Images of isogeny classes on modular elliptic curves

Abstract

Let K be a number field and E/K a modular elliptic curve, with modular parametrization π : X0(N) −→ E defined over K. The purpose of this note is to study the images in E of classes of isogenous points in X0(N). Let π : X0(N)→ E be as above, and denote by K an algebraic closure of K. Theorem 1 Let S ⊂ X0(N)(K) be an infinite set of points corresponding to elliptic curves which all lie in one isogeny class, but which are not isogenous to E itself. Then the subgroup of E(K) generated by π(S) has infinite rank and finite torsion. Proof. Write S = {x0, x1, . . .} and yi := π(xi) ∈ E(K) for i ≥ 0. We first show that 〈π(S)〉 is not finitely generated, and then that it has finite torsion. Suppose that 〈π(S)〉 is finitely generated. Then 〈π(S)〉 ⊂ E(L) for some number field L, which we may extend to include K. Now GL := Gal(L/L) acts on each fiber π (yi), from which follows that |GL · xi| ≤ deg(π), ∀i ≥ 0. (1) Denote by Ei the elliptic curve corresponding to xi for each i ≥ 0. It is isogenous to E0. We now consider two cases. (i) If E0 has complex multiplication, then each End(Ei) is an order of conductor fi in a fixed quadratic imaginary field F . We denote by hF the class number of F . Then we have |GL · xi| ≥ |Pic(End(Ei))|/2[L : Q] (by [2, Chap 10, Theorem 5]) ≥ hF 12[L : Q] · fi ∏ p|fi ( 1− 1 p ) (by [2, Chap 8, Theorem 7]), which tends to ∞ as i→∞, thus contradicting (1). (ii) Now suppose that E0 does not have complex multiplication. We may write Ei = E0/Ci, with Ci ⊂ E0 a cyclic subgroup of order ni. Consider the Galois representations ρni : GL −→ Aut(E0[ni]) ∼= GL2(Z/niZ) ∗The author would like to thank the Max-Planck-Institut fur Mathematik, Bonn, where this paper was written.