Entanglement for elliptic curve group orders

Abstract

For an elliptic curve $E$ defined over a number field $K$ and $\mathfrak{p}$ a prime of good reduction of $E/K$, we let $n_\mathfrak{p}=\#E(k_\mathfrak{p})$ be the order of the group of points over the residue class field $k_\mathfrak{p}$. Divisibility of $n_\mathfrak{p}$ by a prime $\ell$ depends on the splitting behavior of $\mathfrak{p}$ in the $\ell$-division field $K(E[\ell](\bar{K}))$ of $E$ and, generically, divisibility of $n_\mathfrak{p}$ by different primes $\ell_1$ and $\ell_2$ are `independent events' on the set of primes $\mathfrak{p}$ of good reduction. However, there do exist elliptic curves with `entangled division fields' for which infinitely many $n_\mathfrak{p}$ are coprime to $\ell_1$ and infinitely many $n_\mathfrak{p}$ are coprime to $\ell_2$, but no $n_\mathfrak{p}$ is coprime to to $\ell_1 \ell_2$. For $\ell_1=2$, we classify in group theoretical terms the entanglement giving rise to this behavior, and count up to isomorphism the number of `special $2\ell$-division groups' exhibiting the phenomenon. A final section shows how such groups may be realized as the Galois group of $2\ell$ division fields for elliptic curves $E/\mathbf{Q}$.