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There is a higher dimensional correspondence that works for some full-rank lattices $\Lambda \subseteq \mathbb{C}^{g}$, but not all! The reference I know for this is Hindry and Silverman's book "Diophantine Geometry". Theorem A.5.0.1 states that the complex torus $\mathbb{C}^{g}/\Lambda$ is an abelian variety if and only if there exists a positive definite ...

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