Anabelian geometry of punctured elliptic curves - Porowski, Wojciech (2020)
Anabelian geometry of hyperbolic curves has been studied in detail for the last thirty years, culminating in proofs of various versions of Grothendieck Anabelian Conjectures. These results are usually stated as fully faithfulness of a certain functor, which to a hyperbolic curve X associates some type of fundamental group Π_X. Careful inspection of the proofs reveals that in fact quite often we proceed by establishing various reconstruction algorithms, which to a fundamental group Π_X associate some other type of data related to the curve X. In other words, we recover information about the curve X from the topological group Π_X. This algorithmic approach is sometimes called monoanabelian. In this thesis we concentrate on the special case when the hyperbolic curve X is a smooth and proper curve of genus one over a p-adic local field K with one K-rational point removed i.e., elliptic curve E punctured at the origin. We consider the problem of reconstructing the local height of a rational point on an elliptic curve from the fundamental group Π_X equipped with a section of the absolute Galois group GK determined by this point. We provide such construction for the full étale fundamental group of X as well as for its maximally geometrically pro-p quotient in the case when the elliptic curve E has potentially good reduction. Another problem we consider is determining the reduction type of the elliptic curve E from the maximal geometrically pro-p fundamental group of X, equipped with an additional data of the set of discrete tangential sections. Our main result provides such reconstruction when the residue characteristic p is greater than three. Moreover, we study the tempered fundamental group of a Tate curve and prove that a particular torsor of cohomology classes of theta functions admits a natural trivialization, well defined up to a sign, which is compatible with the integral structure coming form the stable model of the Tate curve. Finally, in the last chapter we shift our attention to studying GK- equivariant automorphisms of various multiplicative submonoids of the monoid (Kalg)× and describe their structure.