has only been done in the cases when n =2, 3 (the latter by Smith in Proc. Lon-i. Math. Soc. ix. p. 242). For moderate values of A the difficulty can generally be removed by constructing algebraic functions of j. Suppose we have an irreducible equation

x"'-{-¢:1x""'1+ . . . -I-c, ,.=o,

the coefficients of which are rational functions ofj(w). If we apply any modular substitution w'=S(w), this leaves the equation unaltered, and consequently only per mutates the roots among themselves: thus if x, (<») is any definite root we shall have x1(w') = x;(w), 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(nw), y will satisfy an equation similar to that which defines x, with j' written forj; hence, since ], ]' are connected by the equation, (j, j') =o, there will be an equation m//(x, y) =o satisfied by x and y. By suitably choosing x we can in many cases find gl/(x, y) without knowing fig, j'); and then the equation 1//(x, x) =0 defines a set of singular mo uli, each one of which belongs to a certain value of w and all the quantities derived from it by the substitutions which leave x(w) unaltered. As one of the simplest examples, let n=2, xi*-j(w) =y3-j (w') =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

y”-x”;v2+495xy+x=*-2* - 3” - 5“=0.

and if in this we put x=y=u, the result is 114-2u3-495u2+24.33.5“=o,

the roots of which are 12, 2O, ' 15, - 15. It remains to find the values of w, to which they belong. Writing 'yg(w) = “/j, it is found that we may define 'yz in such a way that 'y2(w+1)=e"2ffi/3'y2(w), 1/2(-w )='y2(w), whence it is found that aw Hg - f?(~/5+1f1+B5-5512)

W2 (~, ¢, ,+5):e 'v2(w).

We shall therefore have 7z(2w) =-y2(w) for all values of such that 2<-> =3%%<15~/37 = I, '15 +ve -l-B5-551250 (mod 3). Putting (a, B, 'y, 5)=(o, -1, 1, 0) the conditions are satisfied, and 2w=i/2. Now j(i)=1725, so that ~/2(z)=12; and since j(w) is positive for a pure imaginary, '~/2(z/ 2) =20. The remaining case is settled by putting

°;=°;+¢',

2 'yw +5

with a, 13, 'y, 6 satisfying the same conditions as before. One solution is (-1, 2, 1, 1) and hence w2+3w-}-4=O, so that 72 =-15. Besides 'Yz, other irrational invariants which have been used with effect are 'v3==~/ (j-1725), the moduli K, »<', their square and fourth roots, the functions f, fi, fz defined by f=2i('<'<')"I”' fi = VK' ~f1 fi = if '<-f» and the function 1;(nw)/q(w) where 1;(w) is defined by +°° $2-l-s I 2 of °°

= nr E (-)S 3 =L9 -, - =q1'2II(I -gh). n(<»> 9 oo I Q, ,311(3 3) I

72. Another powerful method, developed by C. F. Klein and K. E. R. Fricke, proceeds by discussing the deficiency of fi(j, j') =o 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

1= f- 2+'3'+“"lf'2+5*+'>' =¢<f>, f' =¢<f'>, -r-r' = 49.

The corresponding singular moduli are found by solving ¢»§ r) = ¢(-r'). For dehciency I we may find in a similar way two auxiliary functions x, y connected by some simple equation ul/(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 theo ry into connexion with that of- algebraic correspondence and Abelian integrals. A comparatively simple example may help to lndicate the nature of these researches. From the formulae given at the beginning of § 67, we can deduce, by actual multiplication of the corresponding series,

+°° - “l'°° = .;

f;e'uo»=e=we°1e..= is lslqe/.><iq»” “iff .., , "ZX(m)qm/4 ['m='I| 5191 ' ' °

where

x<m> =>2 Isl

extended over all the representations m = E” -|-4112. In a' similar way ? 9'l10l.0 = 0009102601 = 22 ('- I)¥('”'1) X gm/9 E 9'1;901 = 0009109012 = Z( I) f(m“I)x(m) gm/4 If, now, we write

Mm):E (- 1)}(Z1)X(m) gm/2, jim) =22(- I)i(1;;1)X(m)gm/4, j3(w) =2E qm/4

we shall have

di1:dj2:dj3'=610:001:600

where610, 001, 000, are connected by the relation (§ 67) 0104'l'9014 '600 4 =0

which represents, in homogeneous co-ordinates, a quartic curve of deficiency 3. For this curve, or any equivalent algebraic figure, j, (¢0), j2(w) and j3(<») supply an independent set of Abelian integrals of the first kind. If we put x=/»<, y=/f<', it is found that d d d

§§ -=a].<w>. § =a2<<»>. $=iJ.<»»>.

so that the integrals which the algebraic theory gives in connexion with x4-l-y4-I=0 are directly

provided that we put x=/ n(w).

Other functions occur in this theory analogous to j1(w), 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 identified with jifw), j.(<.,), j, <..,), Legendre-]acobi symbol If a, b have a common factor we put (9 =0; while if a is odd and b=2"c, where c is odd, we put n

(lg) = . 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 DE0 or I (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 PEI (mod 4)

=4P, PE - 1 (mod 4)

=8P, PE =f= 1 (mod 4)

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 Do is -fundamental: this being done, we shall have

lf f 'fi (Bti) (Df)

Do

Do

Numbers such as~ D0

T2 2 - 4-Ffhk)

D) T (Dil (Qi)

=l2 E-Fam* bmnc2

2 a b Cl;< + A m, " in < + + n)

wherewearetotake h, k=1,2, 3, . .+°°; m, n=0, =*=I, *2, . .. *oo except that, if D<0, the case m=n=o is excluded, and that, if D>o, (2am+bn)T§ nU Where (T, U) is the least positive solution of T2-DU2=4. The sum 2 applies to a system of representative a, b, c

primitive forms (a, b, c) for the determinant D, chosen so that a is prime to Q, and b, 5 are each divisible by all the prime factors of Q. A is any number prime to 2D and representable by (a, b, 6); and finally -r=2, 4, 6, I according as D'<-4, D= -4, D= 'Eg of 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 S1d€3S are convergent. By putting F(x) =x-I-P, where P is a real positive quantity, it can be deduced from the foregoing that, if D2 is not a square, and if Di is different from 1,

2

fH(DiQ')H(D2Q“)=Lt 2: E (Q) <am2+1>m»+¢»2>-=-P p=o a, b, c A "'~" m

where the function H(d) is defined as follows for any discriminant d:- d = -A <0 fH<d> =-f% h< -A)

hon T+U/d

1> ° Hld) “ 2Ti1Og' l"- Uv E