On Smirnov's approach to the ABC conjecture

TL;DR

This paper employs algebraic geometry over pointed monoids to provide an intrinsic interpretation of compactifications related to number fields, the projective line over algebraic extensions of F_1, and maps induced by elements of K, advancing Smirnov's approach to the ABC conjecture.

Contribution

It introduces an algebraic geometric framework over pointed monoids to interpret compactifications in number theory and algebraic geometry, building on Smirnov's approach to the ABC conjecture.

Findings

Provides an intrinsic interpretation of compactifications over number fields

Connects algebraic geometry over pointed monoids with number theory concepts

Enhances understanding of Smirnov's approach to the ABC conjecture

Abstract

We use algebraic geometry over pointed monoids to give an intrinsic interpretation for the compactification of the spectrum of the ring of integers of a number field $K$, for the projective line over algebraic extensions of $\mathbb{F}_1$ and for maps between them induced by elements of $K$, as introduced by Alexander Smirnov in his approach to the ABC conjecture.