Let e be a CM elliptic curve defined over a number field K, with Weiestrass form y^2 = x^3 + bx or y^2 = x^3 + c. For every positive integer m, we denote by e[m] the m-torsion subgroup of E and by K-m ..= K (e[m]) the m-th division field, i.e. the extension of K generated by the coordinates of the points in E[m]. We classify all the fields K_7. In particular we give explicit generators for K_7/K and produce all the Galois groups Gal(K_7/K). We also show some applications to the Local-Global Divisibility Problem and to modular curves.
On 7-division fields of CM elliptic curves
Alessandri JessicaWriting – Original Draft Preparation
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2023-01-01
Abstract
Let e be a CM elliptic curve defined over a number field K, with Weiestrass form y^2 = x^3 + bx or y^2 = x^3 + c. For every positive integer m, we denote by e[m] the m-torsion subgroup of E and by K-m ..= K (e[m]) the m-th division field, i.e. the extension of K generated by the coordinates of the points in E[m]. We classify all the fields K_7. In particular we give explicit generators for K_7/K and produce all the Galois groups Gal(K_7/K). We also show some applications to the Local-Global Divisibility Problem and to modular curves.File in questo prodotto:
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