Arithmetic complexity revisited

Abstract

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 ${\mathcal{A}}_{\theta }$ is a noncommutative torus corresponding to the rational elliptic curve $\mathcal{E}(K)$, then the rank of $\mathcal{E}(K)$ is given by a simple formula $r(\mathcal{E}(K))=c({\mathcal{A}}_{\theta })-1$, where $c({\mathcal{A}}_{\theta })$ is the arithmetic complexity of $\theta$. We prove that $c({\mathcal{A}}_{\theta })$ is equal to the dimension of the Brock-Elkies-Jordan variety of $\theta$ introduced in [1]. Following Zagier and Lemmermeyer, we evaluate the Shafarevich-Tate group of $\mathcal{E}(K)$.

to appear in the Journal of Analysis

References in corpus (3)

• Conics - a Poor Man's Elliptic Curves
• Comments on a note by M. Waldschmidt
• Periodic continued fractions over $S$-integers in number fields and Skolem's $p$-adic method