1911 Encyclopædia Britannica/Number/Method of Reduction

33. Method of Reduction.—This differs according as $\mathrm {D}$ is positive or negative, and will require some preliminary lemmas. Suppose that any complex quantity $z=x+yi$ is represented in the usual way by a point $(x,y)$ referred to rectangular axes. Then by plotting off all the points corresponding to $(\alpha z+\beta )/(\gamma z+\delta )$, we obtain a complete set of properly equivalent points. These all lie on the same side of the axis of $x$, and there is precisely one of them and no more which satisfies the conditions: (i.) that it is not outside the area which is bounded by the lines $2x=\pm 1$; (ii.) that it is not aside the circle $x^{2}+y^{2}=l$; (iii.) that it is not on the line $2x=1$, or on the arcs of the circle $x^{2}+y^{2}=1$ intercepted by $2x=1$ and $x=0$. This point will be called the reduced point equivalent to $z$. In the positive half-plane $(y>0)$ the aggregate of all reduced points occupies the interior and half the boundary of an area which will be called the fundamental triangle, because the areas equivalent to it, and finite, are all triangles bounded by circular arcs, and having angles ${\tfrac {1}{3}}\pi ,{\tfrac {1}{3}}\pi ,0$ and the fundamental triangle may be considered as a special case when one vertex goes to infinity. The aggregate of equivalent triangles forms a kind of mosaic which fills up the whole of the positive half-plane. It will be convenient to denote the fundamental triangle (with its half-boundary, for which $x<0$) by $\triangledown$; for a reason which will appear later, the set of equivalent triangles will be said to make up the modular dissection of the positive half-plane.

Now let $f'=(a',b',c')$ be any definite form with $a'$ positive and determinant $-\Delta$. The root of $a'z^{2}+b'z+c'=0$ which is represented by a point in the positive half-plane is

$\omega ={\frac {-b'+i{\sqrt {}}\Delta }{2a'}}$,

and this is a reduced point if either

(i.) $|b'|<a'<c'$

(ii.) $b'=a',a'\leq c'$

(iii.) $a'=c',0<b'\leq a'$.

Cases (ii.) and (iii.) only occur when the representative point is on the boundary of $\triangledown$. A form whose representative point is reduced is said to be a reduced form. It follows from the geometrical theory that every form is equivalent to a reduced form, and that there are as many distinct classes of positive forms of determinant $-\Delta$ as there are reduced forms. The total number of reduced forms is limited, because in case (i.) we have $\Delta =4ac-b^{2}>3b^{2}$, so that $b<{\sqrt {}}{\tfrac {1}{3}}\Delta$, while $4a^{2}<4ac<\Delta +b^{2}<{\tfrac {4}{3}}\Delta$; in case (ii.) $\Delta =4ac-a^{2}>3a^{2}$, or else $a=b=c={\sqrt {}}{\tfrac {1}{3}}\Delta$; in case (iii.) $\Delta =4a^{2}-b^{2}>3b^{2},4a^{2}=\Delta +b^{2}<{\tfrac {4}{3}}\Delta$, or else $a=b=c={\sqrt {}}{\tfrac {1}{3}}\Delta$. With the help of these inequalities a complete set of reduced forms can be found by trial, and the number of classes determined. The latter cannot exceed ${\tfrac {1}{3}}\Delta$; it is in general much less.

With an indefinite form $(a,b,c)$ we may associate the representative circle $a(x^{2}+y^{2})+bx+c=0$, which cuts the axis of $x$ in two real points. The form is said to be reduced if this circle cuts $\triangledown$; the condition for this is $a(a\pm {\tfrac {1}{2}}b+c)<0$ which can be expressed in the form $3a^{2}+(a\pm b)^{2}<\mathrm {D}$, and it is hence clear that the absolute values of $a,b$, and therefore of $c$, are limited. As before, there are a limited number of reduced forms, but they are not all non-equivalent. In fact they arrange themselves, according to a law which is not very difficult to discover, in cycles or periods, each of which is associated with a particular class. The main result is the same as before: that the number of classes is finite, and that for each class we can find a representative form by a finite process of calculation.