TY - JOUR

T1 - Rationality of dynamical canonical height

AU - De Marco, Laura

AU - Ghioca, Dragos

N1 - Publisher Copyright:
© Cambridge University Press, 2018.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - We present a dynamical proof of the well-known fact that the Néron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field , where is a curve. More generally, we investigate the mechanism by which the local canonical height for a map defined over a function field can take irrational values (at points in a local completion of ), providing examples in all degrees . Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math. 201 (2014), 631-700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights are rational and positive, for maps and of multiplicatively independent degrees and points , then the orbits and intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture. Invent. Math. 171 (2) (2008), 463-483].

AB - We present a dynamical proof of the well-known fact that the Néron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field , where is a curve. More generally, we investigate the mechanism by which the local canonical height for a map defined over a function field can take irrational values (at points in a local completion of ), providing examples in all degrees . Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math. 201 (2014), 631-700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights are rational and positive, for maps and of multiplicatively independent degrees and points , then the orbits and intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture. Invent. Math. 171 (2) (2008), 463-483].

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U2 - 10.1017/etds.2017.131

DO - 10.1017/etds.2017.131

M3 - Article

AN - SCOPUS:85041589144

SN - 0143-3857

VL - 39

SP - 2507

EP - 2540

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 9

ER -