyz = 1 and x + y + z = k in algebraic integers of small degree, II

H. G. Grundman, L. L. Hall-Seelig (2015)

Acta Arithmetica

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Let k ∈ ℤ be such that $|_{k} (ℚ) |$ is finite, where $_{k} : y ² = 1 - 2 k x + k ² x ² - 4 x ³$ . We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

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...

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...

Solutions to xyz = 1 and x+y+z = k in algebraic integers of small degree, I

H. G. Grundman, L. L. Hall-Seelig (2014)

Acta Arithmetica

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Let k ∈ ℤ be such that $|_{k} (ℚ) | = 3$ , where $_{k} : y ² = 1 - 2 k x + k ² x ² - 4 x ³$ . We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca, Edoardo Sernesi (2014)

Journal de l’École polytechnique — Mathématiques

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We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve $C$ and the stability of the sheaf of logarithmic vector fields along $C$ , the freeness of the divisor $C$ and the Torelli properties of $C$ (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski (2015)

Czechoslovak Mathematical Journal

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We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field $𝔽_{p}$ of $p$ elements, such that the discriminant $D (E)$ of the quadratic number field containing the endomorphism ring of $E$ over $𝔽_{p}$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

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...

Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation

Eric Lombardi, Laurent Stolovitch (2010)

Annales scientifiques de l'École Normale Supérieure

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In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of $ℂ^{n}$ , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part $S$ which ensures that if such a perturbation of $S$ is formally conjugate to $S$ then it is also holomorphically conjugate to it. We study the normal form problem relatively to $S$ . We give a condition on $S$ that ensures...

Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque (2024)

Archivum Mathematicum

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We consider the Lebesgue-Ramanujan-Nagell type equation $x^{2} + 5^{a} 13^{b} 17^{c} = 2^{m} y^{n}$ , where $a, b, c, m \geq 0, n \geq 3$ and $x, y \geq 1$ are unknown integers with $gcd (x, y) = 1$ . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$ -integral points on a class of elliptic curves.

Governing fields for the 2-classgroup of $Q (s q r t - q_{1} q_{2} p)$ and a related reciprocity law

Patrick Morton (1990)

Acta Arithmetica

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Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $ℚ (α)$ and $ℚ (β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $ℚ (β) \to ℚ (α)$ . The algorithm is particularly efficient if there is only one isomorphism.

On the diophantine equation $x^{2} + 2^{a} 3^{b} 73^{c} = y^{n}$

Murat Alan, Mustafa Aydin (2023)

Archivum Mathematicum

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In this paper, we find all integer solutions $(x, y, n, a, b, c)$ of the equation in the title for non-negative integers $a, b$ and $c$ under the condition that the integers $x$ and $y$ are relatively prime and $n \geq 3$ . The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...

Displaying similar documents to “Greatest common divisors of u - 1 , v - 1 in positive characteristic and rational points on curves over finite fields”

Displaying similar documents to “Greatest common divisors of $u - 1$ , $v - 1$ in positive characteristic and rational points on curves over finite fields”