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Canonical heights on projective space. (English) Zbl 0895.14006

Let \(K\) be a number field, and let \(\phi\:\mathbb P^N\to\mathbb P^n\) be a morphism of degree \(d>1\) defined over \(K\). By the theory of canonical heights on varieties relative to an endomorphism as developed by G. S. Call and J. H. Silverman [Compos. Math. 89, No. 2, 163-205 (1993; Zbl 0826.14015)], there is a unique Weil height \(\widehat h\) for \(\mathcal O(1)\) on \(\mathbb P^N\) such that \(\widehat h(\phi(P))=dh(P)\) for all \(P\in\mathbb P^N(\overline{K})\). The present paper shows that this canonical height \(\widehat h(P)\) can be expressed as a sum (over places of \(K(P)\)) of canonical local heights. For non-archimedean places of good reduction for \(\phi\), the local height is shown to be given by a simple formula; for the finitely many remaining places, some series and sequence formulas are given. Additional results are given in the case where there is a hyperplane \(W\) such that \(\phi^{*}(W)=(\deg\phi)W\), and also in the special case of \(\mathbb P^1\). The paper concludes by using the local canonical heights on \(\mathbb P^1\) to define a “\(v\)-adic filled Julia set” of \(\phi\), and to show that a point \(P\in K\) is pre-periodic for \(\phi\) if and only if it lies in the \(v\)-adic filled Julia set for all \(v\).
Reviewer: P.Vojta (Berkeley)

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 0826.14015
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References:

[1] Beardon, A. F., Iteration of Rational Functions. Iteration of Rational Functions, Graduate Texts in Mathematics, Vol. 132 (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0742.30002
[2] Call, G. S.; Silverman, J. H., Canonical heights on varieties with morphisms, Compositio Math., 89, 163-205 (1993) · Zbl 0826.14015
[3] Call, G. S.; Silverman, J. H., Computing the canonical height on K3 surfaces, Math. Comp., 65, 259-290 (1996) · Zbl 0865.14020
[4] Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1989), Addison-Wesley: Addison-Wesley New York · Zbl 0695.58002
[5] S. W. Goldstine, 1993, Canonical Heights on Projective Space, Amherst College; S. W. Goldstine, 1993, Canonical Heights on Projective Space, Amherst College · Zbl 0895.14006
[6] Koblitz, N., \(pp. pp\), Graduate Texts in Mathematics, Vol. 58 (1977), Springer-Verlag: Springer-Verlag New York
[7] Lang, S., Algebraic Number Theory. Algebraic Number Theory, Graduate Texts in Mathematics, Vol. 110 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0601.12001
[8] Lang, S., Fundamentals of Diophantine Geometry (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0528.14013
[9] Lewis, D. J., Invariant set of morphisms on projective and affine number spaces, J. Algebra, 20, 419-434 (1972) · Zbl 0245.12003
[10] Morton, P.; Silverman, J. H., Rational periodic points of rational functions, Internat. Math. Res. Notices, 2, 97-110 (1994) · Zbl 0819.11045
[11] Narkiewicz, W., Polynomial cycles in algebraic number fields, Colloq. Math, 58, 151-155 (1989) · Zbl 0703.12002
[12] Néron, A., Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math., 82, 249-331 (1965) · Zbl 0163.15205
[13] Northcott, D. G., Periodic points on an algebraic variety, Ann. of Math., 51, 167-177 (1950) · Zbl 0036.30102
[14] Pezda, T., Polynomial cycles in certain local domains, Acta Arith., 66, 11-22 (1994) · Zbl 0803.11063
[15] Pezda, T., Cycles of polynomial mappings in several variables, Manuscripta Math., 83, 279-289 (1994) · Zbl 0804.11059
[16] Russo, P.; Walde, R., Rational periodic points of the quadratic function\(Q_c}(xx^2c\), Amer. Math. Monthly, 101, 318-331 (1994) · Zbl 0804.58036
[17] Silverman, J. H., The Arithmetic of Elliptic Curves. The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0585.14026
[18] Silverman, J. H., Computing heights on elliptic curves, Math. Comp., 51, 339-358 (1988) · Zbl 0656.14016
[19] Silverman, J. H., Rational points on K3 surfaces: A new canonical height, Invent. Math., 105, 347-373 (1991) · Zbl 0754.14023
[20] Silverman, J. H., Integer points, Diophantine approximation, and the iteration of rational maps, Duke Math. J., 71, 793-829 (1993) · Zbl 0811.11052
[21] J. T. Tate, Oct. 1, 1979, Letter to J.-P. Serre; J. T. Tate, Oct. 1, 1979, Letter to J.-P. Serre
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