Serre's uniformity problem in the split Cartan case

TL;DR

This paper proves that for sufficiently large primes, the rational points on the modular curve associated with split Cartan subgroups consist only of cusps and CM points, addressing a longstanding question of Serre.

Contribution

It establishes a uniform bound p_0 beyond which the Galois image of non-CM elliptic curves over Q is not contained in the normalizer of a split Cartan subgroup.

Findings

For primes p > p_0, the modular curve X_split(p)(Q) contains only cusps and CM points.

The Galois image for non-CM elliptic curves over Q is not contained in the normalizer of a split Cartan subgroup for large p.

Provides a partial resolution to Serre's uniformity problem in the split Cartan case.

Abstract

We prove that there exists an integer p_0 such that X_split(p)(Q) is made of cusps and CM-points for any prime p>p_0. Equivalently, for any non-CM elliptic curve E over Q and any prime p>p_0 the image of the Galois representation induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.