论文标题
算术复杂性重新审视
Arithmetic complexity revisited
论文作者
论文摘要
算术复杂性计算了定期续分级中代数独立条目的数量$θ= [b_1，\ dots，b_n，\ overline {a_1，\ dots，a_k}] $。如果$ \ MATHSCR {a}_θ$是与有理椭圆曲线相对应的非交换曲线，则$ \ MATHSCR {e}（k）$的等级由简单的公式$ r（\ nathsscr { $ c（\ mathscr {a}_θ）$是$θ$的算术复杂性。我们证明$ c（\ mathscr {a}_θ）$等于[1]中引入的$θ$的Brock-elkies-Jordan品种的尺寸。在Zagier和Lemmermeyer之后，我们评估了$ \ Mathscr {e}（k）$的Shafarevich-Tate组。
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 $\mathscr{E}(K)$.
