LOCAL DATA OF RATIONAL ELLIPTIC CURVES WITH NONTRIVIAL TORSION

By Mazur's torsion theorem, there are fourteen possibilities for the nontrivial torsion subgroup T of a rational elliptic curve. For each T, such that E may have additive reduction at a prime p, we consider a parametrized family ET of elliptic curves with the property that they parametrize all elliptic curves E/Q which contain T in their torsion subgroup. Using these parametrized families, we explicitly classify the Kodaira-Neron-type, the conductor exponent and the local Tamagawa number at each prime p where E/Q has additive reduction. As a consequence, we find all rational elliptic curves with a 2- or 3-torsion point that have global Tamagawa number 1. 1. Introduction 1 2. Preliminaries 4 3. Local data of E-T 9 4. Consequences of main results 35 5. Global Tamagawa numbers 36 Acknowledgments 41 References 41