On the number of connected components of real abelian varieties that admit sufficiently many complex multiplications (Q1343711)

The question for the number of connected components is one of the main problems in the topology of real algebraic varieties. The author solves this problem for a real abelian variety \(X\), which (1) is absolutely simple, i.e. \(X_\mathbb{C} = X \otimes_\mathbb{R} \mathbb{C}\) has no non-trivial complex abelian subvarieties, (2) the ring of endomorphisms \(\text{End} (X_\mathbb{C})\) has rank: \(2 \cdot \dim X\), (3) \(\text{End} (X_\mathbb{C})\) is a maximal order in the field \(\text{End} (X_\mathbb{C}) \otimes \mathbb{Q}\). The known fact is that the number of connected components in this case is a power of 2 between 1 and \(2^{\dim X}\). The main theorem gives an expression of the number of connected components of \(X\) via arithmetic properties of \(\text{End} (X_\mathbb{C})\) and \(\text{End} (X)\). In particular, \(X\) is connected if the discriminant of the ring \(\text{End} (X_\mathbb{C})\) over \(\text{End} (X)\) is odd, and has \(2^{\dim X}\) components if the ramification of \(\text{End} (X_\mathbb{C})\) over \(\text{End} (X)\) is as wild as possible at any prime ideal of \(\text{End} (X)\) dividing 2. This generalizes the classical case of real elliptic curves.