Rational points on \(X^ +_ 0(N)\) and quadratic \(\mathbb Q\)-curves. (English) Zbl 1035.14008
Let \(N\) be an integer \(>1\) and \(X_0(N)\) be the modular curve whose non-cusp points classify the isomorphism classes of isogenies between elliptic curves \(\varphi:E\to E'\) of degree \(N\) with cyclic kernel. We denote by \(X^+_0(N)\) the quotient of \(X_0(N)\) by the Fricke involution \(W_N\). The curve \(X^+_0(N)\) has a model defined over \(\mathbb{Q}\). Rational points of \(X^+_0(N)\) which do not arise from elliptic curves with complex multiplication are called “exceptional”.
F. Momose [J. Fac. Sci., Univ. Tokyo, Sect. 1 A Math. 33, 441–466 (1986; Zbl 0621.14018), J. Math. Soc. Japan 39, 269–286 (1987; Zbl 0623.14009)] has given interesting results on the \(\mathbb{Q}\)-rational points of \(X^+_0(N)\). The author of the paper under review gave a computational study of the case when \(N\) is a prime [S. D. Galbraith, Exp. Math. 8, 311–318 (1999; Zbl 0960.14010)].
In this paper he continues his previous work studying some of the cases not covered by the results of Momose. Exceptional rational points are found in the cases \(N=91\), \(N=125\) and the \(j\)-invariants of the corresponding quadratic \(\mathbb{Q}\)-curves are obtained.
F. Momose [J. Fac. Sci., Univ. Tokyo, Sect. 1 A Math. 33, 441–466 (1986; Zbl 0621.14018), J. Math. Soc. Japan 39, 269–286 (1987; Zbl 0623.14009)] has given interesting results on the \(\mathbb{Q}\)-rational points of \(X^+_0(N)\). The author of the paper under review gave a computational study of the case when \(N\) is a prime [S. D. Galbraith, Exp. Math. 8, 311–318 (1999; Zbl 0960.14010)].
In this paper he continues his previous work studying some of the cases not covered by the results of Momose. Exceptional rational points are found in the cases \(N=91\), \(N=125\) and the \(j\)-invariants of the corresponding quadratic \(\mathbb{Q}\)-curves are obtained.
Reviewer: D. Poulakis (Thessaloniki)
MSC:
| 14G05 | Rational points |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 14G35 | Modular and Shimura varieties |
Keywords:
Heegner point; quadratic \(\mathbb{Q}\)-curves; \(j\)-invariants; modular curve; elliptic curves; exceptional rational pointsSoftware:
ecdataReferences:
| [1] | Atkin, A.O.L., Lehner, J., Hecke Operators on Γ0(N), Math. Ann.185 (1970), 134-160. · Zbl 0177.34901 |
| [2] | Birch, B.J., Heegner points of elliptic curves, AMS Symp. math.15 (1975), Inf. teor., Strutt. Corpi algebr., Convegni1973, 441-445. · Zbl 0317.14015 |
| [3] | Cohen, H., Skoruppa, N.-P., Zagier, D., Tables of modular forms. Preprint, 1992. |
| [4] | Cremona, J.E., Algorithms for modular elliptic curves. Cambridge (1992) · Zbl 0758.14042 |
| [5] | Deligne, P., Rappoport, M., Les schemas de modules de courbes elliptiques. In Modular Functions one Variable II, Springer Lecture Notes Math. 349 (1973), 143-316. · Zbl 0281.14010 |
| [6] | Elkies, N., Remarks on elliptic K-curves, preprint, 1993. |
| [7] | Elkies, N., Elliptic and modular curves over finite fields and related computational issues. In D. A. Buell and J. T. Teitelbaum (eds.), Computational Perspectives on Number Theory, AMS Studies in Advanced Math., 1998, 21-76. · Zbl 0915.11036 |
| [8] | Galbraith, S.D., Equations for Modular Curves. Doctoral Thesis, Oxford, 1996. |
| [9] | Galbraith, S.D., Rational points on X0+(p). Experiment. Math.8 (1999), 311-318. · Zbl 0960.14010 |
| [10] | Galbraith, S.D., Constructing isogenies between elliptic curves over finite fields. London Math. Soc. J. Comp. Math.2 (1999), 118-138. · Zbl 1018.11028 |
| [11] | González, J., Equations of hyperelliptic modular curves. Ann. Inst. Fourier bf41 (1991), 779-795. · Zbl 0758.14010 |
| [12] | González, J., Lario, J.-C., Rational and elliptic parametrizations of Q-curves. J. Number Theory72 (1998), 13-31. · Zbl 0932.11037 |
| [13] | González, J., On the j-invariants of the quadratic Q-curves. J. London Math. Soc.63 (2001), 52-68. · Zbl 1010.11028 |
| [14] | Gross, B.H., Arithmetic on elliptic curves with complex multiplication. 776, Springer, 1980. · Zbl 0433.14032 |
| [15] | Gross, B.H., Heegner Points on X0(N). In Modular Forms, R. A. Rankin (ed.), Wiley, 1984, 87-105. · Zbl 0559.14011 |
| [16] | Gross, B.H., Zagier, D.B., On singular moduli. J. Reine Angew. Math.355 (1985), 191-220. · Zbl 0545.10015 |
| [17] | Hasegawa, Y., Table of quotient curves of modular curves X0(N) with genus 2. Proc. Japan Acad. Ser. A 71 (1995), 235-239. · Zbl 0873.11040 |
| [18] | Hasegawa, Y., Q-curves over quadratic fields. Manuscripta Math.94 (1997), 347-364. · Zbl 0909.11017 |
| [19] | Kenku, M.A., On the Modular Curves X0(125), X0(25) and X0(49). J. London Math. Soc.23 (1981), 415-427. · Zbl 0425.14006 |
| [20] | Lang, S., Elliptic Functions, 2nd edition. Springer GTM112, 1987. · Zbl 0615.14018 |
| [21] | Mazur, B., Modular Curves and the Eisenstein Ideal. Pub. I.H.E.S, 47 (1977), 33-186. · Zbl 0394.14008 |
| [22] | Momose, F., Rational Points on X0+(p r). J. Faculty of Science University of Tokyo Section 1A Mathematics33 (1986), 441-466. · Zbl 0621.14018 |
| [23] | Momose, F., Rational Points on the Modular Curves X +0(N). J. Math. Soc. Japan39 (1987), 269-285. · Zbl 0623.14009 |
| [24] | Murabayashi, N., On normal forms of modular curves of genus 2. Osaka J. Math.29 (1992), 405-418. · Zbl 0774.14025 |
| [25] | Ogg, A., Rational Points on Certain Elliptic Modular Curves. In H. Diamond (ed.), AMS Proc. Symp. Pure Math.24, 1973, 221-231. · Zbl 0273.14008 |
| [26] | Ogg, A., Hyperelliptic Modular Curves. Bull. Soc. Math. France102 (1974), 449-462. · Zbl 0314.10018 |
| [27] | Ribet, K., Abelian varieties over Q and modular forms. Proceedings of KAIST workshop (1992), 53-79. |
| [28] | Shimura, M., Defining equations of modular curves X0(N). Tokyo J. Math.18 (1995), 443-456. · Zbl 0865.11052 |
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.
