Serre’s Uniformity Conjecture over the rational numbers

Establish that there exists a single constant C such that, for every non-CM elliptic curve E defined over the rational numbers and every prime p greater than C, the mod-p Galois representation of E is surjective.

Background

Serre’s Open Image Theorem gives a surjectivity threshold depending on the individual non-CM elliptic curve. Serre’s Uniformity Conjecture asks whether this threshold can instead be chosen uniformly for all non-CM elliptic curves over the rational numbers.

The paper uses partial classification results for mod-p images to obtain finiteness results for rational isolated j-invariants, but explicitly records that the uniformity conjecture itself remains unresolved.