Elliptic Curves with 3-adic Galois Representation Surjective Mod 3 but Not Mod 9

Let E be an elliptic curve over Q, and ρl : Gal(Q/Q)→GL2(Zl) its l-adic Galois representation. Serre observed that for l ≥ 5 there is no proper closed subgroup of SL2(Zl) that maps surjectively onto SL2(Z/lZ), and concluded that if ρl is surjective mod l then it is surjective onto GL2(Zl). We show that this no longer holds for l = 3 by describing a modular curve X9 of genus 0 parametrizing elliptic curves for which ρ3 is not surjective mod 9 but generically surjective mod 3. The curve X9 is defined over Q, and the modular cover X9→X(1) has degree 27, so X9 is rational because 27 is odd. We exhibit an explicit rational function f ∈ Q(x) of degree 27 that realizes this cover. We show that for every x ∈ P(Q), other than the two rational solutions of f(x) = 0, the elliptic curves with j-invariant f(x) have ρ3 surjective mod 3 but not mod 9. We determine all nonzero integral values of f(x), and exhibit several elliptic curves satisfying our condition on ρ3, of which the simplest are the curves Y 2 = X − 27X − 42 and Y 2 + Y = X − 135X − 604 of conductors 1944 = 23 and 6075 = 35 respectively. 0. Introduction. Let E be an elliptic curve over Q, and ρl : Gal(Q/Q)→GL2(Zl) its l-adic Galois representation. Serre observed in 1968 [7, IV, 3.4, Lemma 3] that for l ≥ 5 there is no proper closed subgroup of SL2(Zl) that maps surjectively onto SL2(Z/lZ), and concluded that if ρl is surjective mod l then it is surjective onto GL2(Zl). He noted [7, IV, 3.4, Exercise 3] that for l = 3 there exists a subgroup G ⊂ SL2(Z/9Z) such that the restriction to G of the reduction-mod-3 map SL2(Z/9Z)→SL2(Z/3Z) is an isomorphism. The preimage of G in SL2(Z3) is then a proper closed subgroup that maps surjectively to SL2(Z/3Z). This suggests that there could be curves E for which G is the image of ρ3 mod 9, making ρ3 surjective mod 3 but not mod 9. Serre does not raise this question explicitly, and it does not seem to have been addressed elsewhere in the literature; I thank Grigor Grigorov for drawing my attention to it. In this paper we answer the question by showing that there exist infinitely many j ∈ Q for which an elliptic curve of j-invariant j must have ρ3 surjective mod 3 but not mod 9. The simplest examples are j = 4374, j = 419904, and j = −44789760. In general j is the value of a rational function f(x) of degree 27 at all but finitely many x ∈ P(Q). Such curves E are parametrized by a modular curve X9 = X(9)/G. The natural cover X9→X(1) has degree 27, and our rational function f arises as the pullback to X9 of the degree-1 function j onX(1). It is easy to check from the Riemann-Hurwitz formula that X9 has genus zero. The challenge is to prove that X9 is defined over Q and to compute f(x) for some choice of rational coordinate x on X9. (Once X9 is known to be defined over Q, it is automatically isomorphic with P over Q, becoause it supports the rational function j of odd degree.) We prove the rationality in section 1, and compute x using products of Siegel functions in section 2. Such products are modular 1 Supported in part by NSF grants DMS-0200687 and DMS-0501029.