Arithmetic complexity revisited

Abstract

The arithmetic complexity counts the number of algebraically independent entries in the periodic continued fraction $θ=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. If $\mathscr{A}_θ$ is a noncommutative torus corresponding to the rational elliptic curve $\mathscr{E}(K)$, then the rank of $\mathscr{E}(K)$ is given by a simple formula $r(\mathscr{E}(K))= c(\mathscr{A}_θ)-1$, where $c(\mathscr{A}_θ)$ is the arithmetic complexity of $θ$. We prove that $c(\mathscr{A}_θ)$ is equal to the dimension of the Brock-Elkies-Jordan variety of $θ$ introduced in [1]. Following Zagier and Lemmermeyer, we evaluate the Shafarevich-Tate group of $\mathscr{E}(K)$.