On 5-torsion of CM elliptic curves

2018/07/31 by Paladino, Laura
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1808.00029

Abstract

Let E be an elliptic curve defined over a number field K. Let m be a positive integer. We denote by E[m] the m-torsion subgroup of E and by K_m:=K(E[m]) the number field obtained by adding to K the coordinates of the points of E[m]. We describe the fields K₅, when E is a CM elliptic curve defined over K, with Weiestrass form either y²=x³+bx or y²=x³+c. In particular we classify the fields K₅ in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem, to modular curves and to Shimura curves.

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