Connectedness of families of sphere covers of An-type

Our basic question: Restricting to covers of the sphere by a compact Riemann surface of a given type, do all such compose one connected family? Or failing that, do they fall into easily discerned components? The answer has often been “Yes!,” figuring in such topics as the connectedness of the moduli space of curves of genus g (geometry), Davenport’s problem (arithmetic) and the genus 0 problem (group theory). One consequence: We then know the definition field of the family components. We start with connectedness results for certain spaces – examples of LiuOsserman – of alterating group covers. This implies our spaces are the base level of a sequence spaces – a M(odular) T(ower). Connectedness results ensure certain cusp types – especially H(arbater)-M(umford) – lie on a tower level boundary. Another cusp type (a p-cusp) contributes to the Main MT Conjecture: When all tower levels are defined over some fixed number field K, high tower levels have general type and no K points. Modular curve towers have both pand H-M types, and no others. General MTs can have another cusp type, though – as in our examples – these often disappear at high levels, so giving the Main Conjecture in an infinity of cases. Our cusp description modular representations– rather than semi-simple representations. The sh-incidence matrix, from a natural pairing on cusps, — simplifies displaying results. A lifting invariant — used by the author and Serre —explain both components and cusps. The S(trong) T(orsion) C(onjecture) bounds Q torsion on abelian varieties of a fixed dimension. It implies the Main Conjecture, connecting the former to the R(egular) I(nverse) G(alois) P(roblem). Any Main Conjecture success or failure, can use this cusp description as an explicit test of the STC.