On the rational points of \(y^ 2 = x^ 3 - k\) (Q1210247)

The author considers elliptic curves with \(\mathbb{Z}\left[{-1+\sqrt 3\over 2}\right]\) as its ring of complex multiplications and proves some interesting theorems in support of the Birch and Swinnerton-Dyer conjectures regarding elliptic curves with complex multiplication. More specifically he proves the following three theorems: (i) For primes \(p\equiv 2\), \(5\pmod 9\), \(L_ p(1)\) and \(L_{p^ 2}(1)\) are both nonzero, where \(L_ D(s)\) denotes the Hecke \(L\)-series of the curve \(\Gamma_ D:x^ 3+y^ 3=D\) (which, by a simple change of variables takes the more familiar form \(y^ 2=x^ 3-432 D^ 2)\). (ii) For primes \(p\equiv 2\pmod 9\), \(\text{rank}_ \mathbb{Q}(\Gamma_{2p})=1\) and for primes \(p\equiv 5\pmod 9\), \(\text{rank}_ \mathbb{Q}(\Gamma_{2p^ 2})=1\). (iii) For primes \(p\equiv 7\pmod{24}\), the curve \(y^ 2=x^ 3-p^ 3\) has rank one over \(\mathbb{Q}\). Use of modular functions and class field theory is made and the paper provides excellent examples to the general theory.