D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc., (3) 33 (1976), no. 2, 193--237.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....curve. That is, all elliptic curves over a number eld K with a point of order in E tors (K) can be parameterized by a single variable t 2 K. In each case an explicit equation of the elliptic curve corresponding to t is given by Silverman. Namely, for = 5 12 we have (see [22] p. 278 and [14]) y 2 (t 1)xy ty = x 3 tx 2 ; t 5 (t 2 11t 1) 5) For = 7 we have (see [21] p. 223 and [14] y 2 (1 t t 2 )xy (t 2 t 3 )y = x 3 (t 2 t 3 )x 2 ; t 7 (t 1) 7 (t 3 8t 2 5t 1) 6) When = 11, the modular curve X 1 (11) is itself an ....

....by a single variable t 2 K. In each case an explicit equation of the elliptic curve corresponding to t is given by Silverman. Namely, for = 5 12 we have (see [22] p. 278 and [14] y 2 (t 1)xy ty = x 3 tx 2 ; t 5 (t 2 11t 1) 5) For = 7 we have (see [21] p. 223 and [14]) y 2 (1 t t 2 )xy (t 2 t 3 )y = x 3 (t 2 t 3 )x 2 ; t 7 (t 1) 7 (t 3 8t 2 5t 1) 6) When = 11, the modular curve X 1 (11) is itself an elliptic curve. It has a very simple equation s 2 s = t 3 t 2 : 7) In other words, all elliptic curves over a number ....

D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), 193-237.

....courbe modulaire X 1 (m) est de genre nul, et on peut donc exprimer les quantit es a 1 ; a 2 ; a 3 ; a 4 ; a 6 a l aide des memes param etres. Des param etrisations de quelques courbes elliptiques d efinies sur Q et ayant un sousgroupe E(Q) tors donn e ont et e donn ees (voir par exemple [1] [2], 6] et [7] D autre part, toutes les isog enes des courbes elliptiques de sous groupe de torsion E(Q) tors de la forme Z=mZ, m = 2; 8 et Z=2Z ThetaZ=2mZ, m = 1; 2; 3, correspondant a des isog enies de noyaux form es par des points rationnels, ont et e explicit ees dans [6] Nous ....

D.S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193--237.

.... Gamma 7w: 3. M ethodes utilis ees 3.1 Recherche directe. Soit E une courbe elliptique ayant P 0 pour point de torsion d ordre m 2. D apr es le th eor eme de Mazur (voir [8] m 2 f2; 3; Delta Delta Delta ; 10; 12g. D autre part, ces courbes d ependent de plusieurs param etres (voir [8] [9] ou [10] mais on peut les ramener a deux param etres s et t : E(s; t) y 2 a 1 (s; t)xy a 3 (s; t)y = x 3 a 2 (s; t)x 2 a 4 (s; t)x: 3) COURBES ELLIPTIQUES ET CONJECTURE DE SZPIRO 5 Pour m 4, la courbe de Tate (voir [8] d efinie par : E(A;B;C) y 2 (C Gamma A)xy ....

D.S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193--237.

.... 25) The curve X 0 (5) has two cusps C 1 and C 2 corresponding to the values F 5 = 0 and F 5 = 1 respectively. Hence X 0 (25) has six cusps: a unique one lying above C 1 , corresponding to T 5 = 0; and ve cusps above C 2 , given by T 5 = 1, p 5 5 , p 5 2 5 , p 5 1 5 , p 5 2 5 (cf. [K]) The covering X X 0 (25) is rami ed at the four non rational cusps, and the ber above the cusp T 5 = 1 is composed of rational points (cf. K, p. 226] By proposition 2.1, X can be described by Lehmer s quintic; the roots r 1 ; r 5 of P 5 (Y; T 5 ) are modular functions on X 1 (25) ....

....Z=25Z, and de ne a ( a=25; where (z; 1 z 2 X (m;n)2Z 2 0 1 (z n m ) 2 1 (n m ) 2 is the Weierstrass function. It is well known that the functions a;b ( a ( b ( are modular units on X 1 (25) The divisors of these functions are computed in [K]. In particular, we nd that Divisor 7;9 6;3 1;12 8;4 1;3 7;4 6;7 8;1 = 3P 1 P 2 P 3 2P 4 P 5 ; where the P i denote the cusps on X which are above the cusp 1 of X 0 (25) By expressing the function on the left in terms of so called Klein forms t (a 1 ;a 2 ) cf. ....

Daniel S. Kubert. Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976) 193-237.

....5 Delta 29 2 Delta 37 3 529364550 2 0:4682 Theta 10 Gamma5 0:8653 152 2 1:0004 Gamma5 10 Delta 11 4 Delta 13 32 7. Appendice On donne dans cette partie l expression g en erale des courbes elliptiques d efinies sur Q, suivant la forme de leur sous groupe de torsion (voir [16] [20] ou [25] Les param etres u et v repr esentent des rationnels pour lesquels le discriminant correspondant n est pas nul. Ces discriminants peuvent etre non minimaux. 1. Courbes de sous groupe de torsion de la forme Z=2Z Theta Z=8Z, E 2 Theta8 : Y 2 = X(X 16u 4 v 4 ) X (u 2 Gamma v ....

D.S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193--237.

....that group as torsion subgroup form a 1 parameter family. We will need to have an explicit equation for the universal curve for each family. For N = 3, this universal elliptic curve is y 2 = x 3 (x t) 2 =4 and a 3 torsion point is (0; t=2) For the other cases, we copy 1 Table 3 in [19] to our Table 3. Let E t N denote the elliptic curve with a rational N torsion point with parameter t, and similarly define E t 2;2N . We will need to know something about the field of definition of the 2 torsion points on the curves E t N . Therefore we record the discriminant Delta N (t) ....

.... N (t) is equal (modulo squares) to the discriminant of the cubic field obtained by adjoining the coordinates of one 2 torsion point; if N is even, Delta N (t) is equal 1 Actually, we have done a tiny bit more than copy: we have expanded the implicit expressions for the parameters b and c in [19] to express b and c in terms of a single parameter t. 4 EVERETT W. HOWE, FRANCK LEPR EVOST, AND BJORN POONEN N or (2; 2N) b c 4 t 0 5 t t 6 t 2 t t 7 t 3 Gamma t 2 t 2 Gamma t 8 2t 2 Gamma 3t 1 2t 2 Gamma 3t 1 t 9 t 5 Gamma 2t 4 2t 3 Gamma t 2 t 3 ....

D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc., (3) 33 (1976), no. 2, 193--237.

....60680 USA y Institut National de Recherche en Informatique et en Automatique (INRIA) Domaine de Voluceau, B. P. 105 78153 LE CHESNAY CEDEX (France) z On leave from the French Department of Defense, D el egation G en erale pour l Armement. x AMS Classification: 11Y05, 11G20, 14H52. Kubert [4] gave parametrizations for all these groups. In order to make ECM effective with these curves, we need a further condition beyond Kubert s parametrization, namely that we can exhibit a point on E modulo N . This requires that our parametrization must include a rational point of infinite order on ....

D. S. Kubert. Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. 3, 33 (1976), 193--237.

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D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc., (3) 33 (1976), no. 2, 193--237.

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