Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9

TL;DR

This paper constructs explicit examples of elliptic curves over Q where the 3-adic Galois representation is surjective mod 3 but not mod 9, revealing a deviation from known behavior for primes greater than 3.

Contribution

The authors describe a genus 0 modular curve parametrizing such elliptic curves and provide explicit rational functions and examples demonstrating this phenomenon.

Findings

Existence of elliptic curves with surjective mod 3 but not mod 9 Galois representations.

Explicit rational functions describing the modular cover of degree 27.

Identification of specific elliptic curves with given conductors exhibiting this property.

Abstract

Let E be an elliptic curve over Q, and rho_l: Gal(Q) --> GL_2(Z_l) its l-adic Galois representation. Serre observed that for l>3 there is no proper closed subgroup of SL_2(Z_l) that maps surjectively onto SL_2(Z/lZ), and concluded that if rho_l is surjective mod l then it is surjective onto GL_2(Z_l). We show that this no longer holds for l=3 by describing a modular curve X of genus 0 parametrizing elliptic curves for which rho_3 is not surjective mod 9 but generically surjective mod 3. The curve X is defined over Q, and the modular cover X --> X(1) has degree 27 so X is rational. We exhibit an explicit rational function of degree 27 that realizes this cover, and use it to exhibit several elliptic curves with nonzero j-invariant that satisfy this condition on rho_3, of which the simplest are the curves Y^2 = X^3 - 27X - 42 and Y^2 + Y = X^3 - 135X - 604 of conductors 1944 = 2^3 3^5 and 6075 = 3^5 5^2 respectively.