TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES II

Abstract

Let K be a number field and fix a prime number <TEX>$p$</TEX>. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set <TEX>$B_{K,p}$</TEX> of primes of K satisfying that any elliptic curve over K with <TEX>$B_{K,p}$</TEX>-reduction has no <TEX>$p$</TEX>-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with <TEX>$B_{K,p}$</TEX>-reduction and a <TEX>$p$</TEX>-torsion point. The action of the absolute Galois group on the <TEX>$p$</TEX>-torsion subgroup of E gives its associated Galois representation <TEX>$\bar{\rho}_{E,p}$</TEX> modulo <TEX>$p$</TEX>. We also study the irreducibility and surjectivity of <TEX>$\bar{\rho}_{E,p}$</TEX> for semistable elliptic curves with <TEX>$B_{K,p}$</TEX>-reduction.