Extremal symplectic lattices
type: Item
instance of: scholarly article
title: Extremal symplectic lattices
zbMATH Open document ID: 0922.11058
DOI: 10.1007/BF02783050
author: J. R. Quine
author: Peter Liang Zhang
published in: Israel Journal of Mathematics
publication date: 1999-04-27T00:00:00Z
review text: 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.
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 1270357
zbMATH Keywords: perfect lattices
zbMATH Keywords: extremal Riemann surfaces
zbMATH Keywords: extremal symplectic lattices
zbMATH Keywords: eutactic
zbMATH Keywords: Jacobians
zbMATH Keywords: genus 3 Riemann surfaces
zbMATH Keywords: Klein curve
zbMATH Keywords: Picard curve
zbMATH Keywords: Fermat curve
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02783050
OpenAlex ID: W2021791095
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description: scientific article; zbMATH DE number 1270357