Bielliptic modular curves. (English) Zbl 0964.11029
Let \(C\) be a smooth curve of genus at least 2 defined over a number field \(K\). By Falting’s theorem (ex-Mordell conjecture) \(C\) has only finitely many \(K\)-rational points. In order to generalize this result it is considered the set \(\Gamma_d(C,K)=\{P\in C(L) ; [L:K]\leq d\}\) of points of degree at most \(d\) in \(C\). In the case \(d=2\), D. Abramovich and J. Harris, Compos. Math. 78, 227-238 (1991; Zbl 0748.14010)] show that the existence of infinitely many quadratic points is equivalent to \(C\) being hyperelliptic or bielliptic.
In this paper the author considers the family of modular curves \(X_0(N)\) admitting infinitely many quadratic points. This family consists of all hyperelliptic and bielliptic modular curves. Hyperelliptic modular curves were completely determined by the work of [A. Ogg, [Bull. Soc. Math. Fr. 102, 1-61 (1974; Zbl 0314.10018)]. The author completely determines the classification of modular curves \(X_0(N)\) that are bielliptic.
In this paper the author considers the family of modular curves \(X_0(N)\) admitting infinitely many quadratic points. This family consists of all hyperelliptic and bielliptic modular curves. Hyperelliptic modular curves were completely determined by the work of [A. Ogg, [Bull. Soc. Math. Fr. 102, 1-61 (1974; Zbl 0314.10018)]. The author completely determines the classification of modular curves \(X_0(N)\) that are bielliptic.
Reviewer: Amílcar Pacheco (Rio de Janeiro)
MSC:
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 14G05 | Rational points |
| 14H25 | Arithmetic ground fields for curves |
| 14K05 | Algebraic theory of abelian varieties |
Software:
ecdataOnline Encyclopedia of Integer Sequences:
Numbers n such that the modular curve X_0(n) is bielliptic.Numbers n such that the modular curve X_0(n) has a bielliptic involution of Atkin-Lehner type.
Numbers n such that the modular curves X(n) and X_1(n) are not bielliptic.
Numbers n such that the modular curve X_0(n) contains infinitely many rational points of degree 2.
Numbers n such that the modular curve X_0(n) has genus >= 2 and contains infinitely many points of degree 2 over some number field L.
References:
| [1] | Abramovich, D.; Harris, J., Abelian varieties and curves in\(W_d}(C\), Compositio Math., 78, 227-238 (1991) · Zbl 0748.14010 |
| [2] | Atkin, A. O.L.; Lehner, J., Hecke operators on \(Γ_0N\), Math. Ann., 185, 134-160 (1970) · Zbl 0177.34901 |
| [3] | Cremona, J. E., Algorithms for Modular Elliptic Curves (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0758.14042 |
| [4] | Elkies, N., The automorphism group \(X_0\), Compositio Math., 74, 203-208 (1990) · Zbl 0708.14016 |
| [5] | Harris, J.; Silvermann, J. H., Bielliptic curves and symmetric products, Proc. Amer. Math. Soc., 112 (1991) · Zbl 0727.11023 |
| [6] | Hindry, M., Points quadratiques sur les courbes, C. R. Acad. Sci. Paris, 305, 219-221 (1987) · Zbl 0677.14005 |
| [7] | Kenku, M. A.; Momose, F., Automorphism groups of the modular curves \(X_0N\), Compositio Math., 65, 51-80 (1988) · Zbl 0686.14035 |
| [8] | Kluit, On the normalizer of \(Γ_0N\); Kluit, On the normalizer of \(Γ_0N\) |
| [9] | Mazur, B.; Swinnerton-Dyer, P., Arithmetic of Weil curves, Invent. Math., 25, 1-61 (1974) · Zbl 0281.14016 |
| [10] | Ogg, A. P., Hyperelliptic modular curves, Bull. Soc. Math. France, 102, 449-462 (1974) · Zbl 0314.10018 |
| [11] | A. P. Ogg, Rational points on certain elliptic modular curves, Analytic Number Theory, Proceedings of Symposia in Pure Mathematics XXIV, 221, 232; A. P. Ogg, Rational points on certain elliptic modular curves, Analytic Number Theory, Proceedings of Symposia in Pure Mathematics XXIV, 221, 232 · Zbl 0273.14008 |
| [12] | Corbes modulars: Taules. Notes del seminari de Teoria de Nombres UB-UAB-UPC, Barcelona, 1992; Corbes modulars: Taules. Notes del seminari de Teoria de Nombres UB-UAB-UPC, Barcelona, 1992 |
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