On the Average Value of the Canonical Height in Higher Dimensional Families of Elliptic curves

TL;DR

This paper investigates the average canonical height of specialized elliptic curves over function fields, establishing a uniform non-zero lower bound for the ratio of heights across families in multiple dimensions.

Contribution

It proves the existence of a uniform non-zero lower bound for the average height ratio in higher-dimensional families of elliptic curves over function fields.

Findings

Existence of a uniform non-zero lower bound for height ratios.

Results extend to higher-dimensional parameter spaces.

Provides insights into the distribution of heights in elliptic curve families.

Abstract

Given an elliptic curve E over a function field K=Q(T_1,...,T_n), we study the behavior of the canonical height ^h_(E_w) of the specialized elliptic curve E_w with respect to the height of w in Q^n. In this paper, we prove that there exists a uniform non-zero lower bound for the average of the quotient ^h_(E_w)(P_w)/h(w) for all non-torsion P in E(K).