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Abstract

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 $p \equiv 1 \bmod 4$.