Representations attached to elliptic curves with a non‐trivial odd torsion point

Galois Representations Attached to Elliptic Curves

Construction of high-rank elliptic curves with a nontrivial torsion point

We construct a family of infinitely many elliptic curves over Q with a nontrivial rational 2-torsion point and with rank ≥ 6, which is parametrized by the rational points of an elliptic curve of rank ≥ 1.

To appear in Acta Arithmetica. AVERAGE FROBENIUS DISTRIBUTIONS FOR ELLIPTIC CURVES WITH NONTRIVIAL RATIONAL TORSION

In this paper we consider the Lang-Trotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the Lang-Trotter conjecture holds in an average sense for these families of curves (see Theorem 3). Let E/Q denote an elliptic curve and let ∆E denote its discriminant. As usual, let ap(E) = p + 1 − #E(Fp). Then we have the following...

Elliptic Curves of High Rank with Nontrivial Torsion Group over Q

In order to find elliptic curves over Q with large rank, J.-F. Mestre [1991] constructed an infinite family of elliptic curves with rank at least 12. Then K. Nagao [1994] and S. Kihara [1997b] found infinite subfamilies of rank 13 and 14, respectively. By specialization, elliptic curves of rank at least 21 [Nagao and Kouya 1994], 22 [Fermigier 1997], and 23 [Martin and McMillen 1997] have also ...

The image of Galois representations attached to elliptic curves with an isogeny

giving the action of GQ on Tp(E), the p-adic Tate module for E and for a prime p. If E doesn’t have complex multiplication, then a famous theorem of Serre [Ser2] asserts that the image of ρE,p has finite index in AutZp ( Tp(E) ) for all p and that the index is 1 for all but finitely many p. This paper concerns some of the exceptional cases where the index is not 1. If E has a cyclic isogeny of ...