Genus 3 curves whose Jacobians have endomorphisms by Q(ζ7+ζ¯7). (May 2016)

• Record Type: Journal Article
• Authors: Hoffman, J. William; Liang, Zhibin; Sakai, Yukiko; Wang, Haohao
• Is Part Of: Journal of symbolic computation. Volume 74(2016)
• Page Start: 561
• Page End: 577
• Publication Date: 2016-05
• Subjects: Curves of genus 3 -- Real multiplication -- Abelian variety; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.09.004
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this work we consider constructions of genus 3 curves X such that End ( Jac ( X ) ) ⊗ Q contains the totally real cubic number field Q ( ζ 7 + ζ ¯ 7 ) . We construct explicit two-dimensional families defined over Q ( s, t ) whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied inHoffman and Wang (2013) . We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.