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Minimal models of rational elliptic curves with non-trivial torsion.
- Source :
-
Research in Number Theory . 11/27/2021, Vol. 8 Issue 1, p1-39. 39p. - Publication Year :
- 2021
-
Abstract
- In this paper, we explicitly classify the minimal discriminants of all elliptic curves E / Q 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 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *ELLIPTIC curves
*TORSION theory (Algebra)
Subjects
Details
- Language :
- English
- ISSN :
- 25220160
- Volume :
- 8
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Research in Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 153819295
- Full Text :
- https://doi.org/10.1007/s40993-021-00296-4