VORONOI–ALGORITHM EXPANSION OF A FAMILY WITH PERIOD LENGTH GOING TO INFINITY
Kan Kaneko · SUT Journal of Mathematics · 1998
We consider a family of orders of complex cubic elds which is similar to one introduced by Levesque and Rhim.We nd the Voronoi-algorithm expansions and the fundamental units.AMS 1991 Mathematics Subject Classication.Primary 11R16, 11R27.Key words and phrases.Complex cubic elds.x 1. Introduction Levesque and Rhin 4] introduced two families of complex cubic elds Q(), each of which depends on two parameters.Adam 1] obtained the Voronoialgorithm expansions of the order Z] f o r t h e s e t wo families, for one of which K uhner 3] also found the Voronoi-algorithm expansions.In this paper we shall consider a new family of complex cubic elds, similar but di erent from those families above, i.e.Q(), where is the real root of the irreducible cubic polynomial f(X) in Proposition 1.1.We obtain the following results : the Voronoi-algorithm expansions of the order Z], the period length of these expansions goes to innty, the fundamental units of the order Z] .Our method, in which w e use an isotropic vector of the quadratic form, is due to Adam 1] .Proposition 1.1.Let f(X) = X 3 ; c m X 2 + ( c + 1 ) X ; c m , where m c are intergers such that m=1 and c=2.Then f(X) has only one real root and f(X) is irreducible except the case m = 1 c = 2 .Moreover if m=2 , t h e n satises (1.1) c m ; 1 c m;1 ; 1 c m+2 0 ((m c) 6 = ( 1 2)) c m ; 2 c m;1 < < c m ; 1 c m;1 ((m c) 6 = ( 1 2)) : Therefore if (m c) 6 = ( 1 2) , then f(X) is irreducible.Furthermore we h a ve f(c m ; 1 c m;1 ; 1 c m+2 ) = ; c m;2 + 1 c m;2 ; 1 c m;1 + 3 c m+1 ; 1 c m+2 + 2 c m+4 ; 1 c 3m;3 ; 3 c 3m ; 3 c 3m+3 ; 1 c 3m+6 < 0 (m=2) : Hence if m=2 , t h e n c m ; 1 c m;1 ; 1 c m+2 < < c m ; 1 c m;1 :x 2. Voronoi-algorithm and preliminariesLet K be a cubic algebraic number eld of negative discriminant.Let 1 a 1 2 2 K be rationally independent.We say that R = 1 1 2 ] = Z + Z: 1 + Z: 2 is a lattice of K with basis f1 1 2 g.For ! 2 R we dene F(!) = N K (!) != !0 !00 , w h e r e N K denotes the norm of K over Q, and !0 and !00 the conjugates of !.