Fundamentals of Diophantine Geometry

Lang, Serge

doi:10.1007/978-1-4757-1810-2

Cited by 833 publications

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“…On Abelian varieties defined over a number field, Néron and Tate developed the theory of canonical height functions that behave well relative to the [n]-th power map (cf. [9,Chap. 5]).…”

Section: Introduction and The Statement Of The Main Resultsmentioning

confidence: 99%

Canonical height functions for affine plane automorphisms

Kawaguchi

2006

Math. Ann.

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Abstract. Let f : A 2 → A 2 be a polynomial automorphism of dynamical degree δ ≥ 2 over a number field K. (This is equivalent to say that f is a polynomial automorphism that is not triangularizable.) Then we construct canonical height functions defined on A 2 (K) associated with f . These functions satisfy the Northcott finiteness property, and an Kvalued point on A 2 (K) is f -periodic if and only if its height is zero. As an application of canonical height functions, we give an estimate on the number of points with bounded height in an infinite f -orbit.

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“…On Abelian varieties defined over a number field, Néron and Tate developed the theory of canonical height functions that behave well relative to the [n]-th power map (cf. [9,Chap. 5]).…”

Section: Introduction and The Statement Of The Main Resultsmentioning

confidence: 99%

Canonical height functions for affine plane automorphisms

Kawaguchi

2006

Math. Ann.

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“…The Thue equations are named after A. Thue [31] who proved in the case K = Q, R = Z, m = 2, that if F is a binary form having at least three pairwise linearly independen t linear factors in its factorizatio n over the field of algebraic numbers, then (1) has only finitely many solutions. After several generalizati ons, Lang [13] finally extended Thue's result to the general case considered above (when K is an arbitrary finitely generated extension of Q and R is an arbitrary finitely generated subring of K over Z).…”

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confidence: 99%

Finiteness criteria for decomposable form equations

Evertse¹,

Győry²

1988

Acta Arith.

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“…Hence it follows from a variant of Hilbert's irreducibility theorem (see, e.g., Chapter 9 of Lang (1983)) that there exist infinitely many integers c such that the two conditions in the theorem are satisfied. Indeed, as we will see in the next section, the polynomial HðZ; xÞ is irreducible over the rational function field QðZÞ.…”

Section: Resultsmentioning

confidence: 99%

Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven

Sato

2010

IIS

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We introduce a simple way to construct a family of number fields of given degree with class numbers divisible by a given integer, by using the arithmetic theory of elliptic curves. In particular, we start with an elliptic curve defined over the rational number field with a rational torsion point of order l 2 f3; 5; 7g, and show a way to construct infinitely many number fields of given odd degree d ! 3 with class numbers divisible by l.

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Fundamentals of Diophantine Geometry

Cited by 833 publications

References 0 publications

Canonical height functions for affine plane automorphisms

Canonical height functions for affine plane automorphisms

Finiteness criteria for decomposable form equations

Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven

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