Papers › Serre's uniformity question and proper subgroups of Cₙₛ^+(p)
Serre's uniformity question and proper subgroups of Cₙₛ^+(p)
Lorenzo Furio, Davide Lombardo
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Serre's uniformity question asks whether there exists a bound N>0 such that, for every non-CM elliptic curve E over ℚ and every prime p>N, the residual Galois representation ρ_(E,p):Gal(ℚ/ℚ) →Aut(E[p]) is surjective. The work of many authors has shown that, for p>37, this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup Cₙₛ^+(p). Zywina has further proved that, whenever ρ_(E,p) is not surjective for p>37, its image is either Cₙₛ^+(p) or a certain subgroup G(p) of Cₙₛ^+(p) of index $3$. Recently, Le Fourn and Lemos showed that the index-$3$ case cannot arise for p>1.4 ·10⁷. We strengthen this result by proving that the image of ρ_(E, p) is not conjugate to G(p) for any prime larger than $5$.
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