Anabelian geometry of punctured elliptic curves

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 \Pi_X. Careful inspection of the proofs reveals that in fact quite often we proceed by establishing various reconstruction algorithms, which to a fundamental group \Pi_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 \Pi_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 \Pi_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

Explicit estimates in inter-universal Teichmuller theory

In the final paper of a series of papers concerning inter-universal Teichmuller theory, Mochizuki verified various numerically non-effective versions of the Vojta, ABC, and Szpiro Conjectures over number fields. In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results, we first establish a version of the theory of etale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime “2”. We then proceed to discuss how such a modified version of the theory of etale theta functions affects inter-universal Teichmuller theory. Finally, by applying our slightly modified version of inter-universal Teichmuller theory, together with various explicit estimates concerning heights, the j-invariants of “arithmetic” elliptic curves, and the prime number theorem, we verify the numerically effective versions of Mochizuki’s results referred to above. These numerically effective versions imply effective diophantine results such as an effective version of the ABC inequality over mono-complex number fields [i.e., the rational number field or an imaginary quadratic field] and effective versions of conjectures of Szpiro. We also obtain an explicit estimate concerning “Fermat’s Last Theorem” (FLT) — i.e., to the effect that FLT holds for prime exponents > 1.615·1014 — which is sufficient, in light of a numerical result of Coppersmith, to give an alternative proof of the first case of FLT. In the second case of FLT, if one combines the techniques of the present paper with a recent estimate due to Mihailescu and Rassias, then the lower bound “1.615·1014” can be improved to “257”. This estimate, combined with a classical result of Vandiver, yields an alternative proof of the second case of FLT. In particular, the results of the present paper, combined with the results of Vandiver, Coppersmith, and Mihailescu-Rassias, yield an unconditional new alternative proof of Fermat’s Last Theore

Anabelian geometry of punctured elliptic curves

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 \Pi_X. Careful inspection of the proofs reveals that in fact quite often we proceed by establishing various reconstruction algorithms, which to a fundamental group \Pi_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 \Pi_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 \Pi_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

Isomorphisms of local and global fields and isomorphisms of their absolute Galois groups

Twierdzenia Neukircha-Uchidy i twierdzenie Mochizukiego są przykładami wyników opisujących relacje między izomorfizmami ciał a izomorfizmami ich absolutnych grup Galois. Zaprezentujemy dowód uproszczonej wersji pierwszego z tych twierdzeń oraz dowód drugiego, w sposób bardziej szczegółowy niż w oryginalnym artykule Mochizukiego.Neukirch-Uchida and Mochizuki theorems are examples of results concerning relations between isomorphisms of fields and isomorhisms of their Galois groups. We present proof of a simplified version of the first theorem and a proof of the second one, which is more detailed than in the original article

Explicit Estimates in Inter-universal Teichmüller Theory

In the final paper of a series of papers concerning inter- universal Teichmüller theory, Mochizuki verified various numerically non-effective versions of the Vojta, ABC, and Szpiro Conjectures over number fields. In the present paper, we obtain various numerically effective versions of Mochizuki's results. In order to obtain these results, we first establish a version of the theory of étale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime "2". We then proceed to discuss how such a modified version of the theory of étale theta functions affects inter-universal Teichmüller theory. Finally, by applying our slightly modifed version of inter-universal Teichmüller theory, together with various explicit estimates concerning heights, the j-invariants of "arithmetic" elliptic curves, and the prime number theorem, we verify the numerically effective versions of Mochizuki's results referred to above. These numerically effective versions imply effective diophantine results such as an effective version of the ABC inequality over mono-complex number fields [i.e., the rational number field or an imaginary quadratic field] and an effective version of a conjecture of Szpiro. We also obtain an explicit estimate concerning "Fermat's Last Theorem" (FLT) - i.e., to the effect that FLT holds for prime exponents > 1.615・1014 - which is sufficient to give an alternative proof of the first case of Fermat's Last Theorem