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A dynamical interpretation of the global canonical height on an elliptic curve. (English) Zbl 0927.11009
This paper can be regarded as an introduction to the recent text by the same authors [Heights of polynomials and entropy in algebraic dynamics. Springer, London (1999; Zbl 0919.11064)]. They begin with a leisurely and enlightening review of properties of the (logarithmic) Mahler measure \(m(F(u_1,\dots,u_d))\) and its meaning as the topological entropy of an algebraic dynamical system defined by the polynomial \(F(u_1,\dots,u_d)\). They then turn to the definition of the elliptic Mahler measure of G. R. Everest and B. ni Fhlathúin [Math. Proc. Camb. Philos. Soc. 120, 13-25 (1996; Zbl 0865.11068)] and conjecture that this arises in a similar but more complicated way as the entropy of a suitable algebraic dynamical system. They deduce some necessary properties of this conjectured dynamical system but its existence remains an open question. Table 1 on p. 315 gives a summary of analogous properties of the classical Mahler measure and the elliptic Mahler measure indicating clearly some of the interesting open questions concerning the latter.

MSC:
11C08 Polynomials in number theory
11G50 Heights
22D40 Ergodic theory on groups
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
28D99 Measure-theoretic ergodic theory
37B40 Topological entropy
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