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P-division points on certain elliptic curves

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P-division points on certain elliptic curves COMPOSITIO MATHEMATICA KUANG-YEN SHIH P-division points on certain elliptic curves Compositio Mathematica, tome 36, no 2 (1978), p. 113-129. <> © Foundation Compositio Mathematica, 1978, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 113 P-DIVISION POINTS ON CERTAIN ELLIPTIC CURVES Kuang-yen Shih* 1. Introduction Let p be an odd prime, E = (_1)(p-l)/2@ and k = Q(B/,Ep). Consider an elliptic curve E defined over k. Adjoin to k the x-coordinates of the points of order p on E and denote the resulting field by Fp (E), or simply Fp. It is known (see [7, 6.1]) that Fp is a Galois extension of k and Gal(Fplk) can be identified with a subgroup of GZ.(Z/pZ)/{±l2}, where Note that if Gal(F,/k) is the whole G’Z(Z/pZ)/{±l2}, then Fp contains a subfield F normal over the quadratic field k such that Gal(F/k) is isomorphic to PSL2(Z/p Z). One purpose of this paper is to discuss some conditions on E under which Fp(E) contains a subfield K normal over the rational number field Q so that Gal(K/Q) is isomor- phic to PSL2(Z/pZ). Denote the non-trivial automorphism of k by or. Suppose there are a quadratic non-residue N modulo p, and an N-cyclic isogeny À of E to its conjugate EU such that (1.1) C = ker À is rational over k, and Here E(N) stands for the group of N-division points on E. The existence of such À implies that Fp is not only normal over k, but also over Q. This will be proved in §2. We also determine the

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