The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians

In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy.As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered.Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.

Geometrical Illustrations of Cyclant Substitutions

§1. The following three illustrations of cyclant substitutions are supplementary to those given in § 5 of a previous paper.In the cyclant substitution in three variables with equimodular multipliersin which the coefficients are real and satisfy the relationsthe following expressions, as may at once be verified, are absolute invariants: