Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group actions

Abstract

For any n > 2 we study the group algebra decomposition of an ([n/2] + 1)- dimensional family of principally polarized abelian varieties of dimension n with an action of the dihedral group of order 2n. For any odd prime p, n = p and n = 2p we compute the induced polarization on the isotypical components of these varieties and some other distinguished subvarieties. In the case of n = p the family contains a one-dimensional family of Jacobians. We use this to compute a period matrix for Klein fs icosahedral curve of genus 5.