Serre's uniformity problem in the split Cartan case

By Yuri F. Bilu, Pierre Parent. 2008

89 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e8915e702df8671c9cb7910da0fd2e4b7fbcdc8

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In Geometry and Topology

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Citations in the corpus

Quadratic Chabauty and rational points, I: p-adic heights (2016, methodology): “In recent work with Müller, Tuitman, and Vonk [9], we use the methods described in these papers to determine the rational points on X s (13), the split Cartan modular curve of level 13, the last remaining case of Serre’s uniformity problem for normalisers of split Cartan subgroups, after the work of”
On the field of definition of $$p$$-torsion points on elliptic curves over the rationals (2013, methodology): “It has been solved by Mazur in the Borel case [33], by Serre in the exceptional case [46] and by Bilu and Parent in the split Cartan case [2].”
Rational points on X_0^+ (p^r) (2011, methodology): “e motivated a number of works, dating back at least to Mazur’s foundational paper [19], where the case of X split(p) was tackled. Momose, among others, obtained important results in [23] and [24]. In [4, 5] we proved that for some absolute constant p 0, the only rational points of X+ 0 (pr) with p&g”
Explicit Chabauty—Kim for the split Cartan modular curve of level 13 (2017, unknown): “ Borel and the exceptional cases were handled by Mazur [Maz78] and Serre [Ser72], respectively, and the case of normalizers of split Cartan subgroups (for ℓ>13) follows from results of Bilu-Parent [BP11] and Bilu-Parent-Rebolledo [BPR13], which we now recall. For a prime ℓ, we write Xs(ℓ) for the”