EXCEPTIONAL ELLIPTIC CURVES OVER QUARTIC FIELDS
2012 ◽
Vol 08
(05)
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pp. 1231-1246
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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.
2015 ◽
Vol 18
(1)
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pp. 578-602
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2020 ◽
Vol 16
(06)
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pp. 1307-1323
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2014 ◽
Vol 57
(2)
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pp. 465-473
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2014 ◽
Vol 22
(2)
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pp. 79-90
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1996 ◽
Vol 54
(2)
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pp. 267-274
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2011 ◽
Vol 150
(3)
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pp. 439-458
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2015 ◽
Vol 11
(06)
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pp. 1725-1734
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