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Serre’s uniformity problem in the split Cartan case. (English) Zbl 1278.11065
Summary: We prove that there exists an integer \(p_{0}\) such that \(X_{\mathrm{split}} (p)(\mathbb{Q} )\) is made of cusps and CM-points for any prime \({p>p_0}\). Equivalently, for any non-CM elliptic curve \(E\) over \(\mathbb{Q}\) and any prime \({p>p_0}\) the image of \(\mathrm{Gal} (\overline{\mathbb{Q}} /\mathbb{Q} )\) by the 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.

MSC:
11G15 Complex multiplication and moduli of abelian varieties
11G05 Elliptic curves over global fields
11F80 Galois representations
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