Genus 3 curves whose Jacobians have endomorphisms by Q(ζ₇ + ζ₇)

2014/11/08 by J. William Hoffman, Zhibin Liang, Hoffman, J. William +5
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #math.AG

paper · pdf · doi:10.48550/arxiv.1411.2151

arxiv created 2014/11/08 · arxiv updated 2014/11/11

Abstract

In this work we consider constructions of genus three curves X such that End(Jac(X)) ⊗ Q contains the totally real cubic number field Q(ζ_ 7 + ζ₇). 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 by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.

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