Torsion points of elliptic curves with bad reduction at some primes II

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 P_E,p modulo p. We also study the irreducibility and surjectivity of ρ_E,p for semistable elliptic curves with B_K,p-reduction.