On local-global divisibility by $p^n$ in elliptic curves

Details On local-global divisibility by $p^n$ in elliptic curves On local-global divisibility by $p^n$ in elliptic curves

2011-04-25

arxiv.org/abs/1104.4762v1

Mathematics

Number Theory

21 pages

Scientific paper

Let $ p $ be a prime number and let $k$ be a number field, which does not contain the field $\mathbb{Q} (\zeta_p + \bar{\zeta_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion points of exact order $p$ on $\mathcal{E}$, then the local-global principle holds for divisibility by $p^n$, with $n$ a natural number. As a consequence of the deep theorems of Merel, Mazur and Kamienny we deduce that, for $p$ larger than a constant $C ([k:\mathbb{Q}])$, depending only on the degree of $k$, there are no counterexamples to the local-global divisibility principle. In particular, for the rational numbers $C(1)=7$ and for quadratic fields $C(2)=13$.

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