×

Entropy and the canonical height. (English) Zbl 0994.11022

The authors give an ergodic theoretic interpretation of the canonical height of a point on an elliptic curve. Following R. Bowen [Trans. Am. Math. Soc. 153, 401–414 (1971; Zbl 0212.29201)], the authors define the topological entropy \(h({\mathbf T})\) for a sequence of uniformly continuous maps \(\mathbf T=\{ T_n\}_{n=1}^{\infty}\) on a locally compact metric space \((X,d)\). Then starting with an elliptic curve \(E\) defined over \(\mathbb Q\) and a point \(Q\in E(\mathbb Q)\), the authors associate with \(Q\) a sequence of diagonal transformations \({\mathbf U}=\{ U_n\}_{n=0}^{\infty}\) on the adèle ring \(\mathbb Q_A\) and prove the following result:
If \(Q\) has non-singular reduction modulo \(p\) for all primes \(p\) then the entropy \(h({\mathbf U})\) is equal to the global canonical height \(\hat{h}(Q)\) of \(Q\). Let \(S\) denote the set of primes \(p\) for which \(Q\) has singular reduction modulo \(p\); write \(\mathbb Q_S=\prod_{p\in S} \mathbb Q_p\), let \({\mathbf U}_S=\{ U_{n,S}\}\) where \(U_{n,S}\) is the restriction of \(U_n\) to \(\mathbb Q_S\), and let \({\mathbf U}/{\mathbf U}_S=\{ U_n/U_{n,S}\}_{n=1}^{\infty}\) consist of the quotient maps on \(\mathbb Q_A/\mathbb Q_S\). Then \(h({\mathbf U}/{\mathbf U}_S)= \hat{h}(Q)\). Moreover, both the entropies \(h({\mathbf U})\) and \(h({\mathbf U}_S)\) can be expressed in terms of the local canonical heights of \(Q\).
This result may be viewed as an elliptic analogue of a result of Yuzvinskii for the height on the projective line \(\mathbb P^1(\mathbb Q)\). The authors use lower bounds for linear forms in elliptic logarithms, a strengthening of Siegel’s theorem on integral points on elliptic curves, and an elliptic analogue of Jensen’s formula for zeros of polynomials inside a circle.

MSC:

11G05 Elliptic curves over global fields
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
11G50 Heights

Citations:

Zbl 0212.29201
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Adler, R. L.; Konheim, A. G.; McAndrew, M. H., Topological entropy, Trans. Amer. Math. Soc., 114, 309-319 (1965) · Zbl 0127.13102
[2] Apostol, T. M., Introduction to Analytic Number Theory. Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0335.10001
[3] Bowen, R., Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153, 401-414 (1971) · Zbl 0212.29201
[4] Bowen, R., Periodic points and measures for Axiom \(A\) diffeomorphisms, Trans. Amer. Math. Soc., 154, 377-397 (1971) · Zbl 0212.29103
[5] D’Ambros, P.; Everest, G.; Miles, R.; Ward, T., Dynamical systems arising from elliptic curves, Colloq. Math., 84/85, 95-107 (2000) · Zbl 0965.37020
[6] D’Ambros, P., Algebraic Dynamics in Positive Characteristic (2001), Univ. East Anglia
[7] David, S., Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. France (N.S.), 62 (1995) · Zbl 0859.11048
[8] Dekking, F. M., Some examples of sequence entropy as an isomorphism invariant, Trans. Amer. Math. Soc., 259, 167-183 (1980) · Zbl 0393.28017
[9] Everest, G. R.; Fhlathúin, B. N., The elliptic Mahler measure, Math. Proc. Cambridge Philos. Soc., 120, 13-25 (1996) · Zbl 0865.11068
[10] Everest, G., Explicit local heights, New York J. Math., 5, 115-120 (1999) · Zbl 0994.11024
[11] Everest, G.; Ward, T., A dynamical interpretation of the global canonical height on an elliptic curve, Experiment. Math., 7, 305-316 (1998) · Zbl 0927.11009
[12] Everest, G.; Ward, T., Heights of Polynomials and Entropy in Algebraic Dynamics (1999), Springer-Verlag: Springer-Verlag London · Zbl 0919.11064
[13] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1979), Clarendon Press/Oxford University Press: Clarendon Press/Oxford University Press New York · Zbl 0423.10001
[14] Juzvinskiı, S. A., Calculation of the entropy of a group-endomorphism, Sibirsk. Mat. Z., 8, 230-239 (1967)
[15] Krug, E.; Newton, D., On sequence entropy of automorphisms of a Lebesgue space, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 24, 211-214 (1972) · Zbl 0237.28009
[16] Kušnirenko, A. G., Metric invariants of entropy type, Uspekhi Mat. Nauk, 22, 57-65 (1967) · Zbl 0169.46101
[17] Lind, D. A.; Ward, T., Automorphisms of solenoids and \(p\)-adic entropy, Ergodic Theory Dynam. Systems, 8, 411-419 (1988) · Zbl 0634.22005
[18] Milnor, J., On the entropy geometry of cellular automata, Complex Systems, 2, 357-385 (1988) · Zbl 0672.68025
[19] Shipsey, R., Elliptic Divisibility Sequences (2000), Univ. of London
[20] Silverman, J. H., The Arithmetic of Elliptic Curves (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0585.14026
[21] Silverman, J. H., Computing heights on elliptic curves, Math. Comp., 51, 339-358 (1988) · Zbl 0656.14016
[22] Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0911.14015
[23] Walters, P., An Introduction to Ergodic Theory (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0475.28009
[24] Ward, M., Memoir on elliptic divisibility sequences, Amer. J. Math., 70, 31-74 (1948) · Zbl 0035.03702
[25] Weil, A., Basic Number Theory. Basic Number Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 144 (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0326.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.