On the Picard number and the extension degree of period matrices of complex tori

The rank $\rho$ of the Néron-Severi group of a complex torus $X$ of dimension $g$ satisfies $0\le \rho \le g^2=h^{1,1}\mathrm{.}$ The degree $\mathfrak{d}$ of the extension field generated over $\mathbb{Q}$ by the entries of a period matrix of $X$ imposes constraints on its Picard number $\rho$ and, consequently, on the structure of $X$. In this paper, we show that when $\mathfrak{d}$ is $2$, $3$, or $4$, the Picard number $\rho$ is necessarily large. Moreover, for an abelian variety $X$ of dimension $g$ with $\mathfrak{d}=3,$ we establish a structure-type result: $X$ must be isogenous to $E^g$, where $E$ is an elliptic curve without complex multiplication. In this case, the Picard number satisfies $\rho (X)=\frac{g(g+1)}{2}\mathrm{.}$ As a byproduct, we obtain that if $\mathfrak{d}$ is odd, then $\rho (X)\le \frac{g(g+1)}{2}\mathrm{.}$