Arithmetic complexity revisited

The arithmetic complexity counts the number of algebraically independent entries in the periodic continued fraction theta = [b(1), ... , b(N), (a(1), ... , a(k))]. If A(theta) is a noncommutative torus corresponding to the rational elliptic curve epsilon(K), then the rank of epsilon(K) is given by a simple formula r(epsilon(K)) = c(A(theta)) -1, where c(A(theta)) is the arithmetic complexity of theta. We prove that c(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 epsilon(K).