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On the classification of even unimodular lattices with a complex structure. (English) Zbl 1246.11094

The authors study unimodular positive definite Hermitian lattices $L$ over the ring of integers ${𝒪}_{K}$ in the imaginary quadratic number field $K=ℚ\left(\sqrt{-d}\right)$ where $d$ is a squarefree positive integer. Such a lattice $L$ is a finitely generated projective ${𝒪}_{K}$-submodule of a $K$-vector space $V$ of, say, dimension $r$ with basis ${e}_{1},\cdots ,{e}_{r}$ equipped with the Hermitian form $h$ with $h\left({e}_{i},{e}_{j}\right)={\delta }_{ij}$ such that $L$ contains a basis of $V$ and $h\left(L,L\right)\subseteq {𝒪}_{K}$. $L$ being unimodular means that $L=\left\{x\in V\mid h\left(x,L\right)\subseteq {𝒪}_{K}\right\}$. One has $h\left(x,x\right)\in ℤ$ for all $x\in L$, and $L$ is called even if $h\left(x,x\right)\in 2ℤ$ for all $x\in L$, odd otherwise.

In this paper, one assumes throughout $r\equiv 0\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$. When $d\equiv 3\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$, then there is only one genus ${𝒩}_{r}$ of unimodular such lattices that are necessarily odd, namely that of the standard lattice ${⨁}_{i=1}^{r}{𝒪}_{K}{e}_{i}$. If $d¬\equiv 3\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$, then there is in addition a genus of even unimodular such lattices denoted by ${ℳ}_{r}$. The mass $\mu$ of a genus $𝒢$ of lattices is defined to be $\mu \left(𝒢\right)={\sum |\text{Aut}\left(M\right)|}^{-1}$ where $M$ ranges over a system of representatives of the isometry classes of lattices within the genus $𝒢$. A formula for $\mu \left({𝒩}_{r}\right)$ has been determined by K. Hashimoto and H. Koseki [Tôhoku Math. J., II. Ser. 41, No. 1, 1–30 (1989; Zbl 0668.10029)].

The authors first derive a formula for $\mu \left({ℳ}_{r}\right)$ in terms of $\mu \left({𝒩}_{r}\right)$. Using the trace $\mathrm{Tr}$ of the extension $K/ℚ$, a Hermitian ${𝒪}_{K}$-lattice $\left(L,h\right)$ can be made into a quadratic $ℤ$-lattice $\left(L,{F}_{h}\right)$ of rank $2r$, where ${F}_{h}=\mathrm{Tr}\circ h$ if $d\equiv 3\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$ and ${F}_{h}=\frac{1}{2}\mathrm{Tr}\circ h$ if $d¬\equiv 3\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$. An ${𝒪}_{K}$-lattice $\left(L,h\right)$ is called a theta lattice if $\left(L,{F}_{h}\right)$ is even unimodular. The authors derive a formula for the mass of the genus of theta lattices. They explicitly compute the values for the masses and the numbers of isometry classes of theta lattices in the cases where $K$ has class number one and $r=4,8,12$. The paper finishes with some results on the Hermitian theta series of theta lattices.

##### MSC:
 11E41 Class numbers of quadratic and Hermitian forms 11E39 Bilinear and Hermitian forms 11F03 Modular and automorphic functions 11F27 Theta series; Weil representation; theta correspondences 11H06 Lattices and convex bodies (number theoretic results)