Some special complex tori

Question

Let $T=V/\Gamma $ be a complex torus; so $V$ is a finite-dimensional complex vector space and $\Gamma$ a lattice in $V$. Moreover I have a positive definite hermitian form $H$ on $V$ such that the real part of $H$ takes integral values on $\Gamma $.

If $\dim V=1$, the existence of such a form is equivalent to saying that the elliptic curve $T$ has complex multiplication. What can one say if $\dim V>1$?

Answer 1

Some minor disagreement has arisen in the comments as to `what' an elliptic curve is. Concretely, to perhaps an algebraic geometer the quotient of $\mathbb{C}$ by the lattice $\mathbb{Z}+\mathbb{Z} \sqrt{-D}$ is a very reasonable and simple elliptic curve. However I'm prone to dismiss it (or specifically, rescale it to have covolume one) since $diag(1, \sqrt{D})$ does not lie in $SL_2\mathbb{R}$ but rather $GSL_2$. Likewise, I am prone to only admit principally polarized abelian varieties, i.e. arising from $Sp_{2g}$ and not $GSp_{2g}$. This unfortunately invalidates my perspective on the OP's question, since he/she appears open to arbitrary covolume lattices and my comments below are (i see now) specific to being in the unimodular (normalized covolume one) setting. This confuses the discussion because the CM is not scale-invariant, e.g. the torus $\mathbb{Z} D^{1/4} + \mathbb{Z} iD^{3/4}$ has $\mathbb{Q}(i)$-CM, versus $\mathbb{Q}(\sqrt{-D})$-CM in the case of $\mathbb{Z}[\sqrt{-D}]$.

I'd like to make an extended comment. As I see it, the positive definite hermitian form arising from an elliptic curve (or, principally polarized abelian variety) has the form $H=g_j+i\omega$, where $\omega$ is the standard symplectic form $\omega(x,y)={}^txj_{std}y$ on $\mathbb{R}^{2g}$ and $g_j(x,y):=\omega(jx,y)$ for $j$ an $Sp_{2g}\mathbb{R}$ conjugate of $j_{std}$. In otherwords the data of a complex elliptic curve (or ppav) amounts to an $Sp_{2g}\mathbb{R}$-conjugate of the standard almost complex structure $j_{std}=\begin{pmatrix} 0 & -I_g \\ I_g & 0 \end{pmatrix}$.

In the special case of ppavs, the hypothesis that the real part of $H=H_j$ be integral on the lattice $\mathbb{Z}^{2g}$ (i.e. $j\mathbb{Z}^{2g} \subset \mathbb{Z}^{2g}$) is equivalent to having $j\mathbb{Q}^{2g} \subset \mathbb{Q}^{2g}$. This is because if $j\mathbb{Q}^{2g} \subset \mathbb{Q}^{2g}$, then $j\mathbb{Z}^{2g}$ is a sublattice of some $\frac{1}{N}\mathbb{Z}^{2g} \subset \mathbb{Q}^{2g}$ on which the symplectic form $\omega$ (i.e. imaginary part of $H=H_j$) is integral and unimodular. But $\mathbb{Z}^{2g}$ is the maximal submodule of $\mathbb{Q}^{2g}$ on which this occurs. In otherwords, we know $j$ is defined over $\mathbb{Q}$.

Now the claim that I want to make is this: if $j$ is defined over $\mathbb{Q}$ then the cyclic subgroup $\{ \exp j\theta\}_{\theta}$ of $Sp_{2g}\mathbb{R}$ is defined over $\mathbb{Q}$, where $\exp$ refers to matrix exponential. This would establish that a ppav has CM (in sense of hodge theory) if $j$ is defined over $\mathbb{Q}$. Now the converse of this statement is not clear (and i don't believe true). In an earlier version of this `answer' i wrongly said they were equivalent. In fact $Sp_{2g}\mathbb{R}$ has $\mathbb{Q}, \mathbb{R}, \mathbb{C}$-rank all equal to $g$, and $e^{j\theta}$ is merely a one-dimensional torus. This explains somewhat how the equivalence of $j$ rational and CM degenerates in higher dimension.

Hodge structures (in the sense, say, of Griffiths/Green/Kerr) on ppav's consist of nonconstant linear representations $\rho:\mathbb{S}^1 \to Sp_{2g}\mathbb{R}$. The image of $i \in \mathbb{S}^1$ actually determines the representation, with $\rho(i)$ giving an almost complex structure $j$ and extending to all of the unit circle via the exponential formula $\exp j\theta=(\cos\theta) I_{2g}+(\sin\theta) j$. I first learned of this formula right here on MO thanks to R. Bryant! The so-called Mumford-Tate group $MT(\rho)$ of the representation is the minimal algebraic subgroup of $Sp_{2g}\mathbb{R}$ defined over $\mathbb{Q}$ and containing the image of the representation $\rho(\mathbb{S}^1)$. From this point-of-view, CM is characterized as those representations (i.e. almost complex structures) whose MT-group is a (necessarily compact!) algebraic torus defined over $\mathbb{Q}$ in $Sp_{2g}\mathbb{R}$.