Genus 3 curves whose Jacobians have endomorphisms by Q(ζ7 + ζ-7)
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(ζ7+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 in Hoffman and Wang (2013). We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.
Publication Source (Journal or Book title)
Journal of Symbolic Computation
Hoffman, J., Liang, Z., Sakai, Y., & Wang, H. (2016). Genus 3 curves whose Jacobians have endomorphisms by Q(ζ7 + ζ-7). Journal of Symbolic Computation, 74, 561-577. https://doi.org/10.1016/j.jsc.2015.09.004