The optimality of the Boundedness Height Conjecture

TL;DR

This paper demonstrates that the Boundedness Height Conjecture is optimal by showing varieties not satisfying the hypothesis also do not satisfy the conclusion, confirming the conjecture's precise scope.

Contribution

It proves the optimality of the Boundedness Height Conjecture for varieties in powers of elliptic curves, including examples and remarks.

Findings

The conjecture is optimal; varieties not meeting the hypothesis also fail the conclusion.

Provides explicit examples illustrating the boundaries of the conjecture.

Confirms the conjecture's validity within its specified conditions.

Abstract

We show that the Boundedness Height Conjecture is optimal; all varieties in a power of an elliptic curve which do not satisfy the hypothesis neither satisfy the thesis. The Bounded Height Conjecture is known to hold for varieties in a power of an elliptic curve. We also present some examples and remarks.