Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

Lesieutre, John; Satriano, Matthew

doi:10.1093/imrn/rnz067

Cited by 10 publications

(4 citation statements)

References 45 publications

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“…cit. ), when X is an abelian variety, i.e., when X has the structure of an algebraic group [134,216], and when f is a morphism and either X is surface [163] or X is higher dimensional with additional structure [145].…”

Section: Arithmetic Degrees Of Orbitsmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. 1 2 BENEDETTO, ET AL. 22. Local-Global Questions in Dynamics 61 References 63 TRENDS AND PROBLEMS IN ARITHMETIC DYNAMICS 3background on dynamical systems, number theory, and algebraic geometry that we will need. As noted at the end of Section 3, we have attempted to make this article accessible by generally restricting attention to maps of projective space, thereby obviating the need for more advanced material from algebraic geometry. Our survey of arithmetic dynamics then commences in Section 4 and concludes in Section 22. These nineteen sections are mostly independent of one another, albeit liberally sprinkled with instructive cross-references.A Brief Timeline of the Early Days. The study of Galois groups of iterated polynomials was initiated by Odoni in a series of papers [177,178,179] during the mid-1980s. Beyond this, it appears that the first paper to describe dynamical analogues for a wide range of classical problems in arithmetic geometry did so not for polynomial maps or projective space, but rather for K3 surfaces admitting an automorphism of infinite order. The 1991 paper of Silverman [204] discusses dynamical analogues of various problems, including: (1) Uniform boundedness of periodic points;(2) Periodic points on subvarieties;(3) Lower bounds for canonical heights; (4) Open image of Galois groups; (5) Integral points in orbits. The rest of the 1990s saw an explosion of papers in which numerous researchers began to study a wide variety of problems in arithmetic dynamics. The serious study of p-adic dynamics starts with a 1983 paper of Herman and Yoccoz [107], two of the world's leading complex dynamicists, in which they investigate a p-adic analogue of a classical problem in complex dynamics. Later in the 1980s and 90s, the physics literature includes several papers [6,7,21,230] that discuss p-adic dynamics for polynomial maps, but the deeper study of p-adic dynamics really took off with the Ph.D. theses of Benedetto [24, 25] in 1998 and Rivera-Letelier [190, 191] in 2000. The dynamics of iterated p-adic power series was investigated by Lubin [156] in a 1994 paper. We also mention that there is an extensive literature on ergodic theory over p-adic fields which, although not within the purview of this survey, dates back to at least 1975 [180]. Abstract Dynamical SystemsA discrete dynamical system consists of a set S and a self-map

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Section: Arithmetic Degrees Of Orbitsmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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show abstract

“…Our goal in this article will be to understand both the arithmetic and geometric aspects of f under iteration. Associated to f are two basic numerical invariants that have been well studied, see for example ( [6], [5], [4], [13], [8]). The dynamical degree is defined as (1) λ 1 (f ) := lim n→∞…”

Section: Introductionmentioning

confidence: 99%

Effective Eigendivisors and the Kawaguchi-Silverman Conjecture

Nasserden¹

2020

Preprint

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Let f : X → X be a surjective endomorphism of a normal projective variety defined over a number field. The dynamics of f may be studied through the dynamics of the linear action f * : Pic(X) R → Pic(X) R , which are governed by the spectral theory of f * . Let λ 1 (f ) be the spectral radius of f * . We study Q-divisors D with f * D = λ 1 (f )D and κ(D) = 0 where κ(D) is the Iitaka dimension of the divisor D. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights of Jordan blocks described by Kawaguchi and Silverman. Finally we identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture.

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“…14 ([21],[23, Theorem 2],[31, Theorem 1.2]). Conjecture 2.10 is true for automorphisms on smooth projective surfaces;…”

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confidence: 99%

“…)[31, Proposition 1.7] Let X be a smooth projective variety over Q with dim X " 3 and Kodaira dimension zero. Then Conjecture 2.10 is true for surjective self-morphisms f : X ÝÑ X with deg f ě 2.…”

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confidence: 99%

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

Matsuzawa

2020

Advances in Mathematics

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We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.Cartier divisors D). By definition, N 1 (X) and N 1 (X) are dual to each other.-When X is normal, the Iitaka dimension of a Q-Cartier divisor D on X is denoted by κ(D). • Let M be a Z-module. We write M Q = M⊗ Z Q, M R = M⊗ Z R, and so on.

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Canonical Heights on Hyper-Kähler Varieties and the Kawaguchi–Silverman Conjecture

Cited by 10 publications

References 45 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

Effective Eigendivisors and the Kawaguchi-Silverman Conjecture

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

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