Genus 3 curves whose Jacobians have endomorphisms by Q(ζ7+ζ¯7). (May 2016) - Hoffman, J. William; Liang, Zhibin; Sakai, Yukiko; Wang, Haohao
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.