Quantum arithmetic

TL;DR

This paper formalizes quantum arithmetic by linking number theory and operator algebras, demonstrating that rational projective varieties correspond to $C^*$-algebras with real multiplication, with applications to elliptic curves and related areas.

Contribution

It establishes a formal framework for quantum arithmetic connecting algebraic geometry and operator algebras, confirming conjectures by Manin and others.

Findings

Rational projective varieties are dual to $C^*$-algebras with real multiplication.

The construction satisfies all axioms of quantum arithmetic conjectured by Manin.

Applications to elliptic curves, Shafarevich-Tate groups, and height functions are discussed.

Abstract

We formalize the quantum arithmetic, i.e. a relationship between number theory and operator algebras. Namely, it is proved that rational projective varieties are dual to the $C^*$-algebras with real multiplication. Our construction fits all axioms of the quantum arithmetic conjectured by Manin and others. Applications to elliptic curves, Shafarevich-Tate groups of abelian varieties and height functions are reviewed.