| Let N be a positive integer and let F be any field of characteristic not dividing N. Then for any elliptic curve E defined over F, the absolute Galois group GF of F acts on E[N] (the N-torsion points of E). This induces a representation rho E of GF into GL(2, Z/NZ). The map rho E has cyclotomic determinant. Taking an arbitrary rho: GF → GL (2, Z/NZ) with cyclotomic determinant, we consider the problem of determining when rhoE ≅ rho for some E. This leads to the modular curve XN. For each rho, there is a twist of X N whose non-cusp rational points give solutions to the problem. When N = 6, this curve has genus 1 and we may study it in detail. Then, we take an arbitrary modular curve X of genus 1, and consider a generalization of the above problem for which X is the required moduli space. A general theory is developed, leading to a generalization of the concept of discriminant of an elliptic curve. Finally, we supply a number of examples of the theory by classifying all subgroups of SL(2, Z) of Larcher type having genus 1. |
