×

A family of monogenic \(S_4\) quartic fields arising from elliptic curves. (English) Zbl 1410.11050

The authors give a family of monogenic \(S_4\) quartic fields generated by the \(x\)-coordinate of a \(3\)-torsion point of elliptic curves given by Fueter form. They show that if an elliptic curve \(E\) defined over \(\mathbb Q\) has a twist \(E'\) with a \(\mathbb Q\)-rational \(4\)-torsion point and the twist \(E'\) has reduction types \(I_1^*\) and \(I_1\) only, then the field \(K_3\) generated by the \(x\)-coordinate of a \(3\)-torsion point of \(E\) is a \(S_4\) field and is monogenic with a generator given by a root of the polynomial \(T^4-6T^2-\alpha T-3\) \((\alpha\in\mathbb Z)\). The above condition on reduction types is equivalent to that \(\alpha\pm 8\) are square free. They use the Montes Algorithm to prove that \(K_3\) is monogenic.

MSC:

11G05 Elliptic curves over global fields
11R04 Algebraic numbers; rings of algebraic integers
11R16 Cubic and quartic extensions

Software:

SageMath; PARI/GP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adelmann, Clemens, The Decomposition of Primes in Torsion Point Fields, Lecture Notes in Mathematics, vol. 1761, (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 1019.11031
[2] Alden Gassert, T., Discriminants of Chebyshev radical extensions, J. Théor. Nombres Bordeaux, 26, 3, 607-634, (2014) · Zbl 1360.11117
[3] Alden Gassert, T., A note on the monogeneity of power maps, Albanian J. Math., 11, 1, 3-12, (2017) · Zbl 1392.11082
[4] Bandini, Andrea; Paladino, Laura, Number fields generated by the 3-torsion points of an elliptic curve, Monatsh. Math., 168, 2, 157-181, (2012) · Zbl 1327.11038
[5] Bandini, Andrea; Paladino, Laura, Fields generated by torsion points of elliptic curves, J. Number Theory, 169, 103-133, (2016) · Zbl 1409.11045
[6] Bérczes, Attila; Evertse, Jan-Hendrik; Győry, Kálmán, Multiply monogenic orders, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12, 2, 467-497, (2013) · Zbl 1319.11070
[7] M. Bhargava, A. Shankar, X. Wang, Squarefree values of polynomial discriminants I, ArXiv e-prints, November 2016.; M. Bhargava, A. Shankar, X. Wang, Squarefree values of polynomial discriminants I, ArXiv e-prints, November 2016.
[8] Cali, Élie; Kraus, Alain, Sur la p-différente du corps des points de l-torsion des courbes elliptiques, \(l \ne p\), Acta Arith., 104, 1, 1-21, (2002) · Zbl 1052.11040
[9] Cassou-Noguès, Ph.; Taylor, M. J., Elliptic Functions and Rings of Integers, Progr. Math., vol. 66, (1987), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0608.12013
[10] Cullinan, John; Hajir, Farshid, Ramification in iterated towers for rational functions, Manuscripta Math., 137, 3-4, 273-286, (2012) · Zbl 1235.14023
[11] Fleckinger, V.; Vérant, M., Families of non-Galois quartic fields, J. Number Theory, 54, 2, 261-273, (1995) · Zbl 0852.11063
[12] Fueter, R.; Gut, M., Vorlesungen über die Singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen, B.G. Teubners Sammlung von Lehrbüchern, etc., vol. 41, (1924), Von Dr. R. Fueter (unter Mitwirkung Von Dr. Max Gut) · JFM 50.0101.01
[13] Funakura, Takeo, On integral bases of pure quartic fields, Math. J. Okayama Univ., 26, 27-41, (1984) · Zbl 0563.12003
[14] Gaál, István, Diophantine Equations and Power Integral BasesNew Computational Methods, (2002), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1016.11059
[15] Gras, Marie-Nicole, Z-bases d’entiers 1, θ, \(\theta^2\), \(\theta^3\) dans les extensions cycliques de degré 4 de Q, (Number Theory, 1979-1980 and 1980-1981. Number Theory, 1979-1980 and 1980-1981, Publ. Math. Fac. Sci. Besançon, vol. 6, (1981), Univ. Franche-Comté, Besançon), 14
[16] Gras, Marie-Nicole, Condition nécessaire de monogénéité de l’anneau des entiers d’une extension abélienne de Q, (Séminaire de théorie des nombres, Paris 1984-85. Séminaire de théorie des nombres, Paris 1984-85, Progr. Math., vol. 63, (1986), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 97-107
[17] Gras, Marie-Nicole; Tanoé, François, Corps biquadratiques monogènes, Manuscripta Math., 86, 1, 63-79, (1995) · Zbl 0816.11058
[18] Guàrdia, Jordi; Montes, Jesús; Nart, Enric, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc., 364, 1, 361-416, (2012) · Zbl 1252.11091
[19] Györy, K., Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith., 23, 419-426, (1973) · Zbl 0269.12001
[20] Huard, James G.; Spearman, Blair K.; Williams, Kenneth S., Integral bases for quartic fields with quadratic subfields, J. Number Theory, 51, 1, 87-102, (1995) · Zbl 0826.11048
[21] Kable, Anthony C., Power bases in dihedral quartic fields, J. Number Theory, 76, 1, 120-129, (1999) · Zbl 0934.11051
[22] Kida, M., Ramification in the division fields of an elliptic curve, Abh. Math. Semin. Univ. Hambg., 73, 195-207, (2003) · Zbl 1130.11323
[23] Kraus, Alain, Sur la p-différente du corps des points de p-torsion des courbes elliptiques, Bull. Aust. Math. Soc., 60, 3, 407-428, (1999) · Zbl 0938.11028
[24] Lozano-Robledo, Álvaro, Division fields of elliptic curves with minimal ramification, Rev. Mat. Iberoam., 31, 4, 1311-1332, (2015) · Zbl 1331.11042
[25] Lozano-Robledo, Álvaro, Ramification in the division fields of elliptic curves with potential supersingular reduction, Res. Number Theory, 2, 8, 25, (2016) · Zbl 1405.11076
[26] Motoda, Yasuo, Notes on quartic fields, Rep. Fac. Sci. Eng. Saga Univ., Math., 32, 1, 1-19, (2002) · Zbl 1186.11067
[27] Motoda, Yasuo; Nakahara, Toru; Inayat Ali Shah, Syed; Uehara, Tsuyoshi, On a problem of Hasse, (Algebraic Number Theory and Related Topics 2007. Algebraic Number Theory and Related Topics 2007, RIMS Kôkyûroku Bessatsu, vol. B12, (2009), Res. Inst. Math. Sci. (RIMS): Res. Inst. Math. Sci. (RIMS) Kyoto), 209-221 · Zbl 1251.11073
[28] Narkiewicz, Władysław, Elementary and Analytic Theory of Algebraic Numbers, Springer Monographs in Mathematics, (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1159.11039
[29] Olajos, Péter, Power integral bases in the family of simplest quartic fields, Exp. Math., 14, 2, 129-132, (2005) · Zbl 1092.11042
[30] Silverman, Joseph H., Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 151, (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0911.14015
[31] Smith, Hanson, Two families of monogenic \(S_4\) quartic number fields, Acta Arith., (February 2018), in press, ArXiv e-prints
[32] Spearman, Blair K., Monogenic \(A_4\) quartic fields, Int. Math. Forum, 1, 37-40, 1969-1974, (2006) · Zbl 1190.11057
[33] Stange, Katherine E., Integral points on elliptic curves and explicit valuations of division polynomials, Canad. J. Math., 68, 5, 1120-1158, (2016) · Zbl 1406.11055
[34] The Sage Developers, SageMath, the sage mathematics software system (Version 7.3), (2016)
[35] Univ. Bordeaux. PARI/GP version 2.9.0, 2016, Available from
[36] Verdure, Hugues, A quadratic reciprocity law for elliptic curves, Acta Sci. Math. (Szeged), 75, 3-4, 457-465, (2009) · Zbl 1212.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.