On the Selmer Group of Twists of Elliptic Curves with Q-Rational Torsion Points

Abstract

(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one haswith k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.