Infinite families of elliptic curves over Dihedral quartic number fields

Daeyeol Jeon; Chang Heon Kim; Yoonjin Lee

doi:10.1016/j.jnt.2012.06.014

Infinite families of elliptic curves over Dihedral quartic number fields

Daeyeol Jeon, Chang Heon Kim, Yoonjin Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N = 20, 24. We prove that for each elliptic curve E_t in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D₄ of order 8 for the Galois closure L of K over Q, where K is the defining field of (E_t, Q_t) and Q_t is a point of E_t of order N. We also notice that the plane model for the modular curve X₁(24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 [12]).

Original language	English
Pages (from-to)	115-122
Number of pages	8
Journal	Journal of Number Theory
Volume	133
Issue number	1
DOIs	https://doi.org/10.1016/j.jnt.2012.06.014
State	Published - Jan 2013

Bibliographical note

Funding Information:
✩ The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026917), the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0004284), and the third author was supported by Priority Research Centers Program through the NRF funded by the Ministry of Education, Science and Technology (2010-0028298) and by the NRF grant funded by the Korea government (MEST) (2011-0015684). * Corresponding author. E-mail addresses: [email protected] (D. Jeon), [email protected] (C.H. Kim), [email protected] (Y. Lee).

Keywords

Dihedral quartic number field
Elliptic curve
Modular curve
Torsion

Access to Document

10.1016/j.jnt.2012.06.014

Cite this

@article{fafbdfc24c6949ee8f97facd48a3c0c3,

title = "Infinite families of elliptic curves over Dihedral quartic number fields",

abstract = "We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N = 20, 24. We prove that for each elliptic curve Et in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D4 of order 8 for the Galois closure L of K over Q, where K is the defining field of (Et, Qt) and Qt is a point of Et of order N. We also notice that the plane model for the modular curve X1(24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 [12]).",

keywords = "Dihedral quartic number field, Elliptic curve, Modular curve, Torsion",

author = "Daeyeol Jeon and Kim, {Chang Heon} and Yoonjin Lee",

note = "Funding Information: ✩ The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026917), the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0004284), and the third author was supported by Priority Research Centers Program through the NRF funded by the Ministry of Education, Science and Technology (2010-0028298) and by the NRF grant funded by the Korea government (MEST) (2011-0015684). * Corresponding author. E-mail addresses: [email protected] (D. Jeon), [email protected] (C.H. Kim), [email protected] (Y. Lee).",

year = "2013",

month = jan,

doi = "10.1016/j.jnt.2012.06.014",

language = "English",

volume = "133",

pages = "115--122",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Infinite families of elliptic curves over Dihedral quartic number fields

AU - Jeon, Daeyeol

AU - Kim, Chang Heon

AU - Lee, Yoonjin

N1 - Funding Information: ✩ The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026917), the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0004284), and the third author was supported by Priority Research Centers Program through the NRF funded by the Ministry of Education, Science and Technology (2010-0028298) and by the NRF grant funded by the Korea government (MEST) (2011-0015684). * Corresponding author. E-mail addresses: [email protected] (D. Jeon), [email protected] (C.H. Kim), [email protected] (Y. Lee).

PY - 2013/1

Y1 - 2013/1

N2 - We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N = 20, 24. We prove that for each elliptic curve Et in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D4 of order 8 for the Galois closure L of K over Q, where K is the defining field of (Et, Qt) and Qt is a point of Et of order N. We also notice that the plane model for the modular curve X1(24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 [12]).

AB - We find infinite families of elliptic curves over quartic number fields with torsion group Z/NZ with N = 20, 24. We prove that for each elliptic curve Et in the constructed families, the Galois group Gal(L/Q) is isomorphic to the Dihedral group D4 of order 8 for the Galois closure L of K over Q, where K is the defining field of (Et, Qt) and Qt is a point of Et of order N. We also notice that the plane model for the modular curve X1(24) found in Jeon et al. (2011) [1] is in the optimal form, which was the missing case in Sutherland's work (Sutherland, 2012 [12]).

KW - Dihedral quartic number field

KW - Elliptic curve

KW - Modular curve

KW - Torsion

UR - http://www.scopus.com/inward/record.url?scp=84867312676&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2012.06.014

DO - 10.1016/j.jnt.2012.06.014

M3 - Article

AN - SCOPUS:84867312676

SN - 0022-314X

VL - 133

SP - 115

EP - 122

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 1

ER -

Infinite families of elliptic curves over Dihedral quartic number fields

Abstract

Bibliographical note

Keywords

Access to Document

Fingerprint

Cite this