Page:EB1911 - Volume 19.djvu/885

Now let 𝑓′＝(𝑎′, 𝑏′, 𝑐′) be any definite form with 𝑎′ positive and determinant — 𝚫. The root of 𝑎′𝑧²+𝑏′𝑧+𝑐′＝0 which is represented by a point in the positive half-plane is

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

and this is a reduced point if either

$({\mbox{i.}})\;\;b'<a'<c'$
$({\mbox{ii.}})\;b'=a',\ a'\leq c'$
$({\mbox{iii.}})a'=c',\ 0<b'\leq a'$

Cases (ii.) and (iii.) only occur when the representative point is on the boundary of ∇. 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 —∆ 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 $b<{\sqrt {\scriptstyle {1 \over 3}\Delta }}$, while $4a^{2}<4ac<\Delta +b^{2}<\scriptstyle {4 \over 3}\Delta$; in case (ii.) $\Delta =4ac-a^{2}>3a^{2}$, or else $a=b=c={\sqrt {\scriptstyle {1 \over 3}\Delta }}$; in case (iii.) $\Delta =4a^{2}-b^{2}>3b^{2},4a^{2}=\Delta +b^{2}<\scriptstyle {4 \over 3}\Delta$, or else $a=b=c={\sqrt {\scriptstyle {1 \over 3}\Delta }}$. 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 1/3∆; it is in general much less.

With an indefinite form (𝑎, 𝑏, 𝑐) we may associate the representative circle

𝑎(𝑥²+𝑦²)+𝑏𝑥+𝑐＝0,

which cuts the axis of 𝑥 in two real points. The form is said to be reduced if this circle cuts ∇; the condition for this is $a(a\pm {\tfrac {1}{2}}b+c)<0$, which can be expressed in the form $3a^{2}+(a\pm b)^{2}<{\mbox{D}}$, 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.

34. Problem of Representation.—It is required to find out whether a given number 𝑚′ can be represented by the given form $(a',\ b',\ c')$. One condition is clearly that the divisor of the form must be a factor of 𝑚′. Suppose this is the case; and let 𝑚, (𝑎, 𝑏, 𝑐) be the quotients of 𝑚′ and $(a',\ b',\ c')$ be the divisor in question. Then we have now to discover whether 𝑚 can be represented by the primitive form (𝑎, 𝑏, 𝑐). First of all we will consider proper representations

$m=(a,\ b,\ c)\ (\alpha ,\ \gamma )^{2}$

where α, γ are co-primes. Determine integers β, δ such that $\alpha \delta -\beta \gamma =1$, and apply to (𝑎, 𝑏, 𝑐) the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$; the new form will be (𝑚, 𝑛, 𝑙), where

$n^{2}-4ml={\mbox{D}}=b^{2}-4ac$.

Consequently $n^{2}={\mbox{D}}\ ({\mbox{mod}}\ 4m)$, and D must be a quadratic residue of 𝑚. Unless this condition is satisfied, there is no proper representation of 𝑚 by any form of determinant D. Suppose, however, that $n^{2}={\mbox{D}}\ ({\mbox{mod}}\ 4m)$ is soluble and that 𝑛₁, 𝑛₂, &c. are its roots. Taking any one of these, say 𝑛_𝑖, we can find out whether (𝑚, 𝑛_𝑖, 𝑙_𝑖) and (𝑎, 𝑏, 𝑐) are equivalent; if they are, there is a substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$ which converts the latter into the former, and then $m=a\alpha ^{2}+b\alpha \gamma +c\gamma ^{2}$. As to derived representations, if $m=(a,\ b,\ c)\ (tx,\ ty)^{2}$, then 𝑚 must have the square factor $t^{2}$, and $m/t^{2}=(a,\ b,\ c)\ (x,\ y)^{2}$; hence everything may be made to depend on proper representation by primitive forms.

35. Automorphs. The Pellian Equation.—A primitive form (𝑎, 𝑏, 𝑐) is, by definition, equivalent to itself; but it may be so in more ways than one. In order that (𝑎, 𝑏, 𝑐) may be transformed into itself by the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$, it is necessary and sufficient that

${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}={\begin{pmatrix}{1 \over 2}(t+bu),&-cu\\au,&{1 \over 2}(t-bu)\end{pmatrix}}$

where (𝑡, 𝑢) is an integral solution of

$t^{2}-{\mbox{D}}u^{2}=4$.

If D is negative and $-{\mbox{D}}>4$, the only solutions are $t=\pm 2,\ u=0$; ${\mbox{D}}=-3$ gives $(\pm 2,0),\ (\pm 1,\pm 1)$; ${\mbox{D}}=-4$ gives $(\pm 2,0),\ (0,\pm 1)$. On the other hand, if ${\mbox{D}}>0$ the number of solutions is infinite and if (𝑡₁, 𝑢₁) is the solution for which 𝑡, 𝑢 have their least positive values, all the other positive solutions may be found from

${t_{n}+u_{n}{\sqrt {\mbox{D}}} \over 2}=\left({t_{1}+u_{1}{\sqrt {\mbox{D}}} \over 2}\right)^{n}\ (n=2,\ 3,\ 4\ .\ .\ .)$.

The substitutions by which (𝑎, 𝑏, 𝑐) is transformed into itself are called its automorphs. In the case when ${\mbox{D}}=0\ ({\mbox{mod}}\ 4)$ we have $t=2{\mbox{T}}$, $u=2{\mbox{U}}$, ${\mbox{D}}=4{\mbox{N}}$, and (T, U) any solution of

${\mbox{T}}^{2}-{\mbox{NU}}^{2}=1$.

This is usually called the Pellian equation, though it should properly be associated with Fermat, who first perceived its importance. The minimum solution can be found by converting ${\sqrt {N}}$ into a periodic continued fraction.

The form (𝑎, 𝑏, 𝑐) may be improperly equivalent to itself; in this case all its improper automorphs can be expressed in the form

${\begin{pmatrix}\lambda &,&(\kappa +b\lambda )/2a\\(\kappa -b\lambda )/2c&,&-\lambda \end{pmatrix}}$

where $\kappa ^{2}-{\mbox{D}}\lambda ^{2}=4ac$. In particular, if $b\equiv 0\ ({\mbox{mod}}\ a)$ the form (𝑎, 𝑏, 𝑐) is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.

36. Characters of a form or class. Genera.—Let $(a,\ b,\ c)$ be any primitive form; we have seen above (§ 32) that if $\alpha ,\ \beta ,\ \gamma ,\ \delta$ are any integers

$4(a\alpha ^{2}+b\alpha \gamma +c\gamma ^{2})(a\beta ^{2}+b\beta \delta +c\delta ^{2})=b'^{2}-(\alpha \delta -\beta \gamma )^{2}{\mbox{D}}$

where $b'=2a\alpha \beta +b(\alpha \delta +\beta \gamma )+2c\gamma \delta$. Now the expressions in brackets on the left hand may denote any two numbers 𝑚, 𝑛 representable by the form (𝑎, 𝑏, 𝑐); the formula shows that 4𝑚𝑛 is a residue of D, and hence 𝑚𝑛 is a residue of every odd prime factor of D, and if 𝑝 is any such factor the symbols $\left({m \over p}\right)$ and $\left({n \over p}\right)$ will have the same value. Putting $(a,\ b,\ c)=f$, this common value is denoted by $\left({f \over p}\right)$ and called a quadratic character (or simply character) of 𝑓 with respect to 𝑝. Since 𝑎 is representable by $f(x=1,y=0)$ the value $\left({f \over p}\right)$ is the same as $\left({a \over p}\right)$. For example, if D = −140, the scheme of characters for the six reduced primitive forms, and therefore for the classes they represent, is

	$\left({f \over 5}\right)$	$\left({f \over 7}\right)$
(1, 0, 35) (4, ±2, 9)	+	+
(5, 0, 7) (3, ±2, 12)	−	−

In certain cases there are supplementary characters of the type $\left({-1 \over f}\right)$ and $\left({2 \over f}\right)$, and the characters $\left({f \over p}\right)$ are discriminated according as an odd or even power of 𝑝 is contained in D; but in every case there are certain combinations of characters (in number one-half of all possible combinations) which form the total characters of actually existing classes. Classes which have the same total character are said to belong to the same genus. Each genus of the same order contains the same number of classes.

For any determinant D we have a principal primitive class for which all the characters are +; this is represented by the principal form (1, 0, −𝑛) or (1, 1, −𝑛) according as D is of the form 4𝑛 or 4𝑛+1. The corresponding genus is called the principal genus. Thus, when D=−140, it appears from the table above that in the primitive order there are two genera, each containing three classes; and the non-existent total characters are $+-$; and $-+$.

37. Composition.—Considering X, Y as given lineo-linear functions of (𝑥, 𝑦), (𝑥′, 𝑦′) defined by the equations

${\mbox{X}}=p_{0}xx'+p_{1}xy'+p_{2}x'y+p_{3}yy'$

${\mbox{Y}}=q_{0}xx'+q_{1}xy'+q_{2}x'y+q_{3}yy'$

we may have identically, in 𝑥, 𝑦, 𝑥′, 𝑦′,

$(A,\ B,\ C)\ (X,\ Y)^{2}=(a,\ b,\ c)\ (x,\ y)^{2}\times (a',\ b',\ c')\ (x',\ y')^{2}$

and, this being so, the form (A, B, C) is said to be compounded of the two forms (𝑎, 𝑏, 𝑐), (𝑎′, 𝑏′, 𝑐′), the order of composition being indifferent. In order that two forms may admit of composition into a third, it is necessary and sufficient that their determinants be in the ratio of two squares. The most important case is that of two primitive forms φ, χ of the same determinant; these can be compounded into a form denoted by φχ or χφ which is also primitive and of the same determinant as φ or χ. If A, B, C are the classes to which φ, χ, φχ respectively belong, then any form of A compounded with any form of B gives rise to a form belonging to C. For this reason we write C=AB=BA, and speak of the multiplication or composition of classes. The principal class is usually denoted by 1, because when compounded with any other class A it gives this same class A.

The total number of primitive classes being finite, ℎ, say, the series A, A², A³, &c., must be recurring, and there will be a least exponent 𝑒 such that ${\mbox{A}}^{e}=1$. This exponent is a factor of ℎ, so that every class satisfies ${\mbox{A}}^{h}=1$. Composition is associative as well as commutative, that is to say, (AB)C=A(BC); hence the symbols A₁, A₂,. . . A_ℎ for the ℎ different classes define an Abelian group (see Groups) of order ℎ, which is representable by one or more base-classes B₁, B₂, . . . B_𝑖 in such a way that each class A is enumerated once and only once by putting

$A=B_{1}^{x}B_{2}^{y}...B_{i}^{z}\qquad (x\leq m,\,y\leq n,\,.\,.\,.z\leq p)$

with $mn\,.\,.\,.\,p=h$, and $B_{1}^{m}=B_{2}^{n}=...=B_{i}^{p}=1$. Moreover, the bases may be so chosen that 𝑚 is a multiple of 𝑛, 𝑛 of the next corresponding index, and so on. The same thing may be said with regard