Symplectic group lattices as Mordell-Weil sublattices
The author deals with a problem raised by \textit{B. H. Gross} [J. Am. Math. Soc. 3, 929-960 (1990; Zbl 0745.11035)]. Gross had produced several interesting families of integral lattices in Euclidean spaces, including certain symplectic group lattice some of which had previously been considered by \textit{R. Gow} [J. Algebra 122, 510-519 (1989; Zbl 0679.20039)]. In the present paper it is shown that these lattices may be realized as sublattices of certain Mordell-Weil lattices that are associated with hyperelliptic curves in odd characteristic \(p\). This result was inspired by a similar one of \textit{N. Elkies} on the Barnes-Wall lattices [ Int. Math. Res. Not. 1994, 343-361 (1994; Zbl 0813.52017)], and is obtained by a cohomological description of the Mordell-Weil lattices over the \(p\)-adic numbers. As a consequence a lower bound on the minimal nonzero norm, due to Elkies, applies to the symplectic group lattices.
scientific article; zbMATH DE number 980275