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On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia; Joseph Silverman — 2009

Journal de Théorie des Nombres de Bordeaux

Let $ϕ : X \to X$ be a morphism of a variety defined over a number field $K$ , let $V \subset X$ be a $K$ -subvariety, and let $𝒪_{ϕ} (P) = {ϕ^{n} (P) : n \geq 0}$ be the orbit of a point $P \in X (K)$ . We describe a local-global principle for the intersection $V \cap 𝒪_{ϕ} (P)$ . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V (K)$ are Brauer–Manin unobstructed for power maps on $ℙ^{2}$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical...

The Néron Fiber of Abelian Varieties with Potential Good Reduction.

Joseph H. Silverman — 1983

Mathematische Annalen

Arithmetic Distance Functions and Height Fuctions in Diophantine Geometry.

Joseph H. Silverman

Mathematische Annalen

Rational points on K3 surfaces: A new canonical height.

Joseph H. Silverman — 1991

Inventiones mathematicae

Integral points on abelian surfaces are widely spaced

Joseph H. Silverman — 1987

Compositio Mathematica

The field of definition for dynamical systems on $ℙ^{1}$

Joseph H. Silverman — 1995

Compositio Mathematica

Geometric and arithmetic properties of the Hénon map.

Joseph H. Silverman — 1994

Mathematische Zeitschrift

Heights and the specialization map for families of abelian varieties.

Joseph H. Silverman — 1983

Journal für die reine und angewandte Mathematik

Variation of the canonical height on elliptic surfaces I: Three examples.

Joseph H. Silverman — 1992

Journal für die reine und angewandte Mathematik

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

Joseph H. Silverman — 1987

Journal für die reine und angewandte Mathematik

Integral points on Abelian varieties.

Joseph H. Silverman — 1985

Inventiones mathematicae

Elliptic Carmichael numbers and elliptic Korselt criteria

Joseph H. Silverman — 2012

Acta Arithmetica

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Joseph H. Silverman — 2012

Journal de Théorie des Nombres de Bordeaux

A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to $1$ modulo $m$ . We prove a similar result for polynomials $f (X)$ that are divisible in $(ℤ / m ℤ) [X]$ by a polynomial of the form $1 + X + \dots + X^{n}$ for some $n \geq ϵ deg (f)$ . We also formulate and prove an analogous statement for elliptic curves.

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Currently displaying 1 – 20 of 22

On a dynamical Brauer–Manin obstruction

The Néron Fiber of Abelian Varieties with Potential Good Reduction.

Arithmetic Distance Functions and Height Fuctions in Diophantine Geometry.

Rational points on K3 surfaces: A new canonical height.

Integral points on abelian surfaces are widely spaced

The field of definition for dynamical systems on $ℙ^{1}$

Geometric and arithmetic properties of the Hénon map.

Heights and the specialization map for families of abelian varieties.

Variation of the canonical height on elliptic surfaces I: Three examples.

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

Integral points on Abelian varieties.

Elliptic Carmichael numbers and elliptic Korselt criteria

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Variation of periods modulo $p$ in arithmetic dynamics.

Exceptional units and numbers of small Mahler measure.

Divisibility sequences and powers of algebraic integers.

Common divisors of $a^{n} - 1$ and $b^{n} - 1$ over function fields.

Periodic points, multiplicities, and dynamical units.

The number of elliptic curves over Q with conductor N.

Canonical heights on varieties with morphisms