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 field obtained by adding to K the coordinates of the points of E[m]. We describe the fields K_5, when 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.

On 5-torsion of CM elliptic curves

Paladino L.
2018-01-01

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 field obtained by adding to K the coordinates of the points of E[m]. We describe the fields K_5, when 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.
Elliptic curves; Complex multiplication; Torsion points
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/301942
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