Let K be a field of characteristic char(K) not equal 2,3 and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K) is not equal 0; we denote by E[m] the rn-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]. Let P_i := (x_i, y_i) (i = 1,2) be a Z-basis for E[m]; then K_m = K (x_1, y_1, x_2, y_2). We look for small sets of generators for K_m inside {x_1, y_1, x_2, y_2, zeta_m} trying to emphasize the role of zeta_m (a primitive m-th root of unity). In particular, we prove that K_m = K (x_1, zeta_m, y_2), for any odd m >= 5. When m = p is prime and K is a number field we prove that the generating set {x_1, zeta_p, y_2} is often minimal, while when the classical Galois representation Gal(K_p/K) -> GL(2)(Z/pZ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K_m/K for m = 3 and m = 4.

Fields generated by torsion points of elliptic curves

Paladino L.
2016

Abstract

Let K be a field of characteristic char(K) not equal 2,3 and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K) is not equal 0; we denote by E[m] the rn-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]. Let P_i := (x_i, y_i) (i = 1,2) be a Z-basis for E[m]; then K_m = K (x_1, y_1, x_2, y_2). We look for small sets of generators for K_m inside {x_1, y_1, x_2, y_2, zeta_m} trying to emphasize the role of zeta_m (a primitive m-th root of unity). In particular, we prove that K_m = K (x_1, zeta_m, y_2), for any odd m >= 5. When m = p is prime and K is a number field we prove that the generating set {x_1, zeta_p, y_2} is often minimal, while when the classical Galois representation Gal(K_p/K) -> GL(2)(Z/pZ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K_m/K for m = 3 and m = 4.
Elliptic curves; Torsion points; Galois representations
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/301975
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