à strand, Matti Perttu
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Publication Lifting Problems and Their Independence of the Coefficient Field(2015-01-01) Åstrand, Matti PerttuOur aim is to find out new things about lifting problems in general and Oort groups in particular. We would like to know more about what kind of rings are needed to find liftings to characteristic 0 of covers of curves in characteristic p. For this, we use explicit parametrization of curves and model theory of algebraically closed fields and valued fields. The geometric machinery we need includes local-global principle of lifting problems and HKG-covers of ring extensions. We won’t use formal or rigid geometry directly, although it is used to prove some of that machinery. Also we need some model theoretical results such as AKE-principles and Keisler-Shelah ultrapower theorem. To be able to use model theoretical tools we need to assume some bounds on the complexity of our curves. The standard way to do this is to bound the genus. What we want is that for the finite group G, the curves of a fixed genus can be lifted over a fixed ring extension. This kind of question — where both the curve and the ring are bounded — is well suited for model theoretical tools. For a fixed finite group G, we will show that for genus g and an algebraic integer π, the statement “every G-cover Y → P 1 with genus g has a lifting over W (k)[π]” does not depend on k. In other words, it is either true for all algebraically closed fields k or none of them. This gives some reason to believe that being an Oort group does not depend on the field k. Also it might help in finding explicit bounds on the ring extension needed.