THE MARKOV EQUATION X2+Y2+Z2=AXYZ OVER QUADRATIC IMAGINARY FIELDS

被引：15

作者：

SILVERMAN, JH ^{[1
]}

机构：

[1] BROWN UNIV,DEPT MATH,PROVIDENCE,RI 02912

来源：

JOURNAL OF NUMBER THEORY | 1990年 / 35卷 / 01期

基金：

美国国家科学基金会;

关键词：

D O I：

10.1016/0022-314X(90)90105-Z

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

The Markoff equation Ma: X2 + Y2 + Z2 = aXYZ has been extensively studied. Markoff showed that M3(Z) is generated from (1, 1, 1) by automorphisms, and Zagier gave a precise asymptotic formula for the number of points in Ma(Z) with height less than H. In this paper we prove similar results for Ma(R), where R is an order in a quadratic imaginary field K. In particular, we show that if K ≠ Q(i), and if |a| ≥ 3, then the only solutions come from Markoff's solution. (If |a| > 3, there are no solutions.) And for R = Z[i] we give an asymptotic formula similar to Zagier's result. © 1990.

引用

页码：72 / 104

页数：33