Families of isogenous elliptic curves ordered by height

TL;DR

This paper establishes bounds on the number of special fibers with isogenous elliptic curve factors within families over number fields, introducing a uniform bound dependent only on key parameters.

Contribution

It provides new upper bounds for isogenous fibers in families of elliptic curves, including a uniform bound based solely on the degree, field, and height.

Findings

Upper bounds depend on the number field, family, and height limit

A uniform bound is achieved depending only on degree, field, and height

Method uses Heath-Brown type bounds and optimization techniques

Abstract

Given a family of products of elliptic curves over a rational curve defined over a number field $K$, and assuming that there exists no isogeny between the pair of elliptic curves in the generic fiber, we establish an upper bound for the number of special fibers with height at most $B$ where the two factors are isogenous. Our proof provides an upper bound that is dependent on $K$, the family, and the bound of height $B$. Furthermore, by introducing a slight modification to the definition of the height of the parametrizing family, we prove a uniform bound depends solely on the degree of the family, the field $K$, and $B$. Based on the uniformity, and the fact that the idea of using Heath-Brown type bounds on covers and optimizing the cover to count rational points on specific algebraic families has not been exploited much yet, we hope that the paper serves as a good example to illustrate the strengths of the method and will inspire further exploration and application of these techniques in related research.