Let k be an arbitrary complete discrete valuation field of mixed characteristic whose residue characteristic we denote by p, k an algebraic closure of k, G k def = Gal(k/k), Σ a set of primes that contains a prime l = p, X a proper hyperbolic curve over k. for the geometrically pro-Σ quotient of thé etale fundamental group Π X of X. Thus, we have a natural surjection Π [where i ranges over the ...

Abstract In this chapter, we provide a geometric construction of manifold extending given Galois cover to wreath product, using composita and fiber products. For be possible, certain assumption on the homology, previously called (∗), needs strengthened new condition (∗∗) (equivalent in most cases). To motivate use condition, first recall connection between homology quotient coinvariants. Apart ...

The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of $${\mathsf {A}}_g$$ generically contained in the Jacobian locus. Counterexamples known $$g\le 7$$ . They can all be constructed using families Galois coverings curves satisfying a numerical condition. These already classified cases where: (a) group is cyclic, (b) it abelian and family 1-d...

We define affine pseudo-planes as one class of Q-homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line A1. Moreover, we show that there exists an infinitedimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as th...

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show e...

In this thesis I will describe an explicit method for performing an 8-descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of n-coverings, which suits the needs of higher descent. Then I will sketch the classical method of 2-descent, and the two methods that are known for doing a second 2-descent, also called 4-des...

In this article we explore a non-abelian torsion theory in the category of preordered groups: objects its torsion-free subcategory are partially ordered groups, whereas groups (with total order). The reflector from to has stable units, and prove that it induces monotone-light factorization system. We describe coverings relative Galois structure naturally associated with reflector, explain how t...

We consider equations of the form Bf = g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann’s covering theorem for max-plus linear equations, and give various ...