Quantum dynamics of elliptic curves

TL;DR

This paper explores the quantum dynamics related to elliptic curves by calculating the K-theory of a specific noncommutative algebra, linking it to properties of elliptic curves over number fields.

Contribution

It introduces a novel method to connect noncommutative geometry with the arithmetic of elliptic curves, specifically through K-theory calculations of crossed product $C^*$-algebras.

Findings

Determined the K-theory of the crossed product algebra.

Provided a new approach to evaluate the rank of elliptic curves.

Linked K-theory results to the Shafarevich-Tate group.

Abstract

We calculate K-theory of a crossed product $C^*$-algebra of the noncommutative torus with real multiplication by elliptic curve $\mathscr{E}(K)$ over a number field $K$. This result is used to evaluate the rank and the Shafarevich-Tate group of $\mathscr{E}(K)$.