On a correlation between ranks of elliptic curves and periods of continued fractions

TL;DR

This paper establishes a mathematical relationship linking the rank of elliptic curves with complex multiplication to the arithmetic complexity of associated noncommutative tori with real multiplication.

Contribution

It proves a new correlation between elliptic curve ranks and noncommutative tori complexities, advancing understanding in number theory and noncommutative geometry.

Findings

Rank of elliptic curves with complex multiplication is one less than the arithmetic complexity of corresponding noncommutative tori.

Establishes a theoretical link between elliptic curve properties and noncommutative geometric structures.

Provides a new perspective on the interplay between algebraic and geometric aspects of number theory.

Abstract

It is proved that the rank of elliptic curves with complex multiplication introduced by B. H. Gross is one less an arithmetic complexity of the corresponding noncommutative tori with real multiplication.