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Some uniform bounds for elliptic curves over Q ; Tronto, Sebastiano E-print/Working paper (2021) We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image of ... [more ▼] We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of E, and the Kummer extensions generated by points of infinite order in E(Q). [less ▲] Detailed reference viewed: 69 (1 UL)Reductions of points on algebraic groups ; Perucca, Antonella in Journal of the Institute of Mathematics of Jussieu (2019) Detailed reference viewed: 85 (6 UL)The 1-eigenspace for matrices in GL2(ℤℓ) ; Perucca, Antonella in New York Journal of Mathematics (2017) Detailed reference viewed: 1019 (3 UL)Effective Kummer Theory for Elliptic Curves ; Tronto, Sebastiano E-print/Working paper (n.d.) Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform ... [more ▼] Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform results for the degrees of the Kummer extensions [K(E[N ], N −1 α) : K(E[N ])]. When K = Q, and under a minimal (necessary) assumption on α, we show that the inequality [Q(E[N ], N −1 α) : Q(E[N ])] ≥ cN 2 holds with a constant c independent of both E and α. [less ▲] Detailed reference viewed: 71 (9 UL) |
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