Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

Laura Paladino [a]

On 5-torsion of CM elliptic curves

Pages: 329-350
Received: 4 August 2018
Accepted in revised form: 17 December 2018
Mathematics Subject Classification (2010): 11G05, 11F80, 11G18.
Keywords: Elliptic curves, complex multiplication, torsion points.
Author address:
[a]: University of Calabria, Ponte Bucci, Cubo 30B, Rende, 87036, Italy

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Abstract: Let \(\mathcal{E}\) be an elliptic curve defined over a number field \(K\). Let \(m\) be a positive integer. We denote by \(\mathcal{E}[m]\) the \(m\)-torsion subgroup of \(\mathcal{E}\) and by \(K_m:=K(\mathcal{E}[m])\) the field obtained by adding to \(K\) the coordinates of the points of \(\mathcal{E}[m]\). We describe the fields \(K_5\), when \(\mathcal{E}\) is a CM elliptic curve defined over \(K\), with Weiestrass form either \(y^2=x^3+bx\) or \(y^2=x^3+c\). In particular we classify the fields \(K_5\) in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem and to modular curves.

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