Papers › A uniform bound on the smallest surjective prime of an elliptic curve
A uniform bound on the smallest surjective prime of an elliptic curve
Tyler Genao, Jacob Mayle, Jeremy Rouse
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Let E/ℚ be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the ℓ-adic Galois representation ρ_(E,ℓ^∞) is surjective for all but finitely many prime numbers ℓ. Considerable work has gone into bounding the largest possible nonsurjective prime; a uniform bound of $37$ has been proposed but is yet unproven. We consider an opposing direction, proving that the smallest prime ℓ such that ρ_(E,ℓ^∞) is surjective is at most $7$. Moreover, we completely classify all elliptic curves E/ℚ for which the smallest surjective prime is exactly $7$.
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