2024/12/12 by Igor Nikolaev, Nikolaev, Igor V.
Physics and Astronomy · #11M55 #46L85 #FOS: Mathematics #Number Theory (math.NT) #Operator Algebras (math.OA) #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.2412.09148
openalex publication_date 2024/12/12 · openalex created_date 2024/12/14 · openalex updated_date 2026/07/28
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.