Extremal symplectic lattices
The study of extremal Riemann surfaces motivates the study of extremal symplectic lattices. In the classic theory of lattices, the problem is to maximize the minimum non-zero norm over the space of all \(n\)-dimensional lattices. For Riemann surfaces, the problem is to maximize for a given genus either the minimum non-zero length in the length spectrum or the minimum non-zero norm in the Jacobian. The systole problems for length and Jacobi spectra are closely related in genus \(\leq 3\). For an overview and results refer to the papers of Gromov, Buser, Sarnak, Schmutz, Conway and Sloane. Bergé and Martinet have adapted the concepts of perfect and eutactic to study symplectic and isodual lattices and in general any family of lattices invariant under a subgroup of the linear group. In this paper, the criteria of Bergé and Martinet has been used to study Jacobians of genus 3 Riemann surfaces, the Klein curve, the Picard curve and the Fermat curve. It is shown that the Jacobian of the Klein curve is eutactic in the classical sense. The Jacobian of the Fermat curve is eutactic in the classical sense and therefore also for symplectic lattices, but it is not perfect for symplectic lattices and, in fact, is not a local maximum for the minimum non-zero norm. Further, it is shown that the Jacobian of the Picard curve is perfect and eutactic for symplectic lattices and is therefore a strict local maximum for the minimum non-zero norm.
scientific article; zbMATH DE number 1270357
scientific article