Abstract
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.
Highlights
The use of the Artin–Mazur machinery of étale homotopy theory [2] for Deligne– Mumford stacks was pioneered by Oda [37] following fundamental ideas of Dedicated to Ronnie Brown on the occasion of his 80th birthday and to the memory of Alexander Grothendieck
Using transcendental methods by comparing it with the complex analytic situation Oda showed that the étale homotopy type Mg,n ⊗ Qis given as the profinite Artin–Mazur completion of the Eilenberg-MacLane space K (Mapg,n, 1), where Mapg,n is the Teichmüller modular or mapping class group of compact Riemann surfaces of genus g with n punctures
If we endow the category (Sch/Z[ζN, (N dg)−1]) with the étale topology, AD,[N] is a stack over the big étale site (Sch/Z[ζN, (N dg)−1])et, The moduli stack AD,[N] of abelian schemes with polarisations of type D and level N structures is defined as the stack AD,[N] viewed as a stack over the big étale site (Sch/Z[(N dg)−1])et with structure morphism given by the natural composition
Summary
The use of the Artin–Mazur machinery of étale homotopy theory [2] for Deligne– Mumford stacks was pioneered by Oda [37] following fundamental ideas of Dedicated to Ronnie Brown on the occasion of his 80th birthday and to the memory of Alexander Grothendieck. It turns out that a similar theorem like that of Oda for the moduli stacks of algebraic curves holds with some modifications for the moduli stacks AD and AD,[N] of polarised abelian schemes with and without level structures. We derive similar results for the moduli stacks AD,[N] which are modifications of our main result using appropriate congruence subgroups D(N ) of SpD(Z) which reflect the particular level N structures From these general results we obtain short exact sequences involving the étale fundamental groups of the moduli stacks AD and AD,[N] relating them to the profinite cGoaml(pQle/tQio)n.sInofththisewdiasycrweeteegxrtoeunpdsthSeprDe(sZul)tsanodf. In the final section we relate the étale homotopy types of the moduli stacks of algebraic curves and abelian schemes with principal polarisations by analysing the stacky Torelli morphism
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