\(\mathfrak{o}_{K_0}\)-quasi-Abelian varieties with complex multiplication (Q2844270)

A toroidal group is by definition a connected complex Lie group \(X\) with \(H^0(X,{\mathcal O})={\mathbb C}\). This is the quotient \({\mathbb C}^n/\Gamma\) of a complex linear space by a discrete subgroup \(\Gamma\) such that \(\Gamma\otimes {\mathbb C}={\mathbb C}^n\).NEWLINENEWLINEOne way to construct such a quotient is to fix a number field \(K\) of degree \(n+m\), having \(n-m\geq 0\) real and \(2m\) complex embeddings, giving an embedding \(\Psi: K\hookrightarrow {\mathbb C}^m\times{\mathbb R}^{n-m}\subset {\mathbb C}^n\). If we then choose an order \({\mathfrak o}\) of \(K\), we may take \(\Gamma=\Psi({\mathfrak o})\) and \(X={\mathbb C}^n/\Gamma\).NEWLINENEWLINEIt is shown here that such an \(X\) has complex multiplication. (The idea is that this makes \(K\) play the role of a CM field: it is not a CM field in the usual sense because it is not purely imaginary, but is admissible because \(X\) is only a quasi-abelian variety, not an abelian variety.) If \(K_0\) is the maximal totally real subfield of the join of all the embeddings of \(K\) in \({\mathbb C}\), and \({\mathfrak o}_{K_0}\) is its ring of integers, then \(X\) has a polarisation (Riemann form) taking values in \({\mathfrak o}_{K_0}\): such an \(X\) is called an \({\mathfrak o}_{K_0}\)-quasi-abelian variety. Any two simple \({\mathfrak o}_{K_0}\)-quasi-abelian varieties are shown to be isogenous. These results are proved by using period matrices: along the way, this approach yields some simpler proofs of basic facts about toroidal groups. The analogy between \({\mathfrak o}_{K_0}\)-quasi-abelian varieties and abelian varieties goes slightly further: the spaces \(H^q(X,{\mathcal O})\) are shown to be of finite dimension.