Arithmetic complexity revisited

Igor Nikolaev

Published 2020-02-25Version 1

The arithmetic complexity $c(\mathscr{A}_{\theta})$ is a non-commutative measure of the ranks of elliptic curves $\mathscr{E}(K)=\mathbf{Z}^r \oplus \mathscr{E}_{tors}$. The $c(\mathscr{A}_{\theta})$ is equal to the dimension of a connected component $V_{N,k}^0$ of the Brock-Elkies-Jordan variety associated to a periodic continued fraction $\theta=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. We prove that the $V_{N,k}^0$ is a fiber bundle over the Fermat-Pell conic with the structure group $\mathscr{E}_{tors}$ and the fiber an $r$-dimensional affine space. The result is used to evaluate the Tate-Shafarevich group of $\mathscr{E}(K)$.