The uniform primality conjecture for elliptic curves

TL;DR

This paper proves that elliptic divisibility sequences generated by points related to rational isogenies have a bounded number of prime terms, with results depending on Lang's conjecture over rationals and unconditionally over function fields.

Contribution

It establishes a uniform bound on prime terms in elliptic divisibility sequences, advancing understanding of their primality properties and providing explicit computational methods.

Findings

Bounded number of prime terms over rationals assuming Lang's conjecture

Unconditional uniform bounds over function fields

Explicit methods for computing all irreducible terms

Abstract

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.