Group of the rational points on the unit hyperbole

The group of the rational points on the unit hyperbole consists of the points with rational coordinates for which applies. The group consists of the union of both hyperbolic branches, respectively for and . ${\displaystyle H(\mathbb {Q} )}$ ${\displaystyle (x,y)}$ ${\displaystyle x^{2}-y^{2}=1}$ ${\displaystyle H(\mathbb {Q} )}$ ${\displaystyle x<0}$ ${\displaystyle x>0}$

Group operation

The set of rational points forms an infinite Abelian group . The neutral element is the point . The group operation or "sum" is . ${\displaystyle (1,0)}$ ${\displaystyle (x,y)+(t,u)=(xt+yu,xu+yt)}$

Geometrically, this is the hyperbola angle addition : if and is, as well as and , then their sum is the rational point on the unit hyperbola with the angle in the sense of the usual addition of hyperbola angles . It applies namely and . It should be noted that the "angles" are only to be regarded as parameters and do not correspond to the actual angles of the points on the hyperbola. ${\displaystyle x=\cosh(\alpha )}$ ${\displaystyle y=\sinh(\alpha )}$ ${\displaystyle t=\cosh(\beta )}$ ${\displaystyle u=\sinh(\beta )}$ ${\displaystyle (xt+yu,xu+yt)}$ ${\displaystyle \alpha +\beta }$ ${\displaystyle xt+yu=\cosh(\alpha +\beta )}$ ${\displaystyle xu+yt=\sinh(\alpha +\beta )}$

Group structure

The group is isomorphic to an infinite direct sum of cyclic subsets of : ${\displaystyle H(\mathbb {Q} )}$ ${\displaystyle H(\mathbb {Q} )}$

${\displaystyle H(\mathbb {Q} )\cong H'\oplus \left(\bigoplus _{p\in \mathbb {P} }H_{p}\right),}$

where the subgroup consists of two elements and the subgroups are the infinite cyclic groups each created by the point of the shape . ${\displaystyle H'=\left\{\pm (1,0)\right\}}$ ${\displaystyle H_{p}}$ ${\displaystyle \left({\tfrac {p^{2}+1}{2p}},{\tfrac {p^{2}-1}{2p}}\right)}$

literature

• Lin Tan: The Group of Rational Points on the Unit Circle. In: Mathematics Magazine. Vol. 69, No. 3, June 1996, pp. 163–171, doi : 10.2307 / 2691462 , digitized version (PDF; 792 kB) ( memento of March 8, 2012 in the Internet Archive ) or directly https: //www.maa .org / sites / default / files / pdf / upload_library / 22 / Allendoerfer / 1997 / 0025570x.di021195.02p0087x.pdf

Individual evidence