\'Etale homotopy types of moduli stacks of polarised abelian schemes

We determine the Artin-Mazur \'etale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the \'etale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an \'etale homotopy point of view.