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- Title
EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS.
- Authors
NAJMAN, FILIP
- Abstract
We study the number of elliptic curves, up to isomorphism, over a fixed quartic field K having a prescribed torsion group T as a subgroup. Let T = ℤ/mℤ⊕ℤ/nℤ, where m|n, be a torsion group such that the modular curve X1(m, n) is an elliptic curve. Let K be a number field such that there is a positive and finite number of elliptic curves E over K having T as a subgroup. We call such pairs (T, K)exceptional. It is known that there are only finitely many exceptional pairs when K varies through all quadratic or cubic fields. We prove that when K varies through all quartic fields, there exist infinitely many exceptional pairs when T = ℤ/14ℤ or ℤ/15ℤ and finitely many otherwise.
- Subjects
ELLIPTIC curves; QUARTIC fields; ALGEBRAIC fields; ISOMORPHISM (Mathematics); TORSION theory (Algebra); MODULES (Algebra)
- Publication
International Journal of Number Theory, 2012, Vol 8, Issue 5, p1231
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042112500716