Arithmetic complexity revisited

The arithmetic complexity counts the number of algebraically independent entries in the periodic continued fraction $$\theta =[b_1,\dots , b_N, \overline{a_1,\dots ,a_k}]$$ . If $$\mathscr {A}_{\theta }$$ 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 $$r(\mathscr {E}(K))= c(\mathscr {A}_{\theta })-1$$ , where $$c(\mathscr {A}_{\theta })$$ is the arithmetic complexity of $$\theta $$ . We prove that $$c(\mathscr {A}_{\theta })$$ is equal to the dimension of the Brock–Elkies–Jordan variety of $$\theta $$ introduced in Brock et al. (Acta Arith 197: 379–420, 2021). Following Zagier and Lemmermeyer, we evaluate the Shafarevich-Tate group of $$\mathscr {E}(K)$$ .