Ordinary primes for abelian surfaces
An abelian variety \(A\) over a number field \(K\) is expected to have ordinary reduction at infinitely many places of \(K\). This is known to be true for \(A\) of dimension at most \(2\), and for such \(A\) the set of ordinary places is known to have a positive (natural) density. The goal of the paper under review is to compute this density in the case of an abelian surface. The author achieves a full classification, proving in particular that in the case \(\dim A = 2\) the density of the set of places of ordinary reduction is \(1\), \(\frac{1}{2}\), or \(\frac{1}{4}\), and that either of the latter two possibilities implies that \(A\) has an isogeny factor whose base change to the algebraic closure \(\overline{K}\) has complex multiplication.
scientific article