A uniform bound on the smallest surjective prime of an elliptic curve

Abstract Let $$E/\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -adic Galois representation $$\rho _{E,\ell ^\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:mrow> </mml:msub> </mml:math> is surjective for all but finitely many prime numbers $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> . 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 $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> such that $$\rho _{E,\ell ^\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:mrow> </mml:msub> </mml:math> is surjective is at most 7. Moreover, we completely classify all elliptic curves $$E/\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> for which the smallest surjective prime is exactly 7.