Bulletin of the
Korean Mathematical Society BKMS

Let $K$ be a number field and fix a prime number $p$. 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 $B_{K, p}$ of primes of $K$ satisfying that any elliptic curve over $K$ with $B_{K, p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over $K$ with $B_{K, p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of $E$ gives its associated Galois representation $\REP$ modulo $p$. We also study the irreducibility and surjectivity of $\REP$ for semistable elliptic curves with $B_{K, p}$-reduction.

Keywords: reduction of elliptic curves, torsion points, Galois representation

MSC numbers: Primary 14H52; Secondary 14G05