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THE ELLIPTIC MODULAR FUNCTIONS

ASSOCIATED WITH THE ELLIPTIC

NORM CURVE E^

BT

ROSCOE WOODS

A. B. Georgetown College, 1914 A. M. University of Maine, 1916

>)CT S- ib.

THESIS Submitted in Partial Fulfillment of the Requirements for the

Degree of

Doctor of Philosophy

IN Mathematics

IN

THE graduate SCHOOL OF THE

University of Illinois 1920

Reprinted from the Transactions of the American Mathematical Society, Vol. 23, No. 2, March, 1922.

t

i i

THE ELLIPTIC MODULAR FUNCTIONS

ASSOCIATED WITH THE ELLIPTIC

NORM CURVE E'

BY

ROSCOE WOODS

A. B. Georgetown College, 1914 A. M. University of Maine, 1916

. 'I.

THESIS Submitted in Partial Fulfillment of the ReqVjirements for the

Degree of

Doctor of Philosophy

in mathematics

IN

THE GRADUATE SCHOOL OF THE

University of Illinois 1920

Reprinted from the Tkansactions of the American Mathematical Society, Vol. 23, No. 2, March, 1922.

• • • • • . • '

: •.. . V : !

c^Ki^

X^

ACKNOWLEDGMENT

To Professor A. B. Coble under whose supervision this paper has been written I am especially indebted for his very valuable suggestions, his deep interest and his unfailing kindness and encouragement.

University of Illinois, May, 1920.

RoscoE Woods.

r)20Gl 1

CONTENTS

Page

Introduction 179

I. The groups connected with E'

1. The group Gaj^ of collireations of the E' into itself 180

2. The fixed heptahedra of the 8 cyclic Gy's 181

3. A canonical form of the G2.7^ 182

4. The family of E''s 182

5. The fixed spaces 184

II. The quadrics on E'

1. The pencil of quadrics on E' 185

2. The group on the quadrics 186

3. A Kleinian form 187

III. The interpretation of the form F'

1. Its fundamental elliptic modular functions 187

2. The null-system 188

3. The rational curves in S2 and S3 189

4. The net of quadrics in SsCa) 190

5. The modular line and spread 191

IV. The loci in S,,

1 . The net of quadrics in Ss 193

2. The plane of the half period points 194

3. The locus of the zero point in Su 195

4. Summary 196

THE ELLIPTIC MODULAR FUNCTIONS ASSOCIATED WITH THE ELLIPTIC NORM CURVE E'*

BT

ROSCOK WOODS

Introduction

The elliptic norm curve E' in space 5„., admits a group G^„2 of collineations and there is a single infinity of such curves which admit the same group. A particular £" of the family is distinguished by the coordinates of a point on a modular curv- e, the ratios of these coordinates being elliptic modular functions defined by the modular group congruent to identity (mod n). In the group Ci^t there are certain involutory collineations with two 'fixed spaces. HE' is projected from one fixed space upon the other, a family of rational curves C" mapping the family of E"'s is obtained. The quadratic irrationality separat- ing involutory pairs on E" involves the coordinates of a point on the modular curve and the parameter t on a member of the family C".

Miss B. I. Millerf has discussed the elliptic norm curves for which m = 3, 4, 5. In these cases the genus of the modular group is zero and a point of the mod- ular curve can be denoted by a value of the binary parameter t. The irration- ality separating involutory pairs on E" was used by her to define an elliptic parameter

(^0

■■/

where a is the tetrahedral, octahedral, or icosahedral form. This form of u

T

is invariant under all the cogredient tranformations of t and r which leave a'' unaltered.

The cases considered by Dr. Miller are relatively simple, due to the fact that the genus of the modular group is zero. In this paper, the case m = 7 for which the genus is 3, one which is fairly typical of the general case, is subjected to a

* Presented to the Society, April 14, 1922. t See these Transactions, vol. 17 (1916), p. 259.

(179)

180 ROSCOE WOODS [March

similar investigation. Many of the results may be extended to the case where n is any prime number and in some features to the case where n is any odd num- ber. By methods of geometry and group theory, we derive in this discussion the well known elliptic modular functions attached to this group as well as some new ones and obtain a number of their algebraic properties.* This treatment sug- gests a number of "root functions," i. e., square roots of modular functions which are themselves uniform.

In §1, the groups and subgroups associated with the K' are discussed and thrown into a canonical form. The equations of the transformation from S^ to the fixed spaces Sj, 53, and the equations of the groups of transformations in these spaces are derived. These have been found without the aid of function theory and have been checked with Klein's results in Klein-Fricke's Elliptische Modulfunktionen. In §11, a single Kleinian formf is derived which furnishes the fourteen linearly independent quadrics whose complete intersection is E''. From this form in §111 the fundamental elliptic modular functions ti : t2 : ti are de- termined. Also the families C"^, C of rational curves in 52 and 5,, are found. In §IV, the loci in 53 are discussed. The paper closes with a parametric represen- tation of £'.

I. The groups connected with E''

1 . The group G^-ii of coUineations of E^ into itself. The homogeneous coordi- nates of a point of the eUiptic norm curve £' are Xo : Xi :■■■•. x^ = 1 : p{u) : p'{u) : • • : p^{u). As u runs over the period parallelogram coi, co2 the £' is obtained in a six-dimensional space S^. It is knownj that the only birational transformations of the general elliptic curve into itself are given by m' = ± m -|- b, where h is any constant.! From the parametric representation of the £' as set forth above, it is evident that seven points of the £' on a hyperplane section are characterized by the fact that the sum of their parameters is con- gruent to zero (mod coi, wa) and conversely. In view of this, all transformations for which 76 =; 0 (mod oi, C02) are coUineations. This congruence has three irreducible solutions

(1) 6 = 0, 6 = coi/7, h = W2/7.

* In the case n = 4, Miss Miller has expressed the opinion that the properties of the elliptic integral associated with E* and the Dycjc quartic should apply to Klein's quartic which occurs in this case. This has not been verified.

t By a Kleinian form is meant a form in several variables invariant under isomorphic linear groups on these variables.

% Appell-Goursat, Fonctions Algebriques, p. 474.

§Segre, Mathematische Annalen. vol. 27(1887), p. 296.

Klein-Fricke, Theorie der elliplischen Modulfunktionen, vol. 2, p. 241. Hereafter the 'initials K. F. will be used to refer to this work.

1922] ELLIPTIC MODULAR FUNCTIONS ISl

These furnish the substitutions

5oi : m' = M + woi,

(2) Sio : m' = w + «io, (ijj = ioii/7 + /a)2/7,

V -.u' == -u, {i,j = 0, 1 6).

Soi and Sw are collineations of period seven and generate a group G-;- which is abehan in its elements. F is a colhneation of period two which adjoined to G-ji generates a group G2-!t. This group G2T. of collineations contains all the collin- eations of the general E'' into itself.

The G^t in the G'2.7. contains 8 cyclic Cy's and no other subgroups. These are denoted hy G^^Gi, . . ., Ge where G^ is generated by .Soi and G, by SioSJi {i = 0, 1, . . . , 6). The elements of G271 not in Gji are of the form

(3) Vij :u' = -u + i^ii (i, ;■ = 0, 1, . . . , 6),

and are of period two. The V,j form a conjugate set. Any cyclic G^ with one involution generates a dihedral G2 7 which contains seven involutions. Hence there are 56 dihedral G 's. These with the cyclic 6'2's complete the subgroups of (j2 7»- The relations satisfied by the generators of G'2-72 are

\^) ■-'01 ~ •-'10 ~ 1'

5oi5io = SioSou VSio = -S^iol^i VSoi = SqiV.

2. The fixed heptahedra of the 8 cyclic Gi's. The condition that a hyperplane section touch the £' in seven coincident points is given by

(5) 7m S5 0 (mod 0)1,0)2).

The irreducible solutions of this congruence furnish the 49 parameters coy of the singular points. Under Gx the 49 points w^y separate into 7 sets of seven con- jugate points such that each set is on a hyperplane. Such a set of seven hyper- planes will be called a heptahedron. Since there are 8 cyclic G{s, there are 8 heptahedra which will be designated by Hx, Ho, . . ., Hg.*

The 49 singular points are now arranged in a matrix (using only the subscripts) in such a way that the rows furnish the 7 sets of conjugate points which deter- mine the 7 hyperplanes of Hx, while the columns furnish the 7 hyperplanes of

/30>

* The reason for calling one heptahedron Hx. will appear later. These heptahedra can be determined from the resolvent equation of the 8th degree associated with the Galois problem of degree 168. Compare K. F., vol. 1, p. 732.

182

(6)

		ROSCOE WOODS
00	01	02	03	04	05	06
10	11	12	13	14	15	16
20	21	22	23	24	25	26
30	31	32	33	34	35	36
40	41	42	43	44	45	46
50	51	52	53	54	55	56
«0	61	62	63	64	65	66

[March

Each row of this matrix is transformed into itself by Co,, each column into itself by Gn. Further the seven hyperplanes of each heptahedron are linearly independent. Let us prove this for Gaa- If the seven hyperplanes Xi are not independent, there is a relation among them involving 7 — ^ of these X's such that these 1 — k X's Ao not satisfy further relations. Then the 7 — )fe X's in this relation are all fixed under G^ and meet in an 5^ which is also fixed under Goo. Hence &<» permutes the Ss's on the fixed S^ in such a way that 1 — k oi the Ss's are fixed. Therefore by projection from 5^ upon an 5b_^, we should have in Si_k 7 — k fixed spaces no. 6 — k oi which were related. But such a collineation is the identity in Sb-*- Hence every Sb on 5^ is fixed, contrary to the fact that Gx has only a finite number 7 of fixed spaces.

3. A canonical form of the G2T- Let then the heptahedron //ac withhnearly independent faces be chosen as a reference figure and denote these faces by JY, (t = 0, 1, . . ., 6).* These are determined by the rows of the matrix (6). The reference figure is- completed by choosing a unit-hyperplane. This hyperplane will be chosen as the one containing the singular points of the first column of the matrix (6).

In terms of the coordinates thus defined the generators of the G^-p of collinea- tions of the K' into itself have the form

(7)

Soi : Xi — Xi^i Sio '■ Xi = e Xi V:X'i = X_,

iX, + i^Xi) {i = 0,1, .... 6) (X_i = X,_i)

where « is a seventh root of unity. The formulas (7) constitute a first canonical form of Gj.yi.

4. The family of £''s. The curve E'' depends upon the ratio co = coi/toa. For each value of co, there is an £', hence there is a family F of E'^'s. But the

* Xi is written instead of Xi{u). The X, can be represented as the products of sigma

functions, i. e.,

ny = 6 i = 0 a {u- wii)

where the a,- are constants which insure the double periodicity of the ratios Xi. Compare K. F., vol. 2, p. 238.

19231 ELLIPTIC MODULAR FUNCTIONS 183

group Gjvi of colUneations is the same for each member of the family F since its coefficients are numbers independent of the ratio «. For each curve of F the set of 8 heptahedra is the same, since the heptahedra are determined by their common 63. 72.

All colUneations which leave each member of F unaltered have been deter- mined. If there are further colUneations which interchange the members of F, they must arise from integer period transformations of determinant + 1 . Con- sider then the transformation

(8) ' IS aS — /37 = 1

^ ' Oi2 = 70)1 + OW2

where a, 0, 7, S are integers. The curve as first expressed in terms of p{u) and its derivatives is unaltered by (8). In the new reference system the curve and each Wij are unaltered if (8) is congruent to identity (mod 7) . On the other hand if (8) is not congruent to identity (mod 7), the w,^ are permuted and we may look upon this operation either as merely a change in the coordinate system in which the curve is fixed or as a colUneation in which the reference system is fixed and the E'' passes into a new curve which belongs to F. Therefore all trans- formations (8) which are congruent to identity (mod 7) give rise to the identical colUneation. These transformations constitute a subgroup of (8) of index 2- 168. All elements of (8) in a coset of this subgroup give rise to a colUneation which

-1 0

0 -1

which is the element V.

permutes the curves of F except the element

Hence there are 2-168/2 or 168 colUneations which interchange the members of F* These colUneations may be represented by the elements of (8) reduced modulo 7, that is

(9) 0,; = acoi + /3a„ «5 - ^7 - 1 (mod 7).

It is well known that any transformation of the group (8) is a combination of the transformations

(10) 5:o)' = o> + l T:oi'=-l/o}.

where 5 is of period 7 and T is of period 2 when reduced modulo 7. Since r» = 5' = (ST)^ = (5*7)* = It, these relations define a Gm of colUneations on the reduced periods which permutes the members of the family F- There- fore we have the following theorem :

♦SeeK. F.. vol. l,P-398.

tit should be noted that in homogeneaus form, T is of period 4, (S*T) \s of period 8. Hence V and T^ are the same- Compare Dickson, Linear Groups, p. 303.

184 ROSCOE WOODS [March

Theorem I. The family F of elliptic E'''s, each member of which is unaltered by G2.T1, is unaltered as a whole by a collineation group G'2-7=-i68 for which 6^2-7! *^ <^** invariant subgroup. Under the group of F each curve belongs to a conjugate set of 168 curves.*

The collineation T permutes the Hf (i = 00 , 0, 1, . . . , 6) as follows: ( 00 0), (16), (25), (34), where the subscripts only are used. The collineation S permutes the Hi (i = 0, 1, . . ., 6) cyclically and leaves Hx invariant. Under the group (8) the Hj are permuted like the 8 points 00 , 0, 1, . . ., 6 in a finite geometry modulo 7, there being 8 points on a line.

The equations of the coUineations S and T in terms of A', aref

(11)

S :X'i = e-'Va X<

T -.X'i = cJ^e'^X^ (i ^0,1, ...,6). p = o

5. The fixed spaces. In G2-7». the 7^ involutions Vjj {i,j = 0,l,..., 6) have the form u' = —u + lOfj. The fixed points of these involutions are u ^ w<^/2 + P/2 where P/2 can evidently have the values 0, a)i/2, 0)2/2, and (wi + co2)/2. We consider the simplest set, i. e., the set for which i = j = 0.

Due to the involutory character of V, there are two skew spaces of fixed points in Si, an Si and an 53. If the coordinates of these fixed spaces be denoted by y,-,and Zj (z = 0, 1, 2, 4; / = 1, 2, 4) respectively, the equations of the trans- formation from the coordinates Xi to those of y and z are

Xa = yo,

Xi + X, = 2yi, Xi-X, = 2zi.

^'^''^ X2 + X, = 2y2, X2- X, = 2z2,

Xi+.Xz = 2yi, Xi- X3 = 2z4.

In terms of y and z, V now has the form

(13) y'i = Vi, z] = - Zj (i = 0, 1, 2, 4; ; = 1, 2, 4).

In (12), j'j = 0 determine the 52 of fixed points and Zj = 0 determine the S3 of fixed points. The fixed Si's are either on S2 with equations aojo + onyi + a2j'2 + atyt = 0 or on S3 with equations fiiZi + PiZi + fiiZi = 0. The as may be determined by putting the 55 on Mi, Ui, Ug, three arbitrary points on E'', so that necessarily this S^ ciits E'' in the points —Ui, — M2, — M3. Therefore the S^ con- tains the point u = 0, but no proper half period point. Hence all the fixed S5's on the 52 and therefore 52 itself, contain the point u = 0 but no proper

• See K. P., vol. 1, p. 398.

t Compare K. P., vol. 2, p. 292. The formula for 5 is compatible with Klein's for w a prime number. As we deal with coUineations in homogeneous forms we do not need to keep c of the K. P. formula; it is therefore dropped in the remamder of the work.

1922] ELLIPTIC MODULAR FUNCTIONS 185

half period point. Therefore S3 contains the proper half period points since . they are also fixed points.

The family F of £''s projected from the fixed 52 upon the fixed S^ becomes a family Fi of rational cubics doubly covered, since the pairs (=*=«) corresponding under V each project into the same point. In a similar manner, by projection from 53 upon S2, F becomes a family F2 of conies doubly covered.

It is my purpose to discuss the families Fi, F2, for which the curves in each family will vary with co whereas the points on a particular curve will vary with the pairs (* m) on the original F'. The ^2 yj-ies has now reduced to a Gus in 52 and 53 which leaves Fi and F2 invariant. This Gies is generated by 5 and T whose equations are easily found to be

S : !' _ ^72, ' a = 0, 1, 2, 4; ; = 1, 2, 4)

(14)

T

Zj = ۥ" Zi

y'i = yo + Zi («'' + *"'•'') yj {i = 0, 1, 2, 4)

^* = E, (^'* - *"'*) 'i ^'' k, I =1,2, 4).

Formulas (12), (13) and (14) constitute a second canonical system of coordinates for £'.

II. The quadrics on F^

1. The pencil of quadrics on F'. Hermite has shown that the number of lin- early independent quadrics on F' is fourteen. These fourteen quadrics cut out the F' completely with no extraneous intersection.* In the second system of coordinates a general quadric has the form

6

(15) 9. = X;«.* ^.^* = 0'

where a,fe are constants. Let us suppose that the Ujk are so determined that the quadric contains the curve F'. Under the collineation 5io, F^ is transformed into itself. Hence the quadric (15) is transformed into a quadric on.F'. The transforms of q, under Sw are of the form

(16) qj = J2 «'* *''"^*^ ^•^'* = 0 0' = 0, 1, . . . , 6).

Since each Qj is on F', a linear combination of them will be on the curve. Mul- tiplying each qj by unity and adding we obtain a particular quadric Qo on F' characterized by the fact that it consists only of those terms for which i + fe = 0 (mod 7). Using the multipHers 1, «^ e', e', «', t*, t^, respectively, we obtain a

♦ Compare K. F., vol. 2, p. 245.

186 ROSCOE WOODS [March

second particular quadric Qi on £' characterized by the fact that it consists only of those terms for which i + k ^ I (mod 7) . Proceeding in this way we obtain 7 particular quadrics on £'. They are

(17) Qi = a,o^? + 2a,-,Xi+jX.-_, + 2a,2 X,+2X,_2 + 2a,4Xi+4X,-4

= 0 (i = 0, 1, . . . , 6).

Any quadric on the curve E'' is a linear combination of the Q's, since the seven Q's contain as yet 28 arbitrary coefficients. But since each Q,- is sent into Qi+i by 5oi, these 28 coefficients reduce to four, i. e., ao, ai, a^, 014. From these seven Q's, we know that we must be able to get the 14 linearly independent quad- rics on the E'. The as therefore must contain a parameter linearly and there will be one quadric of the type Q, for which a particular a will vanish.* At most, then, a pencil can arise from the four terms of each Q^. Any one of these seven pencils is defined by the fact that it admits one of the seven dihedral G'2.7's whose cyclic subgroup is 5io. For example Qo admits the dihedral (5ioV'). Since the a's contain a parameter linearly, they may be interpreted as the coordinates of a point on a hne in an S3. By choosing two members from the pencil of quadrics, the line is determined. We shall determine the a's later as functions of w and the parameter just mentioned.

2. The group on the quadrics. Under (72.72 each member of the family F of JS^'s is transformed into itself and the quadrics on each curve are transformed into quadrics on that curve, so that a group of collineations is induced upon the Qi as variables. Moreover since 5 and T interchange the members of the 168 sets of conjugate curves, they will send the quadrics on a given curve into a linear combination of the quadrics on the transformed curve. If we indicate the group Cg.yj.ies on the X/s in (7) and (11) by ^(e), then the induced group on the quad- rics Qi is G'(e^).

In order to express all the quadrics (17) by one equation, consider the general quadric obtained by taking a linear combination of them. Such a quadric has the form.

(18) J2 ^'<^' = 0'

1-0

where the L, are arbitrary constants. On a given curve of F determined by a proper set of values of a,- {i = 0, 1, 2, 4), the bilinear form (18) is an identity in L and u. If we require that this bilinear form be an invariant under G'(e^), there will be a certain group induced upon the L, as variables. This group on the variables L, is G"(«~*).

* Compare K. F., vol. 2, p. 268. Klein obtained the quadrics on the elliptic curves from the three-term sigma relation.

1922] ELLIPTIC MODULAR FUNCTIONS 187

3. A Kleinian form. Since the properties of the groups on the L, and (?, are the same as those on the A',, we isolate one of the involutions in the Lj, (?, groups, i. e., that one induced by V which was isolated in the A', group. We introduce the variables v and u, f and d with Q, and L„ respectively, as y and z were intro- duced with the X,. The equations of the transformations from Q; and L, to V, u, f and t? can be written down as were those for y and z. After this change of variables, (18) has the form

F' = fo[ao/o + 2aiyl + 2aiyl + 2a,y\ - 2aiz\ - 2c^\ - 2a^\\

+ 2 filotoT? 4- 2ajyp>'2 + 2a^\y^ + 2(Myi.y\ + a^\ + 2ai^\Z^ — 2042224] + 2 fsiaoj-j + 2aiyyyx + 2aj>'o3'4 + 2 0L^\yi~2 aiZ-fit, + ooZj + 20421*] (19) +2 f4[ao>'4 + 2axyiyt. + 2atyxyi. + 2<my^y\ + 2ai2s24 - 2aiiZi28 + 0024 ]

+ 4 t>i [ac>'i2i + «ijo22 — <»i{y\Zt. + J'42.) + a4(3'224 — ^422) ] + 4 ty2[ao>'222 + ai()'42i — ^'124) + aiy^x — <x^{y\Zi, + j'221)] + 4 i>4 [aoV424 — q:i(>'224 + >'422) + a2(>'i22 — J'22i) + a\y^^ - 0.

On E' the above form is an identity in f, 1? and can be separated into seven parts. However we shall have occasion to separate it into two parts, P\ and Pz, such that the part P\ contains the coefficients f and the part Pj the coefficients I?. The part P\ is partly symmetrical and partly alternating in the coefficients a and f , hence the f 's can be interpreted as the coordinates of a point on a line in an 53 and are therefore cogredient to the as. Hence we may conclude this section with the theorem

Theorem II. F' is a Kleinian form which remains invariant under the simul- taneous transformation by the isomorphic groups M{t) of (14) on the variables y and 2; M(«~2) on the variables f and a and tJ. The form F' determines the curve E' uniquely when the modular functions a are properly given, i. e., subject to the relation which connects their ratios.

III. The INTERPRETATION OF THE FORM F'

1, Its fundamental elliptic modular fuQctions. Each curve of the family F has on it the point whose parameter is m = 0. As w = &)i/w2 varies this zero point generates a locus. It has already been pointed out that the zero point is in the space 52 of fixed points, i. e., when m = 0 all the y's vanish. Let 2, = ti (i = 1, 2, 4) for M = 0; then F' in (19) reduces to

(20)

ro[0 - 2 ait\ - 2 a^l - 2 a^tl]

+ 2 fi[ao/5 -0 + 2 Oititi - 2 04*2/4] + 2 tiWotl - 2 aititi -0 + 2 04*1/2] + 2.UaQtl + 2 aititi - 2 attitt - 0] sO.

188 ROSCOE WOODS [March

Since (20) is an identity in the fj, their coefficients must vanish. These coeffi- cients are hnear in the a's.all of which do not vanish simultaneously, therefore the determinant of the as must vanish. After removing numerical factors, we find a skew-symmetric determinant of even order. This determinant is a perfect square.* It furnishes in variables ti Klein's quartic, which is denoted as follows :

(21) K = t\t2 + tlu + Ah = 0.

K is the equation of the locus of the zero point of the family of £''s and admits a group des of collineations into itself, cogredient to the group in z in (14). The ratios ti -.ti : U are the fundamental elliptic modular functions of the form F'. The expressions for these ratios as uniform functions of the modulus co may be obtained by setting m = 0 in the expressions for the z's in terms of u, oji, co2. as indicated.

Since the curve £' varies with co, and since each £' possesses a zero point, i. e., a point t which is on K, it is clear that the variation of £' with w may be imaged by the variation of / on K. We shall express other elliptic modular functions associated with the family of £''s in terms of the <,-.

2. The null-system. The form in (20) is a null-system, since it can be written in the form

(22) {aoh)t\ + («or2)'2 + {<^<iU)t\ + 2(a4f2)ii<2 + 2{a,U)hU + 2iaiti)kh = 0,

where (a.ffc) = a,f^ — a^f,. Since (20) vanishes independently of the f's it represents a singular null-system.] Hence (22) is the equation of a line whose coordinates may be taken as

(aofi) = + 2 titi, {aiU) = t\,

(23) {aoh) = + 2 titi, (aif4) = tl, {ot^U) = + 2 tA, {aiU) = tl

where a. is clearly a point on a line. Since the coordinates of the line of the as are functions of t, we shall call it the modular line and denote it by L„. The intersection of the coordinate planes of the reference tetrahedron in the space of the as, an 5'"', with L„ furnishes four convenient sets of values of the a's, which substituted in F' give rise to the 28 quadrics on £', of which only 14 are linearly independent, since any two sets of the a's are linear combinations of the remain- ing two sets. These sets of values are

* Burnside and Panton, Theory of Equations, vol. 2, p. 46. t See Veblen and Young, Projective Geometry, vol. 1, p. 324.

1922]

ELLIPTIC MODULAR FUNCTIONS

189

(24)

ao : ai : at : Ui ==

0

2t3U 2tth

-2hh	-2hh :
0	+t\ :
-tl	0 :
tl	-tl :

-2hU

*5

0

The sets (24) suggest that we make a transformation on the as in F' . Let f be a plane such that it intersects L„ in the point a. From (24) we find this transformation to be

(25)

ao = 0 + 2<i<j?i + 2hUh. + 2Wi|4.

ai = -2hhh + 0 - t% + t%,

ai = -2^2/4^0 + ^4^1 + 0

a4 = -2Wi^, - t\^x + i\h. +0.

<?«4,

If F' is transformed by (25), it will take the form

(26) Y. ^i ^i *'> +E ^' '^' '^^^ = 0 (''• ^' = 0, 1, 2, 4 ; / = 1, 2, 4).

The 28 quadrics on the curve £', of which naturally only 14 are linearly inde- pendent, are found by equating to zero the coefficients of the terms f^ f, and ^, t?j respectively, i. e., the ^^j and (^,j. We shall have occasion to use all of these quadrics, but will refer to them briefly in the above notation.

3. The rational curves in 52 and 53. We have seen that a and f are cogredient variables and that P\ is partly alternating and partly symmetrical in a and f . We now rewrite P\ so as to exhibit this property. It has the form

(27) aofoJo + 4^a,fiyoy2 + 2^(aofi + OL^K^y\ + 4^(a2r4 + ot^^yxj-i

+ 22(aori)25 + 4^(a4f2) 21^2 = 0,

where 2, unless otherwise denoted, refers to the cyclic advance of the subscripts 1, 2, 4. This form furnishes the means by which the projections of the family F of £''s upon the fixed spaces 52 and 53 are found. The second part Pi, bilinear in y and 2, does not enter in these projections, since it vanishes when either space is considered separately.

Since f is perfectly arbitrary, consider it on the modular line L„. Now in- terchange a and f in (27). The new form is similar to the old except that the sign of each term in z is changed. Denote the transformed P\ by Pj. Since Pi in (27) is a quadric on K' and since we consider f on L„, Pi is also a quadric on K'. Whence their sum Pi ■\- Pi and their difference Pi — Pi are quadrics on £'. Consider the former;

(28) a^Uyl + 4^a,f,yo3'2 + 2^(aori + aifo)y' + 4^(a2r4 + "if 2)j'iJ'2 = 0.

190 ROSCOB WOODS [March

The equation (28) for arbitrary a and f on L„ furnishes a system of quadrics in 53 which intersect in a cubic curve. From the symmetry of a and f in (28), we lose no generahty by setting a, »■ f ,. We then have

(29) alyl + 4 ^alyoyt + 4 ^aoaiyl + 8 Y^aiUiyiyt = 0.

Since a is linear in a parameter X on L„, (29) furnishes a system of quadrics quadratic in X. The coefficients of this quadratic system of quadrics ar« func- tions of t, so that as t varies on K, we get a family Fi of cubic curves C in 5s. Hence we may state the following theorem :

(30) Theorem III. The projection C^ of the curve E'' upon S3 is the base curve of the quadratic system of quadrics (29) .

Consider now the difference Pi — Pi. This is a conic in Si. It has the form

(31) Y^iMiVi + 2 Y,{a,U)ziZ2 = 0,

which from (23) may be written as follows :

(32) Y)^UA + Y/\z^Zi = 0.

This shows that the system of conies varies with t on K. It is the polar conic of AT as to 2. Hence the theorem :

(33) Theorem IV. The projection C^ of the family F of E'''s upon S2 is the sys- tem of polar conies of Klein's quartic K.

4. The net of quadrics in S3 ["'. The quadric in (29) will be the square of a plane when the rank of its discriminant is 1. Its discriminant is of rank 1 if only the three relations

(34) aaci2 — a\ = 0, ooUi — aj = 0, ooai — a| = 0,

are satisfied.

Consider now the net of quadrics

(35) tiiomai — al) + tiiaoUi — a]) + ti{aoai — a|) = 0.

From the transformations S and T in (14) we conclude that (35) is a Kleinian form. The discriminant of the net (35) is K. Hence so long as t is on K, the quadric

(35) has a double point. If we border the discriminant (35) with variables J and expand, we find the equation of this double point to be

(36) iaiY = £HW4 +X) (-'!'<«'4) ??+£ (2,«D(2&.fi)+X; (2 <!0(2fife) '0.

1922]

ELLIPTIC MODtJLAR FUNCTIONS

191

Therefore the coordinates of the double-point are

(37)

pao= "^4/1/2*4	2tittU tA	tA	tit, :	t%
pai = ^-tl-tlh	tit.
pa,= ^-tl-tlti	tlti
pa,= <-t\-t%	-tl- tit.

where p is 1, ao, «i, aj, ««, respectively. That is to say, we can express the entire system (37) rationally and without extraneous factors by giving the ten quadratic combinations of the as. These combinations are the coefficients of the terms ^, i, in (36).

The order of the linear modular group in the space of the ^''s and as is double the order of the group* in the space of the z's, that is, the group is a (72.168, due to the fact that the identical coUineation appears in the form Ji = =*= y,- Hence the coordinates of a modular-point or plane in Sz^"' and likewise in Si cannot be expressed rationally in terms of the t, without an extraneous factor. The coordi- nates may however be expressed irrationally in terms of t as above, and it is to be noted that their ratios are uniform functions of co.

A number of such modular root functions are suggested by the geometry of the system of cubic curves C^ in Sj. Thus the locus of the zero point on the curves C, the locus of the plane of the half period points, the locus of the point where the tangent at the zero point meets the half period plane, as well as the transforms of these points and planes in the null-system of C^, give rise to func- tions of this type. Some of these are determined later.

The locus of the double point (36) as t varies on 7v is a well known space curve J of order 6 in 53^"\t whose points are in a one-to-one correspondence with the points of K. If we border the discriminant of (35) with ^ and tj, which are to be thought of as parameters, we have 00 ' curves of the third order in t which intersect K in 12 points' which correspond to the 12 meets of the planes ^, r] with J. Hence when ^ = v the cubic in t will be a contact cubic of K. Thus the system (36) for variable J is a system of contact curves of the third order associated with J. {

5. The modular line and spread . If a point y be taken on /, a quadric of the net (35) has a node at y and the polar plane of this point as to this quadric van- ishes, while the polar planes of the other two quadrics meet in a line. Take the coordinates of the point y on 7 as those in the second column of (37) . The three

* Compare K. F., vol. 2, p. 313.

t Compare Snyder and Sisam, Analytic Geometry of Space, p. 168.

i See K. F., vol. 1, p. 716.

192 ROSCOB WOODS [March

polar planes of this point as to the quadrics in the net (35) are in a pencil, and have the form

aot\ — 0 + 2 02^1^4 — 2 04/2^4 = 0,

(38) aoil - 2 aihh -0 + 2 04^1/2 = 0,

aotl + 2 ai;2/4 - 2 aihk -0 = 0.

The axis of the pencil of planes (38) is the modular line L„. Every point on L„ is in a one-to-one correspondence with the point y on /. Since the coordi- nates of L„ and of the point 5' on 7 are functions of t, the variation of y and of L„ also may be imaged by the variation of t on K. Hence as y generates /, L„ generates a ruled surface of order 8. That M is of order 8 may be shown as follows. The condition that a line / meet L„ is a linear condition on their coordinates, or a conic in t. This conic in t meets K in 8 points to each of which there corresponds a meet of / and M, whence M is of order 8.

Let us now consider the general quadric Q in the net (35), and put on it the condition that it have a node. The four partial derivatives ()Q/bai must then vanish simultaneously. These are

0 + ai^4 + aiti. + a4^2 = 0,

/oq\ «oi4 — 2ai<2 = 0,

^ '' aotl - 2a2/4 = 0,

aati — 204^1 = 0.

The discriminant of these equations is K. If we eliminate t from the equations (39), we find four cubic surfaces on each of which is /. Hence their common intersection is J. The equations of these are obtained from the vanishing of the third order determinants in the matrix of the equations (39). They are

(40)

5i = ao — 8 aia2a4 = 0,

52 = ao«4 + 2 aortj + 4 aial = 0,

53 = alai + 2 aoal + 4 a^al = 0,

54 = ala2 + 2 aoa\ + 4 atal = 0.

The modular spread M multiplied by ao is the following combination of 5 in (40) : (41) Si - 8 S2S3S, = aoM = 0.

From this result it is evident that J is a triple curve on M. Further, it can be shown that through every point of J there pass three trisecants of J and that L„ itself is a trisecant of J*

This section can be partially summarized in the following theorem :

* The equation of M and the facts concerning J are easily obtained from a Cremona trans- formation of the third order.

1922]

ELLIPTIC MODULAR FUNCTIONS

193

Theorem V. Through every paint a ( = ao, a,, a^, Ui) on the octavic ruled surface M there passes a line L„ and the pencil of points a on L„ set in the form F' de- termines the quadrics on the curve E''. As the line L„ varies on M, the E'' varies in the family F. The line L„ {itself a trisecant of f) meets the triple curve f on M in three points which correspond to the three trisecants of J that meet in a point t of J. Thus the points t of J are in a one-to-one correspondence with the curves of the family of E'''s.

This completes the determination of the coefficients a of the quadrics F' which define the curve E''.

IV. The loci IN 53

1. The net of quadrics in S3. In (35) a net of quadrics in Ss^"^ was considered. The modular line L„ and the modular spread were associated with this net. Consider now a similar net of quadrics in plane coordinates U in S3, and let us find the condition that this net have a double plane. From the contragredient transformations S and T on the y's in (14)*, we conclude that the following net is a Kleinian form :

(42)

ti{2 UoUi - V\) + <2(2 C7o[/2 - C/D + «4(2 C/of/4 - C/^ = 0.

The discriminant of this net is K. The bordered form of the discriminant is the square of a plane in point coordinates, i.e.,

(43) tM,y\ + X) (- '2 - hi'^yl + 2 X) i\hym + 2 X) ^Ay^y^ = o.

So long as t is on K, the coordinates of the double plane (43) are

pUo=^ tititi

(44)

pUi = < -tl-t^i:

pU2='^-t\-t4\ pU,=<-t\-t,tl

: tit^ti	tlh	tlh :
: tit.	-tl-tA	tA ■■
: tit.	■ t,tl	■.-tl-tA:
: Hh	: t\u	: tA :

nil,

th

t i^

— t^ — t t^

where p is 1, Ug, Ui, U2, U4, respectively. As in (37), we may express the entire system in (44) by taking the 10 quadratic combinations of the L'''s from (43). The remarks following (37) apply here. The plane coordinates U, taken from the second column of (44) are the modular systems Ay developed by Klein. f

With the net (42) there will be a modular line L„, four cubic surfaces 5,', a modular surface M' and a sextic f. The coordinates of L^ can be developed

• See K. F., vol. 1, p. 719. t See K. F., vol. 1, p. 719.

194 ROSCOE WOODS [MaKJi

in a manner similar to that used in finding those of L„ as the axis of the pencil of planes (38). They are

(UoU{) = Uh, iU,U\) = tl

(45) iUoUl) = hh. {U,U'^ = t\,

To every position of the plane (Uy) = 0 in (43) we have a line L^ whose coordinates are given in (45). Since the coefficients of the plane (Uy) = 0 and L^ are functions of /, the variation of the plane (Uy) and L^ also may be im- aged as the variation of t on K. It should be noted that the space of the a's is different from the space of the y's. Hence the modular lines L„, L^; the curves J, J'; the spreads M, M'; and the cubic surfaces 5,-, 5< are all distinct.

2. Theplaneof the half period points. For the three half period points, the z's all vanish. If in the 14 linearly independent quadrics on £' we set the z's all zero, we then obtain 8 quadrics in y (since 6 of the 14 quadrics are bilinear in y and z and vanish for z set equal zero). These 8 quadrics must pass through the half period points. If we call the plane of these points {Uy), then we should be able to obtain from these 8 quadrics the four combinations yi{Uy) {i = 0, 1, 2, 4). The combinations furnishing these types of quadrics come from the systems

tit^<i>i2 + tit2ti<i)H — tit^chi ~ ilh'hi + t2i^4>^Q = 0,

(46)* tl<l>il + h<i>n - /4<^04 = 0,

hfi>\i + U4>^ — t\4>m = 0,

U^i + <l<^04 ~ ^2002 = 0.

The common factor {Uy) obtained from these equations (46) when the s's are zero is precisely the plane

(47) {Uy) = hUityo + t\t^i + tit ^2 + tlt^y^ = 0,

whose square appeared in (43) . Hence the coordinates of the half period plane are the modular functions set forth in (44).

Since the half period plane is of the form S a^yi = 0 (i = 0, 1, 2, 4), and since it may be considered as an S^ in Se, it contains the point u — Q and three pairs of points ( =*= w) on £', since the three pairs are sufficient to determine the a's. It is therefore a fixed 56 on the fixed 52. Since the pairs (=»= m) are the half period points, they are coincident points in 56, hence the half period plane (47) considered as an S^ is a tritangent hyperplane of E^, tangent at the points oii/2, C02/2 and (wi + co2)/2 and passing through the point m = 0.

* We draw from the entire system of quadrics 00(16 in number) for convenience. These 4>ii are the coefficients of the terms {i{,- in F' after the transformation in (25).

1922] ELLIPTIC MODULAR FUNCTIONS 195

Let us now consider the systems of quadrics in (46) with the z's different from zero. These expressed in terms of y and z are

iiyo(Uy) = + 2t,tlz^ (41) + (- 2lit^iZi - 2tlUz,) (12),

(48) iiyiiUy) = - tlUz^ (12) + (tlt,z^ - t%z,) (41), txy^iUy) = - t\t,z, (24) + {tlhz, - tlt,z,) (12), kytiUy) = - t]t^^ (41) + {t\t2Zi - ilt.z,) (24),

where (ik) = /,Zt — i^z,-. Each of the above quadrics vanishes for Z; = /,, that is each conic on the right in (48) intersects the polar conic C in the zero point. The three remaining variable intersections of these conies and the polar conic correspond to the intersection of the plane («,>',) = 0 and the curve C^ in S3. Hence the system of quadrics (48) give a parametric representation of the curve C To each z in (48) there is a definite point y in Ss except at the base point of the system z, = /,-. This representation can be put in a simpler form if we multiply the quadrics in (48) by tit^, so that each quadric on the left has the common factor tihU{Uy), which may be dropped, leaving the parametric repre- sentation of the curve C as follows :

j-o = p[+ 2t\f^^ (41) + (- 2 i\t\u Zj - 2/i/^/4Z,) (12)],

(49) yi = p[- tAtA^A (12) + ihtlUz^ - t\tlzd (41)], :V2 = P[- htlh^i (24) + {ht\t,z^ - tlt\zi) (12)], y4 = P[- Ut\t^z^ (41) + {Ut\t^, - tltlz,) (24)].

Hence the doubled C^ in S2 is mapped upon the doubled C in the fixed Sz by means of the equations in (49) .

3. The locus of the zero point in S^. In S2 we find K as the locus of the zero point. Each curve of the system C^ has one such point, which generates A' by the variation of w. Each curve of the system C has on it the zero point. What is the equation of its locus? Since z,- = /,• is the base point of the mapping sys- tem which maps C^ upon C^ all the >''s vanish at this point, but as z approaches t the limiting position Of the direction is that of the tangent to the polar conic C^ at z, = ti- If the factors {ik) in (49) are replaced by the coordinates of the tangent to the polar conic at the point z^ = i,-, and if we set z, = <,• in the other factors, the y's do not vanish, and become nonic functions of t which have a common factor ^1/2^4. However, a much simpler way to get this parametric representation of the locus of the zero point in St is to solve the bilinearforms <^'oi = <^'o2 = <^'o4 = 0 for 3/,, and put z,- = /,■ in the result, from which the factor tititA can be removed. These equations are:

j'o = - l^t\tltl

(50) y, = t\t\ - dt\tl - 5t,tlti

y^ = t\t\ - Ztlt\ - bt^t\t\,

J'4 = t\t\ - Ztlt\ - dtfitl

196 ROSCOE WOODS [March

These equations map the locus of the zero point in 52 upon a locus in the space of the j-'s. The order of this locus is 18, for a plane section {U'y) = 0 gives a sextic in t which intersects K in 24 points, but we find that this variable sextic and K have 6 fixed intersections at the flex points tj = 1^ = 0 and consequently IS variable ones. Hence the locus of the zero point in S3 is a curve of order 18 and will be denoted by C*.

It has already been pointed out that the order of the group of the y's is double the order of the group of the z's and that to express a form in y and z covariantly its points and planes in S3 must appear squared. This C* can evidently be rep- resented covariantly if we take the 10 quadratic combinations of the y's from the equations (50) from which we can eliminate the factor tlt2t^ and thereby eliminate the fixed intersections each taken twice, and if we take in primed variables the corresponding quadratic combinations of the U's as the coefi5cients of these quadratic combinations of the y's. This form is

(51) fit", t') = 0,

and is of the third order in t', and of the ninth order in /. The number of var- iable points in which this nonic intersects K is 36, which is double the order of C"'^ since its points appear squared in (51). U t = t' in (51) we find a form of order 12 which is K^ + 16//^, where H is the Hessian of K. We can then say that the form (51) is the third polar of K^ + 16//^ plus covariant terms con- taining the line co5rdinates tt'. To obtain these further terms one would make use of the complete system of invariants and covariants of K which has been calculated and tabulated by Gordan.*

4. Summary. The results obtained may be briefly summarized. The well known elliptic modular functions associated with the elliptic norm curve £' and the algebraic relations connecting them have been readily found from the geometric point of view. The system of contact cubics in (37), the coordinates of the modular lines L„ and L^ and the parametric representation of the locus of the zero point in S3 are new types of functions. The system of modular func- tions By (in Klein's notation)! which define a curve of order 14 has not been found.

If a pair of points in the involution on the curve E'' is isolated, the quadratic irrationality associated with the curve £' is obtained. This irrationality can be obtained from the system y,- in (44). If we substitute the values of these yi in any of the quadrics (19) (except those bilinear in y and z), p is obtained as the square root of the reciprocal of a conic g(<*, z^). This conic has the form

(52) g{i\ z^) = z]{2titlti + t%tl) + zl{2tMt) + zlit%ti) + zMt\tlt4 - 2tltl) + Zi?i{- 4tltl - tlti) - ZiZi{3tltl),

♦Mathematische Annalen, vol. 17 (1880), pp. 217, 359. t See K. F., vol. 2, p. 396-397.

1922] ELLIPTIC MODULAR FUNCTIONS 197

and constitutes the part in 2 of a quadric on the curve £' whose part in y is the square of the half period plane (47).* We can now write down the parametric representation of the curve £'. It is

y^ = ^ 2tY^^ (41) - {2tY4^z^ + 2t^AU^^ (12), y[ = - i,t\hz, (12) + (/,4^z, - t\fifi^) (41), ^2 = - '2^4/1^1 (24) + {U\t^z^ - t\e^z^ (12), (53) y\= - tAhz^ (41) + {iAt^, - tlt\z,) (24),

z[ = zitxhh<g{f\z^, z'i = ZititoMylgjt*, z^, z'i = ^Jltik^lgit^ z^).

If < is on iC the above system maps the doubled C upon the E''. It should be noted that the y's vanish for z, = /, and the z's vanish when 2 is on a half period point, t

• Professor Sharpe of Cornell pointed out this fact to me, as well as a method of eliminat- ing an extraneous factor Ukh from the parametric representation of the curve E'. I append the method in a foot note at the end of the paper.

t All the terms in y; contain the factor IMt except one term in yo *nd this term contains a Zi. If we now find the intersection of the pencil of lines through the point /, Xi(42) -f- Xj(14) = 0 and the polar conic C^, we get the following values for zi :

21 = X?(-2<5fe-«?) +>-ltihk-3tl + /4X1X2, Z2 = \ititl + \l{-2tlu-tl)-{2tit2 + tl)\i\2, Zi = ^ihhh + X2'2'4 h + '4X1X2.

Hence when these values are put in (53) the factor tMt can be removed.

University op Illinois, Urbana, III.

VITA

Born at Mayo, Ky., August 10, 1889, son of Thomas Clinton and Margaret Wheeler- Woods. Received his elementary education in the public school in Hopewell District, Mercer County, Ky. Entered the academy connected with Georgetown College at Georgetown, Ky., in 1908 and graduated from George- town College in 1914 with A.B. degree. Spent the summer of 1914 as a graduate student in the University of Chicago. Held an Assistantship in Mathematics in the University of Maine during the year 1914-15, Instructor, 1915-17. Re- ceived A.M. degree from the University of Maine in 1916. Was an Assistant in Mathematics in the University of Illinois during the years 1917-20.

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