Arithmetic complexity revisited

被引:0
作者
Nikolaev, Igor V. [1 ]
机构
[1] St Johns Univ, Dept Math & Comp Sci, 8000 Utopia Pkwy, New York, NY 11439 USA
关键词
Elliptic curve; Noncommutative torus; Brock-Elkies-Jordan variety;
D O I
10.1007/s41478-023-00554-x
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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).
引用
收藏
页码:2115 / 2126
页数:12
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