Quantum arithmetic

Igor V. Nikolaev

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arXiv · 2412.09148

Quantum arithmetic

Igor V. Nikolaev

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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.

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Igor V. Nikolaev. 2024-12-12. Quantum arithmetic. https://arxiv.org/abs/2412.09148

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