Generators and integral points on elliptic curves associated with simplest quartic fields
-
Sylvain Duquesne
,
Tadahisa Nara
Abstract
We associate to some simplest quartic fields a family of elliptic curves that has rank at least three over ℚ(m). It is given by the equation
Employing canonical heights we show the rank is in fact at least three for all m. Moreover, we get a parametrized infinite family of rank at least four. Further, the integral points on the curve Em are discussed and we determine all the integral points on the original quartic model when the rank is three. Previous work in this setting studied the elliptic curves associated with simplest quartic fields of ranks at most two along with their integral points (see [2, 3]).
Communicated by Milan Paštéka
References
[1] Bosma, W.—Cannon, J.—Playoust, C.: The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265.10.1006/jsco.1996.0125Suche in Google Scholar
[2] Duquesne, D.: Integral points on elliptic curves defined by simplest cubic fields, Exp. Math. 10(1) (2001), 91–102.10.1080/10586458.2001.10504431Suche in Google Scholar
[3] Duquesne, D.: Elliptic curves associated with simplest quartic fields, J. Théor. Nombres Bordeaux 19(1) (2007), 81–100.10.5802/jtnb.575Suche in Google Scholar
[4] Fujita, Y.: Generators for the elliptic curvey2 = x3 – nx of rank at least three, J. Number Theory 133(5) (2013), 1645–1662.10.1016/j.jnt.2012.10.011Suche in Google Scholar
[5] Fujita, Y.: Generators for congruent number curves of ranks at least two and three, J. Ramanujan Math. Soc. 29(3) (2014), 307–319.Suche in Google Scholar
[6] Fujita, Y.—Nara, T.: Generators and integral points on twists of the Fermat cubic, Acta Arith. 168(1) (2015), 1–6.10.4064/aa168-1-1Suche in Google Scholar
[7] Fujita, Y, —Terai, N.: Generators for the elliptic curvey2 = x3 – nx, J. Théor. Nombres Bordeaux 23(2) (2011), 403–416.10.5802/jtnb.769Suche in Google Scholar
[8] Fujita, Y, —Terai, N.: Generators and integral points on the elliptic curvey2 = x3 – nx, Acta Arith. 160(4) (2013), 333–348.10.4064/aa160-4-3Suche in Google Scholar
[9] Gras, M. N.: Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de ℚ, Publ. Math. Besançon Algébre Théorie Nr. 2 (1977/1978).10.5802/pmb.a-17Suche in Google Scholar
[10] Gusić, I.—Tadić, P.: Injectivity of specialization homomorphism of elliptic curves, J. Number Theory 148 (2015), 137–152.10.1016/j.jnt.2014.09.023Suche in Google Scholar
[11] Kim, H. K.: Evaluation of zeta functions ats = –1 of the simplest quartic fields, Proceedings of the 2003 Nagoya Conference “Yokoi-Chowla Conjecture and Related Problems”, Saga Univ., Saga, 2004, pp. 63–73.Suche in Google Scholar
[12] Lazarus, A. J.: Class Numbers of Simplest Quartic Fields, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 313–323.10.1515/9783110848632-027Suche in Google Scholar
[13] Lazarus, A. J.: On the class number and unit index of simplest quartic fields, Nagoya Math. J. 121 (1991), 1–13.10.1017/S0027763000003378Suche in Google Scholar
[14] Louboutin, S.: The simplest quartic fields with ideal class groups of exponents less than or equal to 2, J. Math. Soc. Japan 56(3) (2004), 717–727.10.2969/jmsj/1191334082Suche in Google Scholar
[15] Olajos, P.: Power integral bases in the family of simplest quartic fields, Exp. Math. 14(2) (2005) 129–132.10.1080/10586458.2005.10128916Suche in Google Scholar
[16] Saad, Y.: Numerial Methods for Large Eigenvalue Problems, 2nd ed., SIAM, Philadelphia, 2011.10.1137/1.9781611970739Suche in Google Scholar
[17] Sage software, available at http://sagemath.orgSuche in Google Scholar
[18] Siksek, S.: Infinite descent on elliptic curves, Rocky Mountain J. Math. 25(4) (1995), 1501–1538.10.1216/rmjm/1181072159Suche in Google Scholar
[19] Silverman, J.: Computing heights on elliptic curves, Math. Comp. 51(183) (1988), 339–358.10.1090/S0025-5718-1988-0942161-4Suche in Google Scholar
[20] Silverman, J.: The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 2009.10.1007/978-0-387-09494-6Suche in Google Scholar
[21] Washington, L. C.: Elliptic Curves: Number Theory and Cryptography, 2nd ed., CRC Press, Taylor & Francis Group, Boca Raton, FL, 2008.10.1201/9781420071474Suche in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Relatively residuated lattices and posets
- Strongly s-dense injective hull and Banaschewski’s theorems for acts
- Wild sets in global function fields
- Generators and integral points on elliptic curves associated with simplest quartic fields
- New Filbert and Lilbert matrices with asymmetric entries
- Returning functions with closed graph are continuous
- On sets of points of approximate continuity and ϱ-upper continuity
- Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
- On solvability of some nonlocal boundary value problems for biharmonic equation
- The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
- Strongly increasing solutions of higher-order quasilinear ordinary differential equations
- Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
- Filippov solutions of vector Dirichlet problems
- Sequences of positive homoclinic solutions to difference equations with variable exponent
- Modified Lupaş-Jain operators
- New fixed point results in bv(s)-metric spaces
- Improved Young and Heinz operator inequalities for unitarily invariant norms
- Scrutiny of some fixed point results by S-operators without triangular inequality
- The lattices of families of regular sets in topological spaces
- Iterated partial summations applied to finite-support discrete distributions
- Hamiltonicity of a class of toroidal graphs
Artikel in diesem Heft
- Regular papers
- Relatively residuated lattices and posets
- Strongly s-dense injective hull and Banaschewski’s theorems for acts
- Wild sets in global function fields
- Generators and integral points on elliptic curves associated with simplest quartic fields
- New Filbert and Lilbert matrices with asymmetric entries
- Returning functions with closed graph are continuous
- On sets of points of approximate continuity and ϱ-upper continuity
- Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
- On solvability of some nonlocal boundary value problems for biharmonic equation
- The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
- Strongly increasing solutions of higher-order quasilinear ordinary differential equations
- Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
- Filippov solutions of vector Dirichlet problems
- Sequences of positive homoclinic solutions to difference equations with variable exponent
- Modified Lupaş-Jain operators
- New fixed point results in bv(s)-metric spaces
- Improved Young and Heinz operator inequalities for unitarily invariant norms
- Scrutiny of some fixed point results by S-operators without triangular inequality
- The lattices of families of regular sets in topological spaces
- Iterated partial summations applied to finite-support discrete distributions
- Hamiltonicity of a class of toroidal graphs
