Fields generated by torsion points of elliptic curves

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• 作者：Andrea Bandini^a ; ^{andrea.bandini@unipr.it" class="auth_mail" title="E-mail the corresponding author} ; Laura Paladino^b ; ¹ ; ^{paladino@mat.unical.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11G05 ; 11F80
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：169
• 期：Complete
• 页码：103-133
• 全文大小：523 K

文摘

Let K   be a field of characteristic char(K)≠2,3$\mathrm{char}(K)\ne 2,3$ and let E$\mathcal{E}$ be an elliptic curve defined over K. Let m   be a positive integer, prime with char(K)$\mathrm{char}(K)$ if char(K)≠0$\mathrm{char}(K)\ne 0$; we denote by E[m]$\mathcal{E}[m]$ the m  -torsion subgroup of E$\mathcal{E}$ and by K_m:=K(E[m])$K_m:=K(\mathcal{E}[m])$ the field obtained by adding to K   the coordinates of the points of E[m]$\mathcal{E}[m]$. Let P_i:=(x_i,y_i)$P_i:=(x_i,y_i)$ (i=1,2$i=1,2$) be a Z$\mathbb{Z}$-basis for E[m]$\mathcal{E}[m]$; then K_m=K(x₁,y₁,x₂,y₂)$K_m=K(x_1,y_1,x_2,y_2)$. We look for small sets of generators for K_m$K_m$ inside {x₁,y₁,x₂,y₂,ζ_m}$\{x_1,y_1,x_2,y_2,{\zeta }_m\}$ trying to emphasize the role of ζ_m${\zeta }_m$ (a primitive m  -th root of unity). In particular, we prove that K_m=K(x₁,ζ_m,y₂)$K_m=K(x_1,{\zeta }_m,y_2)$, for any odd m⩾5$m⩾5$. When m=p$m=p$ is prime and K   is a number field we prove that the generating set {x₁,ζ_p,y₂}$\{x_1,{\zeta }_p,y_2\}$ is often minimal, while when the classical Galois representation Gal(K_p/K)→GL₂(Z/pZ)$\mathrm{Gal}(K_p/K)\to {\mathrm{GL}}_2(\mathbb{Z}/p\mathbb{Z})$ is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K_m/K$K_m/K$ for m=3$m=3$ and m=4$m=4$.