## 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

### Keywords:

elliptic curves; canonical height; entropy

Zbl 0212.29201
Full Text:

### References:

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