Elliptic curves and continued fractions

TL;DR

This paper establishes a relationship between the rank of elliptic curves and the arithmetic complexity of associated non-commutative tori, with specific examples involving elliptic curves with complex multiplication.

Contribution

It introduces a novel connection between elliptic curve ranks and non-commutative geometry, specifically linking arithmetic complexity to curve rank.

Findings

Rank of elliptic curve equals one less than the arithmetic complexity of the related non-commutative torus

Illustrative example using elliptic curves with complex multiplication

Provides a new perspective on elliptic curve invariants through non-commutative geometry

Abstract

It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.