Torsion points and reduction of elliptic curves

Masaya Yasuda¹

¹ Institute of Mathematics for Industry Kyushu University 744 Motooka Nishi-ku Fukuoka 819-0395, Japan

Acta Arithmetica, Tome 176 (2016) no. 1, pp. 89-100.

Résumé

Let $E$ be an elliptic curve over a number field $K$. Given a prime $p$, the $K$-rational $p$-torsion points of $E$ are the points of exact order $p$ in the Mordell–Weil group $E(K)$. In this paper, we study relations between torsion points and reduction of elliptic curves. Specifically, we give a condition on the pair $(K, p)$ under which there do not exist $K$-rational $p$-torsion points of any elliptic curve over $K$ with bad reduction only at certain primes. Let $\zeta_p$ denote a primitive $p$th root of unity. Our result shows that any elliptic curve over $\mathbb Q(\zeta_p)$ with everywhere good reduction has no $\mathbb Q(\zeta_p)$-rational $p$-torsion points for the regular primes $p \geq 11$ with $p \equiv 1 \bmod 4$.

Détail

DOI : 10.4064/aa8425-6-2016

Keywords: elliptic curve number field given prime k rational p torsion points points exact order mordell weil group paper study relations between torsion points reduction elliptic curves specifically condition pair under which there exist k rational p torsion points elliptic curve bad reduction only certain primes zeta denote primitive pth root unity result shows elliptic curve mathbb zeta everywhere reduction has mathbb zeta rational p torsion points regular primes geq equiv bmod