On the average value of the canonical height in higher dimensional families of elliptic curves
Volume 166 / 2014
Acta Arithmetica 166 (2014), 101-128
MSC: Primary 11G05; Secondary 11G50, 14G40.
DOI: 10.4064/aa166-2-1
Abstract
Given an elliptic curve $E$ over a function field $K=\mathbb {Q}(T_1, \ldots , T_n)$, we study the behavior of the canonical height $\hat{h}_{E_\omega }$ of the specialized elliptic curve $E_\omega $ with respect to the height of $\omega \in \mathbb {Q}^n$. We prove that there exists a uniform nonzero lower bound for the average of the quotient ${\hat{h}_{E_\omega }(P_\omega )}/{h(\omega )}$ over all nontorsion $P \in E(K)$.
