Connectedness of families of sphere covers of Atomic-Orbital 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. Our connectedness story considers the existence of unramified p-group extensions attached to a compact Riemann surface cover of the sphere. This translates to existence of a sequence of spaces – a M(odular) T(ower) whose levels correspond to a power pk, k ≥ 0, of p. Connectedness results ensure certain cusp types lie on a tower level boundary. One cusp type – H(arbater)M(umford) – guarantees the full sequence of these spaces (and so the group extensions) are nonempty. Another, called 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 these often disappear at high levels. This happens in examples of Liu-Osserman, so directly giving the Main Conjecture in an infinity of cases. A combinatorial cusp description enables a different type of group theory–modular representations–than used by representation/automorphic function people. 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) — bounding Q torsion on abelian varieties of a fixed dimension — 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.