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If we observe the correspondence $$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$ we see the relationship between weight 4 and weight 6 modular forms, the coefficients of an elliptic curve, and a lattice $\Lambda$, i.e. a free $\mathbb{Z}$ module of rank 2.

Is there a higher dimensional correspondence between $\mathbb{C}/\Lambda$ and an abelian surface or abelian variety when $\Lambda$ is a free $\mathbb{Z}$-module of rank $d$? Do(es) the equation(s) describing such an object include coefficients with higher weight modular forms like $g_{4}(\Lambda),g_{5}(\Lambda),...?$ Is there a way to embed odd-dimensional spaces in $\mathbb{C}^{2g}$, where $g$ is the genus, and then mod; I'm particularly interested in the case of free $\mathbb{Z}$-modules of rank $3$.

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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 Hermitian form on $\mathbb{C}^{g} \times \mathbb{C}^{g}$ whose imaginary part takes integer values when restricted to $\Lambda \times \Lambda$. They give examples to show that this is sometimes true, and sometimes false. (In this construction, $\Lambda$ must be a lattice of full rank in $\mathbb{C}^{g}$, and hence $\Lambda$ must have even rank.)

There are some analogues in these higher-dimensional cases of the results for $g = 1$. There is a vector space of Riemann theta functions that gives an embedding of $\mathbb{C}^{g}/\Lambda$ into projective space. The coefficients of the resulting equation for the abelian surface will involve Siegel modular forms. This is rather painful to do explicitly. (In Cassels's and Flynn's book "Prolegomena to a middlebrow arithmetic of curves of genus 2", they embed abelian surfaces in $\mathbb{P}^{15}$, and the equations defining them are a system of 72 quadrics. In the paper "Defining equations of the universal abelian surfaces with level three structure" by Gunji (Manuscripta Math, 2005), abelian surfaces are embedded in $\mathbb{P}^{8}$.)

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  • $\begingroup$ So, there is no hope for understanding this correspondence for free $\mathbb{Z}$-modules of rank 3? What if I embed $\Lambda = \bigoplus_{i=1}^{3} \omega_{i}\mathbb{Z}$ into a free $\mathbb{Z}$-module of rank 4, where the 4th dimensional basis vector is trivial? Then could I take $\mathbb{C}^{2}/\Lambda$? I need the odd-dimensional case, and I'm not sure how this generalizes to that, although thank you for the response and references. $\endgroup$ – Samuel Reid Jul 9 '15 at 20:50
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    $\begingroup$ It is known that $\mathbb{C}^{g}/\Lambda$ can be realized as a subvariety of a projective space if and only if $\Lambda$ has a Riemann form. Is it possible that instead of working with abelian varieties, what you really want is to study the moduli space of lattices up to homothety? If so, the right object is ${\rm SL}(n,\mathbb{R})/{\rm SL}(n,\mathbb{Z})$, where $n$ is the rank. $\endgroup$ – Jeremy Rouse Jul 10 '15 at 1:13
  • $\begingroup$ Exactly! I was using this to construct the moduli space of lattices and then wanted to investigate extremized quantities in moduli subspaces. Do you have a reference for any review/open problems related to this? I was wanting to explore this connection very deeply for my masters thesis. $\endgroup$ – Samuel Reid Jul 10 '15 at 13:55
  • $\begingroup$ The paper here might be a good start for some prior results and some ways to think about questions like this, but I'm not entirely sure what exactly you're interested in investigating. $\endgroup$ – Jeremy Rouse Jul 10 '15 at 14:26

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