$E_{6}$ AND THE ARITHMETIC OF A FAMILY OF NON-HYPERELLIPTIC CURVES OF GENUS 3. (1st January 2015)

• Record Type: Journal Article
• Authors: THORNE, JACK A.
• Is Part Of: Forum of mathematics. Volume 3(2015)
• Publication Date: 2015-01-01
• Subjects: 14G05 (primary), -- 14L30 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2014.2
• Languages: English
• ISSNs: 2050-5086

Abstract

Abstract : We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field $\mathbb{Q}$ of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type $E_{6}$ . We show that average size of the 2-Selmer sets of these curves is finite (if it exists). We use this to show that a positive proposition of these curves (when ordered by height) has integral points everywhere locally, but no integral points globally.