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.


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


Zbl 0212.29201
Full Text: DOI Link


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