Symplectic lattices (Q2704194)

This paper presents a survey on isodual (positive definite) lattices and their connections with other interesting classes of lattices, Riemann surfaces, and modular forms. The interest in the subject, in particular in symplectic lattices, is motivated by the work of \textit{P. Buser} and \textit{P. Sarnak} [Invent. Math. 117, 27-56 (1994; Zbl 0814.14033)], where the density of the associated sphere packings is used to locate the Jacobians of curves among principally polarized abelian varieties. The author explains this connection between symplectic lattices, Jacobians, and abelian varieties.NEWLINENEWLINENEWLINEThe symplectic lattices form a subclass of the isodual lattices (lattices that are isometric to their duals), whose systematic investigation was initiated by \textit{J. H. Conway} and \textit{N. J. A. Sloane} in [J. Number Theory 48, 373-382 (1994; Zbl 0810.11041). If an isodual lattice can be rescaled to be integral, then it gives rise to a modular lattice, a generalization of the notion of unimodular lattices. Rich theory of (strongly) modular lattices has been developed by \textit{H. G. Quebbemann} [J. Number Theory 54, 190-202 (1995; Zbl 0874.11038); Enseign. Math., II. Ser. 43, 55-65 (1997; Zbl 0898.11014)].NEWLINENEWLINENEWLINEMuch efforts have recently been devoted to finding or classifying dense strongly modular forms, also see the recent paper of \textit{R. Scharlau} and \textit{R. Schulze-Pillot} [Extremal lattices, in: Algorithmic algebra and number theory, B. H. Matzat et al. (ed.) Springer-Verlag, 139-170 (1999; Zbl 0944.11012)]. The author describes various results on Voronoi's theory of isodual lattices. A number of constructions of isodual lattices has also been discussed, including Hermitian lattices, quaternionic lattices, and using finite group representations.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].