On rational torsion points of central $\mathbb{Q}$-curves

Sairaiji, Fumio; Yamauchi, Takuya

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 465-483

Let $E$ be a central $\mathbb{Q}$-curve over a polyquadratic field $k$. In this article we give an upper bound for prime divisors of the order of the $k$-rational torsion subgroup $E_{tors}\left(k\right)$ (see Theorems 1.1 and 1.2). The notion of central $\mathbb{Q}$-curves is a generalization of that of elliptic curves over $\mathbb{Q}$. Our result is a generalization of Theorem 2 of Mazur [12], and it is a precision of the upper bounds of Merel [15] and Oesterlé [17].

Soit $E$ une $\mathbb{Q}$-courbe centrale sur un corps polyquadratique $k$. Dans cet article, nous donnons une borne supérieure des diviseurs premiers de l’ordre du sous-groupe de torsion $k$-rationnel $E_{tors}\left(k\right)$ (voir Théorèmes 1.1 et 1.2). La notion de $\mathbb{Q}$-courbe centrale est une généralisation de celle de courbe elliptique sur $\mathbb{Q}$. Notre résultat est une généralisation du Théorème de Mazur [12], et c’est une précision des bornes supérieures de Merel [15] et Oesterlé [17].

[1] J. Cremona, Algorithms for modular elliptic curves. Cambridge: Cambridge University Press, 1992. | Zbl | MR

[2] P. Deligne, M. Rapoport, Schémas de modules de courbes elliptiques. Lecture Notes in Math. 349, Springer, Berlin-Heiderlberg-New York, 1973. | Zbl | MR

[3] N.D. Elkies, On elliptic $K$-curves. In Modular curves and abelian varieties, Birkhäuser, 2004. | MR

[4] M. Furumoto, Y. Hasegawa, Hyperelliptic quotients of modular curves $X_0\left(N\right)$. Tokyo J. Math. 22 (1999), 105–125. | Zbl | MR

[5] A. Grothendieck, Groupes de monodromie en géometrie algébrique. In Séminaire de Géometrie Algébrique, Springer, 1972/3. | MR

[6] Y. Hasegawa, $\mathbb{Q}$-curves over quadratic fields. Manuscripta Math. 94 (1997), 347–364. | Zbl | MR

[7] G. Karpilovsky, Group representations, Vol. 2. Elsevier, Amsterdam, 1993. | Zbl | MR

[8] S. Kamienny, On the torsion subgroups of elliptic curves over totally real field. Invent. Math. 83 (1986), 545–551. | Zbl | MR

[9] S. Kamienny, Torsion points on elliptic curves and $q$-coefficients of modular forms. Invent. Math. 109 (1992), 221–229. | Zbl | MR

[10] M. Kenku, F. Momose, Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109 (1988), 125–149. | Zbl | MR

[11] D.S. Kubert, Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. 33 (1976), 193–237. | Zbl | MR

[12] B. Mazur, Rational points on modular curves. In Modular functions of one variable V, Springer, Berlin, 1977. | Zbl | MR

[13] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHES 47 (1978), 33–186. | Zbl | MR | Numdam

[14] B. Mazur, Rational isogenies of prime degree. Invent Math. 44 (1978), 129–162. | Zbl | MR

[15] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), 437–449. | Zbl | MR

[16] K. Murty, The addition law on hyperelliptic Jacobians. Currents trends in number theory (Allahabad, 2000), 101–110, Hindustan Book Agency, New Delhi, 2002. | Zbl | MR

[17] J. Oesterlé, Torsion des courbes elliptiques sur les corps de nombres, preprint.

[18] P. Parent, Borne effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. Reine Angew Math. 503 (1999), 129–160. | Zbl | MR

[19] E.E. Pyle, Abelian varieties over $\mathbb{Q}$ with large endomorphism algebras and their simple components over $\overline{\mathbb{Q}}$. In Modular curves and abelian varieties, Birkhäuser, 2004. | Zbl | MR

[20] J. Quer, $\mathbb{Q}$-curves and abelian varieties of GL${}_2$-type. Proc. London Math. Soc. 81 (2000), 285–317. | Zbl | MR

[21] M. Raynaud, Schémas en groupes de type $(p,...,p)$. Bull. Soc Math. Fr. 102 (1974), 241–280. | Zbl | MR | Numdam

[22] K. Ribet, A modular construction of unramified $p$-extension of $\mathbb{Q}\left({\mu }_p\right)$. Invent. Math. 34 (1976), 151–162. | Zbl | MR

[23] J.-P. Serre, J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1968), 492–517. | Zbl | MR

[24] J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil. In Modular Function of One Variable IV, Springer-Verlag, 1975. | MR

[25] T. Yamauchi, On $\mathbb{Q}$-simple factors of Jacobian varieties of modular curves. Yokohama Math. J. 53 (2007), no. 2, 149–160. | Zbl | MR

[26] L.C. Washington, Introduction to cyclotomic fields. Springer-Verlag New-York, 1997. | Zbl | MR