Counterexamples to the local-global divisibility over elliptic curves

2017/05/04 by Ranieri, Gabriele
#11G05 #11R34 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1705.01880

Abstract

Let p ≥ 5 be a prime number. We find all the possible subgroups G of \rm GL₂ ( ℤ / p ℤ ) such that there exists a number field k and an elliptic curve E defined over k such that the \rm Gal ( k ( E[p] ) / k )-module E[p] is isomorphic to the G-module ( ℤ / p ℤ )² and there exists n ∈ ℕ such that the local-global divisibility by pⁿ does not hold over E ( k ).

Related

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• On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility