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.File in questo prodotto:
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