Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

Robert Auffarth; Paweł Borówka

paper

Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

arXiv:2111.11799

Abstract

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs $(m, n)$ the locus of polarised abelian surfaces of type $(1, d)$ that contain two complementary elliptic curve of exponents $m, n$ , denoted $E_{d} (m, n)$ is non-empty. We show that if $d$ is square-free, the locus $E_{d} (m, n)$ is an irreducible surface (if non-empty). We also show that the loci $E_{d} (d, d)$ can have many components if $d$ is an odd square. As an application, we show that for a genus $3$ curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings $f_{i} : C \to E_{i}, i = 1, 2, 3$ satisfy $l c m (d e g (f_{1}), d e g (f_{2})) = l c m (d e g (f_{1}), d e g (f_{3})) = l c m (d e g (f_{2}), d e g (f_{3}))$ .

arXiv commentA previous result was incorrect and has been changed in this new version. Application to genus 3 curves added. Any comments welcome!