Minimal models of rational elliptic curves with non-trivial torsion

摘要

In this paper, we explicitly classify the minimal discriminants of all elliptic curves E / Q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$E/{mathbb {Q}}$$end{document} with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves E / Q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$E/{mathbb {Q}}$$end{document} with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for E . We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which E has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterlé pertaining to the primes at which an elliptic curve over a number field K with a non-trivial K -torsion point can have additive reduction.