Elliptic curves with rational 2-torsion and related ternary Diophantine equations

Our main result is a classification of elliptic curves with rational 2-torsion and good reduction outside 2, 3 and a prime p. This extends the work of Hadano and, more recently, Ivorra. A key factor in doing this is to have a method for efficiently computing the conductor of an elliptic curve with 2-torsion. We specialize the work of Papadopolous to provide such a method.

Next, we determine all the rational points on the hyper-elliptic curves y² = x⁵ ± 2 ^a3^b. This information is required in providing the classification mentioned above. We show how the commercial mathematical software package MAGMA can be used in solving this problem.

As an application, we turn our attention to the ternary Diophantine equations xⁿ + yⁿ = 2 ^apz² and x³ + y³ = ±p^mz ⁿ, where p denotes a fixed prime. In the first equation, we show that for p = 5 or p > 7 the equation is unsolvable in integers (x, y, z) for all suitably large primes n. In the second equation, we show the same conclusion holds for an infinite collection of primes p. To do this, we use the connections between Galois representations, modular forms, and elliptic curves which were discovered by Frey, Hellegouarch, Serre, and Wiles.