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

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