Concerning Certain Elliptic Modular Functions of Square Rank

Concerning Certain Elliptic Modular Functions of Square RankAuthor(s): John A. MillerSource: American Journal of Mathematics, Vol. 27, No. 1 (Jan., 1905), pp. 47-68Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370095 .

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Concerning Certain Elliptic lModular Functions of Square Rank.

BY JOHN A. MILLER.

INTRODUCTION.

fn his " Vorlesungen* fiber die Theorie der Elliptischen Modulfunctionen," Professor Felix Klein has defined the functionst

n-1

Xa (Uh I Cal (i)2) = Ct | a+ (Ulal | J1 i2) ,( n %u=O v-n n

where u is the fundamental variable of the elliptic function; (a,, c, are the periods of the elliptic integral of the first kinid; e 0 or g according as n is odd or even, Ca is a quantity independent of u, and

or (UI, C Cal) = e n 2n(u) ) G (%-G1+[8X%1| ) (2)

where a (u I ca,, C) is the ordinary Weierstrassian sigma-function ; nl, n2, are the periods of elliptic integrals of the second kind and a, n, X, y are integers. He has shown that these functions are of rank (stufe$) n.

[ have, in what follows, investigated certain properties of these functions, and functions derived from them, when n is a square number, and have treated in detail the particular cases where n = 9 and n = 4.

In the future, when there is no possible ambiguity, I shall write, instead of X. (it I l, 2), the briefer forms X,a (u I ,, c2) or X,a (u); and instead of

n

* Hereafter I shall refer to this treatise as " Klein-Fricke." f Klein-Fricke, Vol. II, p. 261. t Klein, " Ueber die Normal Curven der nter Ordnung."



48 MILLER: Concerning Certain Elliptic

rA . (u I(cul, ' )2) the briefer forms or,, (u) and A (0) =A,. I shall enumerate n n

without proof certain properties of the functions defined by (1), viz.

Xa + n (U) Xa (u), (3)* X U(u) - 1))n Xy ((U), (4)*

L12tS + y + = (- i)n (+Y)We( 171 + -172)( + X1 +()2) (7r5)(A-2a)

a (U4)=(1)na e a1(u" 2n)X(' u - )n (6)*

There are n distinct functions Xa (u), and n only, and they are linearly independent.

There are two cases to be considered: Case I, n_ 1 (mod 2); Case II, nO0 (mod 2).

Case I.

1. Transformation of the X (u) by the Modular Substitutions when n is Odd.

The arbitrary constant Ca in Xa (u) may be so chosen that we may write, in case n is odd,

Xa (U I Ca 1 ( '72) =(1)V/Z e or 0n O t/ I (x7 n) (7)t

or

X-(U 16)1, 2) =(-1)a e n 6l (- (n-a&1) e-'o) (8)

twhereA =AQ1, '2)and

G nl-n1 - 2- n2

2w, 2%2

where, owing to the cogrediencell of 1, 6?2 and VlZ n2l

w1-w and - - n' n

* Klein-Fricke, Vol. II, p. 264, et seq.

f Klein-Fricke, Vol. II, p. 277. t For definition of A, see Klein-Fricke, Vol. I, p. 15. 1I Klein-Fricke, Vol. I, p. 122.



Modular Functions of Square Rank. 49

Professor Klein has shown* that there exists a finite group of linear substitutions on Za (u), of order 2Lt (n) which may be generated by repetitions and combinations of the operators S and T, which are defined as follows:

S. Ci) = 01 + )2, T: 01 = 6- 2 I) I) 6 C) = 6)2 G)2 = ?1 X (9)

27ri and that if e en,

-(n-a) a S X;t (u) E 2 Xa (U) i

n-I n-1 (10)

T: Xa(ub) = 2 E Ea t) V%In P =

Where S: X,, (u) means that X,a (u) has been subjected to the operation S. We shall now define a set of (n --1)/ 2 distinct functionst of c and ?2 by the

following equation: Xa (0) = Za ()1, ')2) = Za (say). (11)

If we put u = 0 and n = 9 in equations (10), we obtain a group of quater- nary substitutions on the Za generated by

-(9 - a) a

S: Z.' 2 Za}

T: Za E_ 1 .... 4). P=o

The order of this group is (9) p (9) (9) = 648. We shall call it G648.

2.- The Biquadratic Relations of the Za-

If we substitute in the well-known a-relation, t

a (Ul + U2) a (U3 + U4) a (Ul - U2) S (U3 -U4)

+a (Ul + U3) a (U1 -u3) a (U4 + U2) Ca (U4- U2)

+a (Ul + Uq4) (7t1 - U4) a (U2 + u3) a (U2 - U3) = 0,

* Klein-Fricke, Vol. II, p. 296.

f To show that there actually exists a set of functions of o, ?2 such as we have defined, see Klein- Fricke, Vol. II, p. 267, equation (12).

1 Schwarz. Formeln und Lehrsiitze zum Gebrauche der Elliptischen Fuxnctionen, p. 47.

7




= a __ , we obtain, after reduction,

2 n

Xal + a2 Xa3 + a4 7a2-a, Za4 -- a3 + Xal + a3 Xa2 + a4 Oa3-al Za2-a4

+KXal + a4 a2 + a3 Za4 - al Za8- 0. (a)*

Put in this equation u = 0, and we obtain

Zat + a2 Za3 + a4 Zag-al Za4 -a3 + Zal + as Za4 + a2 Za3-a, Za2-a4

+ zal + a4 Za2+a3 Za4 - al Za3. a2- 0. (b)

The left side of equation (b) vanishes for all values of ca, Ca2. It is our pur- pose to find all such expressions for the case n = 9. Inspection shows that the left side of equation (b) vanishes identically in Zal if any subscript of Za is con- gruernt zero modulo n.

If we write a, + a2 + a3 + a4 =8, then any pair of X's occurring in (a) may be put in the form X; . X, -i. Therefore, by virtue of equation (3), s runs through the residue system, modulo 9.

By virtue of equation (a), any X.-pair XYXX -i can be expressed in terms of two pairs, XkX -,kXlX8 -, arbitrarily chosen; and any other relation can be obtained by the elimination of one of these pairs from two independent equations, provided the determinant of the coefficients of XkX8 and XX8- does not vanish.

Choose then: the pairs X1X-1, X2X82; whence

al + a3 1,

a2 + a4 =s-1, (mod 9).

a3+a2=s - 21

Making substitutions in equation (b) consistent with the foregoing, we obtain all relations except in the case s = 3, when the equation vanishes identically in Za. They are

z2 Z1 Zs Z2 + Z4 Z = 0 = S, (say),

zlz4+ z2z4 -z3z= = S2 (say),

-Z Z2 + Z2 Z4 + Z Z4 = 0 = S(say).

* Here, and throughout this section, Xa means X. (u).




Since no relation exists in case s = 3 involving the X.-pair X1X2, there remains only one relation, and it is found to be

-Z2Zl+ZSZ4+Z2Z4=O=84 (say).

If we regard zl, z2, z3, z4 as homogeneous coordinates in a space of three dimensions, and if 8 -Z4 84 Z= 85, then the curve

84=0, 85=0

is the complete intersection of the four surfaces

S?i = 0 = S2= 8S = 84.

The curve is of ninth degree.

3.-The Invariant Tetrahedron. The Functions ta.

Regarding again zl, Z2, z3, z4 as homogeneous coordinates in a space of three dimensions, we slhall show first, that no plane is invariant under the substitutions of

G848. Let A1z1 + A2.z2 + A3Z3 + A4Z4 = 0

be the equation of any plane. Then

*S: A1z1 + A2Z2 + A3Z3 + A4z4 = e5A1zl + e2A2z2 + A3Z3 + ,cA4z4.

Hence the only planes unchanged by S are z- = 0, z2 = 0, Z3 = 0 Z4 = 0. None of these are unchanged by T. This proves the theorem.

Let us find now the equations of all tetrahedrons invariant under the group of substitutions G648.

Any tetrahedron invariant under S is evidently invariant under S>. Let

Alzl + A2Z2+ A3z3 + A4z4 = 0 = P

be the equation of a plane of the invariant tetrahedron.

8s: Alz + A2z' + A3z' + A4z4 = Aje6z1 + A2e6z2 + A3z3 + A4e0z4 = Q 0,

As: Alzl + A252 + A3z' + A44 = A1e3zl + A2e3z2 + A3Z3 + A4e3z4 = R = 0.

* By this notation is meant that we apply the operation S to YAizi and derive IZA.z. But we

know that this operation yields the expression on the right of the equation. We shall hereafter use

this notation.




Since (S3)3 = 1,

the operation S3 either leaves the planes P = 0, Q = 0 and R = 0 each unchanged or permutes them. Hence S leaves at least one plane of the tetrahedron unchanged.

Suppose P1 = Alz1 + A2Z2 + A3z3 + A4z4 =0

be the equation of this plane, it is invariant under S if, and only if,

or=A1 = 2A4=- (A) or A3- =O.J

Let us choose first, the first set of relations; then

ZS= 0

is one of the planes of the tetrahedron. The equation of the other three planes may be found, if an invariant tetrahedron exists, by subjecting Z3 to any three operations of the group such as TvSu, v * 0. The equation of the tetrahedron is found to be

Z3(Z1 -Z2 + Z4)(E Z4 - Z2 + E Z4)(ZI - 63Z2 + e6Z4) =0 (13)

The plane A1z1 + A2z2 + A4z4 = 0 = L (say),

is unchanged by S3, and hence, reasoning as before, one plane at least of the tetrahedron is unchanged by S. The only planes unchanged by S are

Z-=0; Z2 = z4=0 and z3=O.

The last one we have discussed. Let us examine the others: Let

T1: z: L, TS: zI M, TS2: z11.V.

Since S: N produces neither L, M nor z1, the tetrahedron

z1LMN= 0

is not invariant under the substitutions of the group. Hence z1 = 0 is not one of the planes of the tetrahedron. In a similar way, it may be shown that

-= 0 and z4 = 0 are not planes of the tetrahedron. Hence, the only tetrahe-




dron invariant under C648 is given by equation (13). Put now

3t, = Zl -Z2 + Z4 1

3t2 -=5z - E2Z2 + E8Z4 (14) 3t3 = 8 Z1- 64Z2 + 47Z4j

3t4 = (F3 -_ 6) Z3. We have immediately

S t= t2,i T: t - - t4

t28-= tS t2' _-ee3t} (16)

3-- (4 t4 t

The substitutions defined by (15) and (16) form a group of unary substitutions, holoedric isomorphic with G648. I shall call this group G648.

The functions a, ,, when subjected to S and T defined by (9) are inter-

changed among themselves as follows:

fig at,1 (T 01 T: ao,' = 1, (T/ o l, 1) (T/ - - , 1,0= 1,12 10 I

1,2 61, 0 i's 1 1

Comparing the substitutions on the or, with those* on t.a, we see that aYf, , 3' 3 ~~~~~~~9

a1, 2, i, are cogredientt with ti, t2, t, tL.

The functions a, ,, and t{ are functions of cal, W/.

We may find the relations connecting them by considering the theory of the transformation of the 0-functions in terms of which both may be expressed. They are:

04, + = J (a) AtX ,

* Since ta are derived from za and za from Xa (u), when it is especially desirable to emphasize the relation of a to 9, we shall write t a instead of ta and z a instead of Za.

9 _

t Burnside and Panton, Theory of Equations, p. 384.




4.-The Invariant Functions of ta.

By an invariant function we shall mean a rational integral function of ta defined by equation (14), which does not vanish identically in taI with degree greater than zero, and which remains unchanged by all the substitutions of G648. Such functions exist, for the elementary symmetric functions of t', t2, t6, t4 are invariant under this group.

An i-lettered term is one containing i letters. An i-lettered function is a rational integral function containing only i-let-

tered terms. A form is a homogeneous integral function. We shall now seek all inva-

riant functions of ta 4

Any invariant integral functionFmay be written F=X a,F,, where F, are K=O

i-lettered functions and ak are independent of ta and do not all vanish. Since, under any substitution of G648, we change neither the number of letters nor the degree of any term, it follows that each 1F. consists of the sum of i-lettered forms, and is, therefore, itself an invariant form. Any four-lettered form is a rational integral function of t1t2t3t4 and one-, two- and three-lettered forms.

We shall now derive all three-lettered invariant forms. Choose arbitrarily one of the terms RX, where

R1=ttt, R2 = ta t t, R3 = t tt, R4 = tt4Y, (a)

where oa, , r are integers. Call the term so chosen the originiating term and denote it by R. Let the terms derived by subjecting R to the substitutions of G648 be denoted by Ri. It is evident that R1, R2,R3 and R4 will be of their number. Let Fa, be an invariant three-lettered function of degree a + 1 +'y. Then Fa, , , its the simplest symmetric function of RB; for, since it is unchanged by S and T, it is unchanged by S T". Therefore, the RB are all found among the terms of Fa, 8, . Moreover, F,,,,,, is of the form

Fi, ,O, y-aRi,

where ai is independent of t,. For, F is a form, and any operation except the addition of terms such as a,RI either destroys its homogeneity or changes its




degree. Some substitution o of G648 exists that changes Ri into Rj, and the neces- sary and sufficient condition that a: 1 ,., = , is that ai = aj.

Now, if Ri be either of the terms (a), then

T2: R -(- 1)(O0+ + Y) Rt

Hence, when we form the simplest symmetric function of the originating term and those obtained by operating with T2, this function vanishes identically unless

a+ + y O (mod 2). (b)

Hence, we conclude that this congruence holds.

Also, if R = ta to t3,y S3 RI = 6(a++Y)RI S6: R'I-=63fE r +P+v)R

Henice, the simplest symmetric function formed from the originating term and the terms resulting from the repeated applications of 83, vanishes identically unless

a + d + y_ O (mod 3). (c)

Hence, we conclude that congruence (c) holds; or that no three-lettered invariant function exists. But we have shown that some do exist.

If we choose R3 as the originating term, we may show in a similar manner that

a + 8 = O (mod 3),

therefore, yO0 (mod 3).

In a similar way we can show that

ac-0 (mod 3)

and 8 O0 (mod 3).

We conclude, therefore, that in all three-lettered forms the exponents satisfy the congruences

a +3++ O (mod 6), (d) at-n- _t O (mod 3)w .

Choosing as our originating term tl t t., vve obtain as a three-lettered inva-




riant form

Fa y ta t, t3 + tn t, ty + ta t, t2y

+ ( 1 )[tl t 4 + 2 t t+ 3 2 4]

+ (- 1) [t4 tg t3 + t4a t t'y + t4m tg t-]( 7

+( ),Y [ta t, ty + ta t tY+ ta p tflt]

Giving to a, 13, y in (17) all values satisfying the congruences (d), we obtain all possible three-lettered invariant forms. For, by making suitable interchanges of a, (3, y, we can find among the terms of (17) any of the four possible originating terms.

If we put in (17) one exponent, y 0, we get all two-lettered invariant functions,

Fa , =tla tg + t2a tO + t3a tg

' )a (ta tg + t2 t + 3 t2 ) 8 +(1), (tX t, + t'X t, + t4 3$ (

+(t3 tg + t1X tg + t1X t )

Put in (18),, d- and F.,t -Fx, we obtain

F.a= 3 (ta + ta + ta + ta) (19)

5.- The Basis of the System of Forms.

Giving to a, (, y all possible values satisfying the congruences (d), we obtain from equations (17)-(19) an infinity of forms, all invariant under the group G648. And, as we have shown that they are the only forms so constituted, we have a totality of forms definitely defined. We shall designate this totality of forms by n; By Hilbert's *Law we can express any form, F, of H, in the following form:

F E XF, where du is finite, i=l

tand A, are forms of H. The forms F, is called the basis of the system H.

* Hilbert, " Ueber die Theorie der Algebraischen Formen,"' in Mathematische Annalen, Band 86. Weber, Lehrbuch der Algebra, Band 2, p. 165.

t Weber, Lehrbuch der Algebra, Band 2, p. 170.



Modular Functions of Square Rank. .57

Problem: To find the basis of H. In order to solve the problem, we need the set of identities (a) .... (g), the

truth of which may be verified by conmputation. The summation under the summation signs, and the products under the product signs, are mnade from I .... 4, although it is not so indicated in the equations of the present section.

Xti6Fa, ,y = Fd,p+6,)y+ Fa,y+ 6 + +6, , y + 1t7. Fa-6,-6,y (a) Xtit6 Fa,p,= Fa,+e,V+G + Fa+6,,y 6 + Fa+6,+ 6,y

+ IIt' [Fa, P6.y-y.6

+ Fa-6,p,y.6 +Fa-B,-6,y], (b)

6t6 =F -L+6,P+6,y+6 + "ti [Fa_6,S,y + ?_ 6, y i -6 (C)

Therefore,

Xtq.* F. _ y-ttk* F,-6, , y + 5t6t6t' . F,-p12, 1, y F, - F (d) Xt I Fas fyEtq Y as6y+Eikt -2y ilt k IaX1,-F a + 61 e} (e

ti6- F ti6k* F" * , -12 ti. FF F, - + fi-F (g)

a, P,.y 6,.y + itlt I.~~as,3 y Fl3sa, 9 2 - y s a, Al - 18,

Making a = 0 in the equations (a), (b), (c), (d), (e), (f) and (g), we derive similar equations for F, ,

Assuming a 0 (mrod 6), the discussion is divided into two cases, according as

d-3_rO0 (mod 2) or {3-y_ 1 (mod 2).

Case I.

F fl and Fa, are symmetric with regard to t6; and each is accordingly a rational integral function* of the elementary symmetric functions of t?.

Case IL. We shall consider

a. Two-lettered forms.

Making a = 0 in equation (e), we can express every F,, as an integral function of F,,,, and Ff,+3,3,3' where 3' < 15. Since F ,,,= F- , we have

* Chrystal's Algebra, Vol. I, p. 440.

8




an expression for every F , in terms of F,, y, and F + 3, 3, 3,F + 3, 3, 3, where ', y' < 15. Whence ,B', yI can have the following sets of values:

3, 9; 3, 15; 9, 15.

But by (b), F9, , can be expressed as an integral function of F3,9 and certain

F,81y which shall be discussed under b.

b. Three-lettered forms.

By successive applications of the recursion formula3 (d), (e), (f) and formula (a), we can express any 1i's as an integral function of Fg,3, F15,3, the elementary symmetric functions of tB and F1,, 3 ,,,, where the simultaneous values of at, /1, y7 are given by

a! 6; 6; 6; 6; 6; 6; -3'=3; 3; 3; 15; 15; 15;

y'-3; 9; 15; 3; 9; 15.

We shall now show that every FI, fl, y can be expressed as an integral func- tioIi of the elementary symmetric function of ti, F9, 3, F15,3, FO,3,3, and F6, 9,3, and that none of the functions just named can be expressed as integral functions of forms of H of lower degree.

To express rationally and integrally one of these forms in terms of forms of lower degree, it must be involved in an equation with them; its coefficient must be independent of ti, and, since the equation must be homogeneous, the coefficients of the forms of lower degree are functions of t. The method is suffi- ciently well illustrated by solving one case-that in which the degree of the form is 30.

We have the following set of equations:

(E) F6, 9,9 F6,15, 9 + F12, 9,9 + F6, 9,15, (Dt t6) F6,3,3 - F1t (F6,9, 3-F6, a, 9) = F12,9,9, (Z ti t6 t6) F9, 3 - 1itF12, 3, 3 =F6, 15,9 F6, 9, 15

which enable us to solve for all forms of degree 30. Accordingly the elementary symmetric functions of t6, F9, 3, F15, F6,3,3, and

F6, 9, 3 constitute the bastIs of the System H.




6.- The Invariant Forms as Rational Functions of g2, Y3 and A.

Each ilnvariant function FI,. is a rational* function of J, where J

and, therefore, a rational function of g3 and a, where g3 has the usual signification.t

By equationis (ao, p. 12), and the relations found in Klein-Fricke, Vol. II, p. 30, we obtain

A33- 3 _=l 3

_4

til-

iV

3t3- 3 -Pi3 ~4

t3t <y3 + -_ _ 04 3

2ar whiere p-e

$Whence S =t6 0,

18 t6 t6 18

ti tk to, 216g3 (20)

O ts 3,%A/3

12 t3 4- iAais

anld F36 F9,3 - - ' I 1

F153 3, 2 16,-/3g3

F, 3,3 36 iA93p

F61 ,93= 3p2l6.q3 AA20

* Weber, Elliptische Functionen., p. 149. f Klein-Fricke, Vol. I, p. 15. t Klein-Fricke, Vol. I, p. 630.




From equations (20), it is evident that t6, t6, t6, t' are the roots of the equation

y4 +18y+216qs 27=0. Y' +ALY Y +26-

Case II. n-0 (mod 2).

7.-The Functions Xa (u I cl, ca2). The Functions Za .for Case n_ 0 (mod 2).

The arbitrary constant Ca, in the definition of Xa (b), may be determined so that we may write,* in case n is even,

Xa (u I (,), 6))= e ( I GnU2a n

or,

X ( | ) ^/2n7% n,%2

-a 2a

nzt awl24 n) Xa (U I (i, 6h) )~6)Ve 22.Z15 03 ____ r

C2 W

There are in this case also n linearly independent functions Xa (u) and no more. Professor Klein has shownt that the linear substitutions on Xa (u) corre-

sponding to the modular substitutions S and T (defined in equation (9)), to be

n +a2

S: Xa (?b = Xa(8?XG)) l(1

T:Xa (u) = 2 X ~a (U (61,6) 2 l i2

T:X,,u) - _apXO(t Cl 602. (21)

If weput Za---Xa(o)I

there are in the case n = 4 three distinct Zal Viz. zo, z1, z2. Putting n 4 and

u 0 in equation (21) and writing e= e 8 , we obtain

S: zo/ = CZ,,

ZI --

5Z

z = 662 (2

T: zj- (zo + 2z1 + z2), (22)

2 - (zo-Z2),

=3 - (zo - 2z1 + Z2) J

* Klein-Fricke, Vol. II, p. 285. t Ib.




We shall designate these substitutions on za as S and T. By composition of S and T, we obtain a group of linear substitutions on three variables of order 4+ (8) ik(8) 192. We shall call this group G*.

If we regard zo, zl, z2 as homogeneous coordinates in a plane, we may show exactly as in Case I, that no straight line is unchanged under the sutbstitutions S and T, and that

Z1 (Zo-Z2)(zO + Z2) = 0

is a triangle invariant under G192, and the only one.

Put uo zo + Z2, U1 -Zo Z2

u2= 2z,.

Then, from equations (22), we derive the substitution group of the ut. It is given by

: uM= EU1,

u'.2 = 2z,(23) T: uo'-- bO, ut=

I= U2,

U2 =- U1.

We shall call the group generated by equations (23) G192. By a consideration of the theory of transformation of the 0-functions, we may

find the relation between ua and aA, a They are

(1 +i) (U2) 1 92

2V%/2 4'

2V 2 )1 A'h (24)

C (i+i) eu2 ;f t i2,V2 4',,AJ

8.-The Invariant Functions of ua.

We shall now seek the functions of ua which are invariant uinder the substitutions of C192.



62 MILLER: Concerning Certatn Elliptic

Reasoning, as in Case I, we can show that every invariant form is an integral function of two-lettered invariant forms, one-lettered invariantfornms, and (uou1u2)a, where a is a positive integer. Choosing as our originating term a u z4 and forming the simplest symmetric function of terms arising from subjecting it to the operators of G192, we may show by reasoning, as in Case I, that every two- lettered invariant form may be obtained by giving to a, d all values in Fa, where

a-0 (mod 4), 3o (mod 4),

and where

FaIS= _?a4u + a+ e+UagUg + Ua 6 ? EL+ UaU + aaUa4 + (25)

Making i = 0 in (25), we get all invariant one-lettered functions, and if we write Fa, =o Fa we get

Fa= 2 (uoa + 6aUa + Ea'a)* (26)

We shall now seek the basis of this system of forms obtained by giving a, 3 all values in (25) and (26).

By calculation it is shown that

F4,4 Fa,t Fa, +4 F4 =Fa, + 8 + (U0 1U 2)4 (Fa, -4)* (a)

This is a recursion formula for /3, and since

F.,B 6+ ' FB, a}I

it is also a recursion formula for a. By the use of (a), we can express every Fa,, . as an integral function of

,at, O where a', /3 < 8. We have left, then, the form FX, ,, where a', j31 may have the pairs of values 0, 4; O, 8; 4, 4; 4, 8; 8, 8. But

2F4 '=F8- 2F4,4

F4F4, 4=4F8, 4- 21t4,

F2= 2F88 + fIU4. F4.

The basis of the system is, therefore, F4, F4, 4 and (It u1 u2)2.

Since these forms are invariant under S and T, each of these (and, therefore, every invariant function) can be expressed as a rational function of g2, g3 and A . We shall now find F4, F4,4, (u0o U1 U22 in terms of these invariants.




j Since 14,o + ao, 1 + (40 = 0,

P4 0.

t Now, 4 8X4 4 8%

4= 8 (%3- 4).

Therefore, from equations (24),

u4= 24A8 (X3-X4), Ub4 24A823,

'24 - 24A8X4.

t Therefore, F44 214Ag2 $4 u441 22 =22

Hence ul, ut, u2 satisfies the equation

X3 + 214Ag2x- 222A2 = 0.

Now, r0 u2 uf = ( ra, all multiplied into a power series of integral

powers of r with real coefficients A(?U) into a power series of integral

powers of r with real coefficients. If we now approach infinity along positive y-axis, we see that i2u,2tu has

the same sign as A. Therefore, we may write

uo u6 ulu= 211A.

9.-Linear Uonnection between a , (n?u) and X a (7.6). n n a

We shall show now that a , , (nu) is a linear homogeneous function of n'n

Xa (u) for any finite value of n.

_ (nu) = F( ~) e[(x + n(n -)1)) + (A + n2 n ])) (u-(X+ (n -) a) w+?2(L + (n -1)n)co2)

*-Ilay (7b nAD 1(2 n t n(Ml Ir M2(2)l =0 ... n- 1) . Ml1, 7a2

* Klein-Fricke, Vol. II, p. 29. t Ib., Vol. I, p. 628. t Klein-Fricke, Vol. I, p. 629.




This is a a-product of n2 factors multiplied by e with an exponent in which the

coefficient of u [ ? n (n - 1) + + 2 1)] r2, and whose residual

sum s l + 8Y2 + n (n- ) + n (n 1) W2; moreover, there are n2 differ- 2 2

ent quantities or, (nu). n n

If we define n2 functions by the equations

n-1

Xi Ci][Jr(u- a,k). 0 n2 1), k=O

where ci is independent of u and the residual sumn,

n2 -1

s = a,k o , (a) k=O

then by *Hermites' Law a, (nu) can be expressed as a linear homogeneous n n

function of xi. Also, if n is odd,

X (u) f (o, )e( o u I + 2

From this it appears that Xa (U) is a a-product of n2 factors multiplied by e W2

with an exponent which has the coefficient of u = an, + 22 and whose re-

sidual sum

n2 1

There are n9 different quantities X,a (u), and hence, according to Hermite's Law, ii2

Xa (u) can be expressed as a linear homogeneous function of xi defined by equa- n2

* Klein, Elliptischen Normal Curven der nter Ordnung, p. 354.




tion (a), and hence a, (nu) can in general be expressed as a linear homoge- n'

neous function of Xa (u).

If n is even,

n2~~~~~~~~~~~2+ +11

n2~ ~ ~ (? [(n + n)!1 fn' ?

Reproducing the argument used above, it follows that, in this case also, or, (nu) n' n

can be expressed as a linear homogeneous function of X, (u), and hence for any

finite value of n we may write

-1

afoi1(nu') =zN'AaXa (U)t (27)

az=1

where Aa is independent of u. We shall now determine A., of equation (27) in the case n is even. The

theory of the transformation of the theta-functions gives

1 +it

CT (nu) = V2 aeXn (U). (28)

n n

By equation (7) (Klein-Fricke, Vol. II, p. 27), the a(. ,a( nu) is transformed 2, 2 n n

into another ar, ,(nu) by both Sk and T. We shall indicate this transforma- _, n

tion by marking the transformed quantities with a prime. If we operate on the left side of equation (28) with T, we obtain

n2-1

T: X'fl (u.f) = 1 f/npyl (%), n2 Y=? n2

2vi

where '-e .

9




Whence, n-I n-i n2-1

T: EE-a3 X' =X a E-a ef- nlyX , (u) =03 =oO n =2 - YQ W2

n-1 n-1

= X (u) (1E a) + fXr (U),. (r + a)

ii2 P=0 W2 P=0

n-1

+ Xn2--1 (U)( P(2-1+a)) (29) O ,B=O

It is evident that n-I

E- (rj+a) 0

9 =0

if [(r + a), n] 1; for, if f is the least number such that

E-(r+a)f/ 1 =E-,n

then, f= + Hence, since f_ n, s = r + a. Therefore, f = n.

Hence u- (r + a) appertains t to the exponent n, and hence - ((r + a is an nth

root of unity, and if a * a/, E-a(r+ a) * E-a (+ a), where (a, a' 0 .... n- 1).

Moreover, if [(r+), n]=d, d <n,

E-(r + a) is an nth root of unity which appertains to the exponent c Hence -d

there are -a- different quantities E-k(r+ i, = .k . .. (j -I)). Therefore,

d

E-k(r+a) - 0. (a) k=O

Since E-k(r+a)- E-I(r?a)

if, and only if, k 1 (mod n) ,and since there are just d quantities < n satisfy-

ing this coongruence, the quantities -,I(r+a) may be arranged in d sets, each of

* This notation has the ordinary signification that the G. C. D. of r + a, and n is 1. f Mathews, Theory of Numbers, Part I, p. 185.




which is defined by the summation (a). Therefore,

(r + a) =)-o if [(r +a), n] =d:, d <n.

If r + a- 0(mod n), thene- & (r + a) 1 for'all values of P. We have now shown that the coefficients of XI in equation (29) vanishes

n,2

except when r + a 0 (mod n). Hence we may write

1 +i

T: a (nu) = V2 (30) n nV

where y satisfies the congruence

y + a= 0(mod n). (b)

Now there are just n quantities less than n2 satisfying this congruence, and since a is less than n, and, therefore, not divisible by n, neither is y. Hence, in the summation on the left of equation (29), there are n quantities, X, (u),

W2

not one of which is found in summation (28). Giving a all values from 0 to

n-1 ,we get n equations, each containing n quantities, X, (u). Moreover, an ,Yy (zu)

occurring in one equation derived from (29), by giving a all values, occurs in no other, for if so,

XI (ff) -X,(u) . n2 n2

Whence y +a=_ + a (mod n2). That is, y-y'+ a- a _O(mod n), which is impossible. We now apply S to one of the equations obtained by giving a all values in (30):

1+i Sk: ) 2 2 n2

+ 2

)kX (31) a',, n(nu) k _ - 'vu

Let us now enquire for a value of k such that by an application of Sk to the summation on the left side of one of the equations (29), produces the same summation multiplied by a constant.

The evident conditions are that any X, (u) that occurs in the given summa- n2

tion also occurs in the resulting summations, and that any two quantities,



68 MILLER: Concerning Certain Elliptic, etc.

XL (u), X, (u), X, (u) shall have the same coefficient. The last condition is W2 W2 nsii

equivalent to El n y2 8k + 2

We may now show that this is equivalent to

k=O (mod n).

Hence, giving k in equation (31) all values, we can derive n, and only n, different equations.

Hence, the values of Aa in equation (27) may be obtained by giving to a all values from 0 to n - 1, in equation (29), and to k all values from 0 to n - 1 in

equation (31). We shall, at the same time, arrange these n2 equations in n sets, each set containing n equations, and in every equation of each set, the left side is a linear homogeneous expression of the same n quantities, XZ (u), while some

a,u (nu) constitutes the right side.

BLOOMINGTON, INDIANA.



Concerning Certain Elliptic Modular Functions of Square Rank

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