Page:EB1911 - Volume 19.djvu/893

has only been done in the cases when 𝑛＝2, 3 (the latter by Smith in Proc. Lond. Math. Soc. ix. p. 242).

For moderate values of 𝚫 the difficulty can generally be removed by constructing algebraic functions of 𝑗. Suppose we have an irreducible equation

𝑥^𝑚＋𝑐₁𝑥^𝑚－1＋ . . . ＋𝑐_𝑚＝0,

the coefficients of which are rational functions of 𝑗(𝜔). If we apply any modular substitution 𝜔′＝𝖲(𝜔), this leaves the equation unaltered, and consequently only permutates the roots among themselves: thus if 𝑥₁(𝜔) is any definite root we shall have 𝑥₁(𝜔′)＝𝑥_𝑖(𝜔), where 𝑖 may or may not be equal to 1. The group of unitary substitutions which leave all the roots unaltered is a factor of the complete modular group. If we put 𝑦＝𝑥(𝑛𝜔), 𝑦 will satisfy an equation similar to that which defines 𝑥, with 𝑗′ written for 𝑗; hence, since 𝑗, 𝑗′ are connected by the equation 𝑓₁(𝑗, 𝑗′)＝0, there will be an equation 𝜓(𝑥, 𝑦)＝0 satisfied by 𝑥 and 𝑦. By suitably choosing 𝑥 we can in many cases find 𝜓(𝑥, 𝑦) without knowing 𝑓₁(𝑗, 𝑗′); and then the equation 𝜓(𝑥, 𝑥)＝0 defines a set of singular moduli, each one of which belongs to a certain value of 𝜔 and all the quantities derived from it by the substitutions which leave 𝑥(𝜔) unaltered.

As one of the simplest examples, let 𝑛＝2, 𝑥³－𝑗(𝜔)＝𝑦³－𝑗(𝜔′)＝0. Then the equation connecting 𝑥, 𝑦 in its complete form is of the ninth degree in each variable; but it can be proved that it has a rational factor, namely

𝑦³－𝑥²𝑦²＋495𝑥𝑦＋𝑥³－2⁴ . 3³ . 5³＝0,

and if in this we put 𝑥＝𝑦＝𝑢, the result is

𝑢⁴－2𝑢³－495𝑢²＋2⁴.3³.5³＝0,

the roots of which are 12, 20, － 15, － 15. It remains to find the values of 𝜔, to which they belong. Writing 𝛾₂(𝜔)＝∛𝑗, it is found that we may define 𝛾₂ in such a way that 𝛾₂(𝜔＋1)＝𝑒^{－2𝜋𝑖/3}𝛾₂(𝜔), 𝛾₂(－𝜔^－1)＝𝛾₂(𝜔), whence it is found that

$\gamma _{2}\left({\alpha \omega +\beta \over \gamma \omega +\delta }\right)=e^{-{2\pi i \over 3}(\gamma \delta +\gamma \alpha +\beta \delta -\beta \delta \gamma ^{2})}\gamma _{2}(\omega )$.

We shall therefore have 𝛾₂(2𝜔)＝𝛾₂(𝜔) for all values of [𝜔?] such that

$2\omega ={\alpha \omega +\beta \over \gamma \omega +\delta }$, 𝛼𝛿－𝛽𝛾＝1, 𝛾𝛿＋𝛾𝛼＋𝛽𝛿－𝛽𝛿𝛾²≡0 (mod 3).

Putting (𝛼, 𝛽, 𝛾, 𝛿)＝(0, －1, 1, 0) the conditions are satisfied, and 2𝜔＝𝑖√2. Now 𝑗(𝑖)＝172𝛿, so that 𝛾₂(𝑖)＝12; and since 𝑗(𝜔) is positive for a pure imaginary, 𝛾₂(𝑖√2)＝20. The remaining case is settled by putting

${\omega \over 2}={\alpha \omega +\beta \over \gamma \omega +\delta }$,

with 𝛼, 𝛽, 𝛾, 𝛿 satisfying the same conditions as before. One solution is (－1, 2, 1, 1) and hence 𝜔²＋3𝜔＋4＝0, so that $\gamma _{2}\left({-3+i{\sqrt {7}} \over 2}\right)=-15$.

Besides 𝛾₂, other irrational invariants which have been used with effect are 𝛾₃＝√(𝑗－172𝛿), the moduli 𝜅, 𝜅′, their square and fourth roots, the functions 𝑓, 𝑓₁, 𝑓₂ defined by

$f=2^{1 \over 6}(\kappa \kappa ')^{-{1 \over 12}}$, $f_{1}={\sqrt[{4}]{\kappa '}}\ .f,\ f_{2}={\sqrt[{4}]{\kappa }}.\,f$,

and the function 𝜂(𝑛𝜔)/𝜂(𝜔) where 𝜂(𝜔) is defined by

$\eta (\omega )=q^{1 \over 12}\sum _{-\infty }^{+\infty }(-1)^{s}q^{3s^{2}+s}={1 \over {\sqrt {3}}}\theta _{11}\left({2 \over 3},{\omega \over 3}\right)=q^{1 \over 12}\prod _{1}^{\infty }(1-q^{2s})$.

72. Another powerful method, developed by C. F. Klein and K. E. R. Fricke, proceeds by discussing the deficiency of 𝑓₁(𝑗, 𝑗′)＝0 considered as representing a curve. If this deficiency is zero, 𝑗 and 𝑗′ may be expressed as rational functions of the same parameter, and this replaces the modular equation in the most convenient manner. For instance, when 𝑛＝7, we may put

𝑗＝(𝜏²＋13𝜏＋49)(𝜏²＋5𝜏＋1)³/𝜏＝𝜙(𝜏), 𝑗′＝𝜙(𝜏′),

𝜏𝜏′＝49.

The corresponding singular moduli are found by solving 𝜙(𝜏)＝𝜙(𝜏′). For deficiency 1 we may find in a similar way two auxiliary functions 𝑥, 𝑦 connected by some simple equation 𝜓(𝑥, 𝑦)＝0 not exceeding the fourth degree, and such that 𝑗, 𝑗′ are each rational functions of 𝑥 and 𝑦.

Hurwitz has extended this field of research almost indefinitely, not only by generalising the formulae for class-number sums, such as that in § 69, but also by bringing the modular-function theory into connexion with that of algebraic correspondence and Abelian integrals. A comparatively simple example may help to indicate the nature of these researches. From the formulae given at the beginning of § 67, we can deduce, by actual multiplication of the corresponding series,

${1 \over \pi }\theta '_{11}\theta _{00}=\theta ^{2}{}_{00}\theta _{01}\theta _{10}=\sum _{-\infty }^{+\infty }\left({-1 \over \xi }\right)|\xi |q\xi ^{2}/_{4}\times \sum _{-\infty }^{+\infty }q\eta ^{2}\quad {\begin{cases}\xi =\pm 1,\ \pm 3,...;\\\eta =0,\ \ \pm 1,\ \ \pm 2,...\end{cases}}$

＝𝚺𝜒(𝑚)𝑞^𝑚/4    {𝑚＝1, 5, 9, . . .

where

$\chi (m)=\sum \left({-1 \over \xi }\right)|\xi |$

extended over all the representations 𝑚＝𝜉²＋4𝜂². In a similar way

1/𝜋𝜃′₁₁𝜃₁₀＝𝜃₀₀𝜃₁₀² 𝜃₀₁＝2𝚺(－1)^{1/4(𝑚－1)}𝜒(𝑚)𝑞^𝑚/2

1/𝜋𝜃′₁₁𝜃₁₀＝𝜃₀₀𝜃₁₀𝜃₀₁² ＝2𝚺(－1)^{1/4(𝑚－1)}𝜒(𝑚)𝑞^𝑚/4

If, now, we write

$j_{1}(\omega )=\sum {(-1)^{{1 \over 4}(m-1)}\chi (m) \over m}q^{m/2},\ \ \ j_{2}(\omega )=2\sum {(-1)^{{1 \over 4}(m-1)}\chi (m) \over m}q^{m/4}$,

$j_{3}(\omega )=2\sum {\chi (m) \over m}q^{m/4}$

we shall have

𝑑𝑗₁:𝑑𝑗₂:𝑑𝑗₃＝𝜃₁₀:𝜃₀₁:𝜃₀₀

where 𝜃₁₀, 𝜃₀₁, 𝜃₀₀ are connected by the relation (§ 67)

𝜃₁₀⁴＋𝜃₀₁⁴－𝜃₀₀⁴＝0

which represents, in homogeneous co-ordinates, a quartic curve of deficiency 3. For this curve, or any equivalent algebraic figure, 𝑗₁(𝜔), 𝑗₂(𝜔) and 𝑗₃(𝜔) supply an independent set of Abelian integrals of the first kind. If we put 𝑥＝√𝜅, 𝑦＝√𝜅′, it is found that

$\int {dx \over y^{3}}={1 \over 2}j_{3}(\omega ),\quad \int {dx \over y^{2}}={1 \over 2}j_{2}(\omega ),\quad \int {x\,dx \over y^{3}}={1 \over 2}j_{1}(\omega )$,

so that the integrals which the algebraic theory gives in connexion with 𝑥⁴＋𝑦⁴－1＝0 are directly identified with 𝑗₁(𝜔), 𝑗₂(𝜔), 𝑗₃(𝜔) provided that we put 𝑥＝√𝜅(𝜔).

Other functions occur in this theory analogous to 𝑗₁(𝜔), but such that in the 𝑞-series which are the expansions of them the coefficients and exponents depend on representations of numbers by quaternary quadratic forms.

73. In the Berliner Sitzungsberzchte for the period 1883–1890, L. Kronecker published a very important series of articles on elliptic functions, which contain many arithmetical results of extreme elegance; some of these Kronecker had announced without proof many years before. A few will be quoted here, without any attempt at demonstration; but in order to understand them, it will be necessary to bear in mind two definitions. The first relates to the Legendre-Jacobi symbol $\left({a \over b}\right)$. If 𝑎, 𝑏 have a common factor we put $\left({a \over b}\right)=0$; while if 𝑎 is odd and 𝑏＝2^ℎ𝑐, where 𝑐 is odd, we put $\left({a \over b}\right)=\left({2^{h} \over a}\right)\left({a \over c}\right)$. The other definition relates to the classification of discriminants of quadratic forms. If 𝖣 is any number that can be such a discriminant, we must have 𝖣≡0 or 1 (mod. 4), and in every case we can write 𝖣＝𝖣₀𝖰², where 𝖰² is a square factor of 𝖣, and 𝖣₀ satisfies one of the following conditions, in which 𝖯 denotes a product of different odd primes:—

		𝖣₀	＝	𝖯,	with	𝖯	≡	1 (mod 4)
		𝖣₀	＝	4𝖯,		𝖯	≡	－1 (mod 4)
		𝖣₀	＝	8𝖯,		𝖯	≡	±1 (mod 4)

Numbers such as 𝖣₀ are called fundamental discriminants; every discriminant is uniquely expressible as the product of a fundamental discriminant and a positive integral square.

Now let 𝖣₁, 𝖣₂ be any two discriminants, then 𝖣₁𝖣₂ is also a discriminant, and we may put 𝖣₁𝖣₂＝𝖣＝𝖣₀𝖰², where 𝖣₀ is fundamental: this being done, we shall have

$\tau \sum _{h=1}^{h=\infty }\sum _{k=1}^{k=\infty }\left({{\mbox{D}}_{1}{\mbox{Q}}^{2} \over h}\right)\left({{\mbox{D}}_{2}{\mbox{Q}}^{2} \over k}\right){\mbox{F}}(hk)$

$={1 \over 2}\sum _{a,b,c}\left[\left({{\mbox{D}}_{1} \over {\mbox{A}}}\right)+\left({{\mbox{D}}_{2} \over {\mbox{A}}}\right)\right]\sum _{m,n}\left({{\mbox{Q}}^{2} \over m}\right){\mbox{F}}(am^{2}+bmn+cn^{2})$

where we are to take ℎ, 𝑘＝1, 2, 3, . . .＋∞ ; 𝑚, 𝑛＝0, ±1, ±2, . . . ±∞ except that, if 𝖣＜0, the case 𝑚＝𝑛＝0 is excluded, and that, if 𝖣＞0, (2𝑎𝑚＋𝑏𝑛)𝖳⩾𝑛𝖴 where (𝖳, 𝖴) is the least positive solution of 𝖳²－𝖣𝖴²＝4. The sum $\sum _{a,b,c}$ applies to a system of representative primitive forms (𝑎, 𝑏, 𝑐) for the determinant 𝖣, chosen so that 𝑎 is prime to 𝖰, and 𝑏, 𝑐 are each divisible by all the prime factors of 𝖰. 𝖠 is any number prime to 2𝖣 and representable by (𝑎, 𝑏, 𝑐); and finally 𝜏＝2, 4, 6, 1 according as 𝖣＜－4, 𝖣＝－4, 𝖣＝－3 or 𝖣＞0. The function 𝖥 is quite arbitrary, subject only to the conditions that 𝖥(𝑥𝑦)＝𝖥(𝑥)𝖥(𝑦), and that the sums on both sides are convergent. By putting 𝖥(𝑥)＝𝑥^－1－𝜌, where 𝜌 is a real positive quantity, it can be deduced from the foregoing that, if 𝖣₂ is not a square, and if 𝖣₁ is different from 1,

$\tau {\mbox{H}}({\mbox{D}}_{1}{\mbox{Q}}^{2}){\mbox{H}}({\mbox{D}}_{2}{\mbox{Q}}^{2})=\lim _{\rho \rightarrow 0}\sum _{a,b,c}\left({{\mbox{D}}_{1} \over {\mbox{A}}}\right)\sum _{m,n}\left({{\mbox{Q}}^{2} \over m}\right)(am^{2}+bmn+cn^{2})^{-1-\rho }$

where the function 𝖧(𝑑) is defined as follows for any discriminant 𝑑:—

	𝑑＝－𝚫＜0	$\tau {\mbox{H}}(d)={2\pi \over {\sqrt {\Delta }}}h(-\Delta )$
	𝑑＞0	${\mbox{H}}(d)={h(d) \over 2{\sqrt {d}}}\log {{\mbox{T}}+{\mbox{U}}{\sqrt {d}} \over {\mbox{T}}-{\mbox{U}}{\sqrt {d}}}$