Shortest vectors in Hermitian lattices

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathcal O$ be the ring of integers of an imaginary quadratic field, embedded in $\mathbb C$, and suppose $\mathcal O$ is a principal ideal domain. For $n\ge1$ and $d\in\mathbb Z_{>0}$, let $M\in M_n(\mathcal O)$ be Hermitian ($M=\overline M^{\mathsf T}$), positive definite and of determinant $d$. The squared length of $v\in\mathcal O^n$ is $v^*Mv$, where $v^*=\overline v^{\mathsf T}$. Define $m(M)=\min_{0\ne v\in\mathcal O^n}v^*Mv$.

Statement recorded in the supplied source.

Determine the sharp upper bound, as a function of $\mathcal O,n,d$, on $m(M)$ among all such matrices $M$ of determinant $d$; equivalently, determine the optimal guaranteed upper bound on the shortest nonzero vector length $\sqrt{m(M)}$. [S2]

Short English statement

Find the optimal guaranteed short-vector bound in integral positive-definite Hermitian lattices with fixed rank and determinant over an imaginary quadratic PID.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800051 (AMR-087-0051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Everett W. Howe, Low-genus curves over finite fields: Problems and variations, IPAM Number Theory and Cryptography: Open Problems workshop (9--13 October 2006), PDF pages 42--45. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6237.pdf.