The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields

Kim, Byeong; Kim, Ji; Park, Poo-Sung

doi:10.1090/S0025-5718-09-02287-X

The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields
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by Byeong Moon Kim, Ji Young Kim and Poo-Sung Park PDF

Math. Comp. 79 (2010), 1123-1144 Request permission

Abstract:

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields $\mathbb {Q}(\sqrt {-m})$ for all $m$. For each imaginary quadratic field $\mathbb {Q}(\sqrt {-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice $L$ represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then $L$ is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger’s fifteen theorem and ours is the number 13. In addition, we determine the minimal rank of universal Hermitian lattices for all imaginary quadratic fields.

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