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Shibata, Takahiro

Ample canonical heights for endomorphisms on projective varieties. (English) Zbl 1442.37124

J. Math. Soc. Japan 71, No. 2, 599-634 (2019).
Summary: We define an “ample canonical height” for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call-Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell-Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety.

Cited in 3 Documents

MSC:

37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
14G05 Rational points

Keywords:

canonical height; dynamical degree
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Full Text: DOI arXiv Euclid

References:

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