Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9
| dc.creator | Elkies, Noam D. | |
| dc.date | 2006-12-23 | |
| dc.date.accessioned | 2026-07-07T07:36:59Z | |
| dc.date.available | 2026-07-07T07:36:59Z | |
| dc.description | 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. | |
| dc.description | 8 pages | |
| dc.identifier | https://arxiv.org/abs/math/0612734 | |
| dc.identifier | http://arxiv.org/abs/math/0612734 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/120619 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G05, 11G18 | |
| dc.title | Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9 | |
| dc.type | text |