Let $V$ be a complex vector space of dimension $g$, and let $\Lambda\subseteq V$ be a full rank lattice endowed with a Riemann form $E\colon \Lambda\times\Lambda\to \mathbb Z$. Then the pair $(V/\Lambda,E)$ is a polarized abelian variety. Now, I know that one can always find a $\mathbb Z$-basis $B=\{\lambda_1,\ldots,\lambda_g,\mu_1,\ldots,\mu_g\}$ of $\Lambda$ such that, with respect to $B$, $E$ takes the form $\left(\begin{array}{cc}0 & D\\-D & 0\end{array}\right)$, where $D$ is a diagonal $g\times g$-matrix with integer positive entries $d_1,\ldots,d_g$ and $d_i\mid d_{i+1}$ for every $i$. I read several times that if $d_1=d_2=\ldots =d_g$, then the polarization induced by $E$ is principal, but what about the converse? Is there an easy way in general to see whether $E$ induces a principal polarization? Or can anyone point me out to a good reference for that?
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2$\begingroup$ Essentially by definition, the polarization is principal if and only if $E$ is unimodular, i.e. $d_1=\ldots =d_g=1$. What you write is not correct: if $d_1=\ldots =d_g=k$, the polarization induced by $E$ is $k$ times a principal polarization. $\endgroup$– abxCommented Sep 28, 2017 at 19:55
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