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On Smirnov's approach to the ABC conjecture

Manoel Jarra

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.

On 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 , for the projective line over algebraic extensions of and for maps between them induced by elements of , as introduced by Alexander Smirnov in his approach to the ABC conjecture.
Paper Structure (9 sections, 14 theorems, 60 equations)

This paper contains 9 sections, 14 theorems, 60 equations.

Table of Contents

  1. Monoid schemes
  2. Basic definitions
  3. The geometry of monoids
  4. Base extension to rings
  5. Strong congruence spaces
  6. The map gamma: XZ ---> SCong X
  7. The projective line over algebraic extensions of F1
  8. The compactification of Spec OK
  9. Maps from the compactification of Spec OK to the projective line over F1 induced by elements of K

Key Result

Theorem 1

If the conjecture above holds, then the ABC conjecture is true.

Theorems & Definitions (31)

  • Conjecture : Smirnov93
  • Theorem : Smirnov93
  • Remark
  • Theorem A: \ref{["Smirnov's projective line as strong congruence space F1"]}
  • Theorem B
  • Proposition 1: \ref{["Smirnov's projective line as strong congruence space F1infty"]}
  • Theorem C
  • Proposition 1
  • proof
  • Definition 1
  • ...and 21 more