The arithmetic complexity $c(mathscr{A}_{theta})$ of a noncommutative torus $mathscr{A}_{theta}$ measures the rank $r$ of a rational elliptic curve $mathscr{E}(K)cong mathbf{Z}^r oplus mathscr{E}_{tors}$ via the formula $r= c(mathscr{A}_{theta})-1$. The number $c(mathscr{A}_{theta})$ 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 $theta=[b_1,dots, b_N, overline{a_1,dots,a_k}]$ of the period $(a_1,dots, a_k)$. We prove that the component $V_{N,k}^0$ is a fiber bundle over the Fermat-Pell conic $mathscr{Q}$ with the structure group $mathscr{E}_{tors}$ and the fiber an $r$-dimensional affine space. As an application, we evaluate the Tate-Shafarevich group $W (mathscr{E}(K))$ of elliptic curve $mathscr{E}(K)$ in terms of the group $W (mathscr{Q})$ calculated by Lemmermeyer.
تحميل البحث