The Automorphic Transformations of the Binary Quartic

The Automorphic Transformations of the Binary QuarticAuthor(s): A. H. WilsonSource: American Journal of Mathematics, Vol. 33, No. 1/4 (Jan., 1911), pp. 29-36Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369985 .

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The Automorphic Transformations of the Binary Quartic.

By A. H. WILSON.

The transformations which are the subject of this paper have been discussed from several points of view. They form groups which are holohedrically iso- morphic with certain substitution groups of four letters; and they have been given a picturesque geometric interpretation by use of the regular polyhedrons. When, however, the analytical expressions of these transformations have been obtained, it has always been by a specialization of the system of coordinates, which reduces the quartic to a canonical form. It is the object here to derive them for the general quartic.

1. Substitution groups. If, as usual, we let [ABCD] denote a definite double ratio of four distinct points on a line, then the following permutations will in every case have the same value:

[ABOD], [BADC], [CDAB], [DCBA].

If the four points have the harmonic position, so that [ABCDJ = -1, there are eight permutations which have the same value:

[ABCD], [BADC], [CDAB], [DOCBA] [BACD], [ABDC], [CDBA], [DCAB].

For the equianharmonic position the letters may be permuted in twelve ways without changing the value of the double ratio:

[ABCD], [BADC], [CDAB], [DCBA], [BCAD], [CBDA], [ADBC], [DACB],

[CABD], [ACDB], [BDCA], [DBAC].

The three sets of permutations just given correspond to the substitution groups of four letters denoted, respectively, by C4, the "Vierergruppe"; G8,

a dihedron group; and G12, the tetrahedron group.



30 WIMSON: Automorphic Transformations of the Binary Quartic.

In the symbolism of these groups we have:

G4 1, (AB)(CD), (AC)(BD), (AD)(BC);

G8: 1, (AB)(CD), (AC)(BD), (AD)(BC),

(AB), (CD), (ACBD), (ADBC);

G12: 1, (AB)(CD), (AC)(BD), (AD)(B C), (ABC), (A CD), (BDC), (ADB), (ACB), (B CD), (ABD), (ADC).

If the substitutions of 04 be denoted by IO, 61, 12, t3, respectively, then G8 is obtained by adjoining to 64 the transposition a = (AB), and 012 is obtained by adjoining to 04 the circular substitution =(ABC). This composition of the groups will be useful in the following discussion:

04 ?, ?1, ?2 ?3. (1)

08: ?o, ?, ?2, ?3, } (2) or1to O?1, cY2, l Or3.

012:, to ' ?, ?2, t3, 3

,2 2 t?1, r?2, T?3.r (3) 1z1 to, 26 t2 626 t2 3*J

2. The decomposition of the quartic into factors. The transformations are considered here as those which carry a binary quartic into a multiple of itself. In a paper published in the AMERICAN JOURNAL, VO1. XVII, page 1855, Professor STUDY has discussed fully the irrational covariants of the quartic, and has inci- dentally derived the expressions of thie tranisformations of G4. His further results enable us to write the transformations of G8 and G12. For the sake of clearness I quote as much of the invariant theory of the quartic and of Pro fessor STUDY'S paper as is necessary for the purpose, adopting throughout the notation of the latter.

The rational covariant and invariant system of the quartic:

f (ax)4 (a'x)4 . . . . (a,x2 - a2x1)4, 1 h = (hx)4 (hfx)4 . .. . - i (aa') 2 (ax)2 (a'X)2 (4) t = (tx) 6 (tfx)6 _ ....

2 (ah) (ax) (hx)3,( 92 = (aa')4a g3 1- (ah)4 - (aa')2 (aa")2 (alaff)2. J



WiLSON: Automorphic Transformations of the Binary Quartic. 31

The discriminant takes the form

G = 27g (5)

16 and the cubic resolvent,

4 e3 - 92 e - 93 0 (6)

with roots denoted by e,, e, e,. The forms h + ejf, h + e,f and h + ejf are perfect squares, and they are factors of t2. The quadratic factors of t are, however, given the following form for the sake of certain invariant properties:

I =(Ix)2 = V-h-ejf,

1 m =(mMx)2 -= -h-el., (7)

n =(nx)2 =- -h- eVf;

8v~~~~~~~~~~~8 where

s, --S/ e -ex . s/ e, e,W

s,- Ve e,-e, *Se -e (8-)

8v /e, eV * eV ex * Then is

2 = (e,-ev)(ev - e;) (e;,- e,.) Imn =- ,/ G . Imn.(9 2 /j (9)

For the invariants and covariants of the quadratics 1, m, n, we have

(mn) = (imn) (mx) (nx) = - (IX)2, 1

(n), = (nl) (nx) (lx) - (Mx)2, (10)

(Im) = (im) (lx) (mx) = - (nx)2;

i (111)2 = j (mm')2 =i (nnl)2 1,

(mn)2 = (nl)2 (Im)2 = O .

The decomposition of f into quadratic factors is derived from equations (7), and in fact in three forms corresponding to the three ways of pairing the linear factors:

f=- 1

m + 8 n)(s, mm-9vn)

-e-e (8sn +sl)(V-SAI) s

(12)

- e e (s e +s,.m)(sl-s,,).I



32 WILSON: Automorphic Transformations of the Binary Quartic.

The decomposition of f into linear factors is given as follows. We set

f = 4 (rO x) (r x) (r2 ) (r3 x), (13)

and have for the squares of these linear factors (see the paper cited, p. 210):

-2 (rox)2 = V e,-eV. *+ V e,-e. rn + m eA- el. n, 1 - 2(r,x)2= -.e,l-eV. l+eV-e-e.m-VeX-e,L.n, (14) -2 (r,x)= -Ve,-eV. l e+eV-eX. m+ eA-eL. n J

2 (r. x)2 e_ - - /J e. . I - s -e.-e, . rn + -s+J eA - e.J . n .

The simultaneous invariants of these forms are given by the formule

(ro rx) (r,L r) -(e,L -eV); (ro rA) = (r, rV)2 e -1e

(r.rV) (rVrX) (rXr,.) = (rxrO) (rO r,L) (r,L rX) = 4/V, (15)

(r. r.) (rA ro) (rO r.) = (ro r,.) (rh. r.) (rv ro) = - -

G J

In the formulh (14) the signs of the radicals may be arbitrarily chosen, while that of v, G is then determined by

4/ G = %/ e. ev, V/, e^. /e- e,, .

3. Transformations expressed in symbolic notation. The binary bilinear form in cogredient variables, (dx) (Sy), set equal to zero, will, if the discriminant does not vanisb, establish a projective correspondence between the points of the binary domain. To a point x (that is, x1: x32), there corresponds a point x', the vanishing point of (dx) (Sy) = (x'y) = 0. The linear invariant of the form is i = (dS), and the discriminant, j = i (dd') (&3'). The identity transformation is given by (xy) = 0. If s, = (d1x) (SIy) = 0 and S2 = (d2x) (2Y) = 0 are two non- degenerate transformations, then their product is given by

'1 92 = (d1 x) (d2. 1) (S2Y) = 0. The condition that a transformation is involutory or of period 2, is i =0;

of period 3, i2_j_ 0; of period 4, i2 2j= 0; and so on. 4. The transformations of the general case are, as stated above, derived in the

paper from which I have quoted. It is a well-known property of the covariant t of the quartic that the vanishing points of each of the quadratic factors are the double points of an involution, of which every form of the pencil xf + > h furnishes two pairs of points. In particular, to each of the three ways of pairing the points of f correspond in this manner, as double points of an involution, the root point of a quadratic factor of t.



WMSON: Automorphic Transformations of the Binaryj Quartic. 33

The polar form (ax) (ay), set equal to zero, is, if the discriminant i (aau)2 does not vanish, a transformation pairing the points of the line according to an involution wbose double points are given by the equation (ax)2 =O. We have in this way for the bilinear forms representing the transformations of the group leaving the general quartic unaltered:

G4: to = (xy), tl = (1X) (ly), ?2 = (mX) (my), t3= (nx) (ny). (16)

5. The transformations for the harmonic case may be derived from the con- sideration of the quadratic factors of f, (12). If the points off form a harmonic quadruple, the invariant g3 vaniishes, and conversely. The cubic resolvent (6) then becomes

4 e3 g2 e = O,

with roots 0, i V7gj and -i V/g2. According to the naming of the roots, we have different forms of the transformations. Setting

e, = 0 e, =V/g2, e r-*Vg2,

the decomposition (12) becomes

:= i V2 (Y + m) (l-m)

92 + -(n+V 2l(n -7_21) p (17)

= Wg2(l+V-2m)(l-V-2m); J where now

2 I -V-^/h,

Vg2

m g2 1%/-A- g */ (18)

~ 2 n :- - v- h + i ^/g2 -f

The second and third decompositions off in (17) represent a pairing of the linear factors corresponding to the double ratio * or 2. In the first this double ratio is -1, and from this we derive the desired transformations. The involutory transformation, namely, whose double points are given by m + n = 0, as well as that whose double points are given by m - n =0, will interchange one pair

5



34 WILSON: Automorphic Transformations of the Binary Qaartic.

of the harmonic points, leaving the other points unaltered. For the group G8 we have then, by (2), only to adjoin to G4 the transformation

= - (mx) (my) + (nx) (ny).

G8: to $y? = (xy), - (ix) (ly), 2 = (mx) (my), t3 (nx) (ny),

a (o = mx) (my) + (nx) (ny), at, = (mx) (my) - (nx) (ny), F (19) t2 =(Ix) (ly) + (Xy), - at3 =- (lx) (y) + (xy) J

The transformations ato and alt are involutory, the linear invariant i vanishing by virtue of (1 1); a 2 and ort are of period 4, as for each i =-2 and j= 2, and hence L2- 2j=o.

5. lthe transformations of the equ-ianhiarmonic case are derived from the decomposition of f into linear factors. We know that for this case there exist transformations which permute cyclically any three of the points, leaving the fourth unaltered.

We first derive a general formula for the transformation which takes the points xl, X2, X3, into Y', y2, y3 * This is given by the elimination of the coefficients dC{ <, from the equations

(dx) (ay) = 0, (dxl) (3y1) = 0, (dX2) (OY2) = 0, (dX3) (3y3) = 0;

which gives as a result the transformation

(dX) (Sy) = (X1X3) (Y1Y2) (x2X) (Y3Y) - (X1 2) (Y1 Y) (x3X) (Y2Y) = 0. (20) This is a statement of the equality of the double ratios [xix2x3 x] and [yiy2y3y], and as such may be given other equivalent forms. Setting now Y1 = 2 Y2 = = 3) y3 = X1, we have for the cyclic transformation,

(dX) (by) = (Xl x2)2 (X3 X) (x3 y) - (X3 X1) (x2 X3) (x2 X) (X1 Y)

This may be given a symmetric form. Permuting cyclically the letters x1, X2, X3

and adding the results, we obtain

3 (dx) (by) = (x2 x3)2 (xI X) (xI y) + (a;3 x1)2 (X2 X) (X2 Y) + (X1 X2)2 (X3 X) (X3 y) - (x3 X1) (X1 X2) (X3 X) (X2 Y) (X1 X2) (X2 X3) (X1 X) (X3 y) - (X2 X3) (x3 XI) (X2 X) (X1 Y).

The last three terms may be easily reduced in pairs by use of the fundamental identity of binary forms. Thus, for example,

- (Xl X2) (X2 x3) (X1 X) (X3 Y) - (X2 X3) (X3 X1) (X2 X) (X1 y)

= (XI x2)2 (X3 X) (X3 y) + (X2 X3) (X3 x1) (XI X2) . (Xy).

* By xl, x2, x3. yj y2, are meant ll:X12t X21:X22, 31:X32 Y11:Y127 Y21: Y221 Y.1:Y32



WILSON: Automorphic Transformations of the Btnary Quartic. 35

Substituting these results, we have finally

2 (dx) (by) = (x2 x3)2 (x1 x) (x1 y) + (X3 X)2 (X2 X) (X2 y) (21) + (l X2)2 (X3 X) (X3 y) + (X2 X3) (X3 X1) (X X2) * (XY)-

This is the general form of a transformation determined by the passing of

x1, x'2, a3 into x'2, ax3, xl, respectively. Writing in this formula r,, r,, r, [see (13)] for x', a2, X3, we have the

transformation sought:

2 (dx) (Sy) = (r,. rV)2 (r, x) (r, y) + (r,, r,)2 (rvw x) (r,, y) 1 22 + (rkr,,)2 (rx X) (r, y) + (r,rv) (r rA) (r r,,) . (xy). e (2)

By use of (14) and (15) this may be written in terms of 1, m, n, eA, e,, e,. From these formula we have

- 2 (rOTrA)2 (r, x) (rx y)

(e,,,- e") [V e,, -ce . (lx) (ly) - AVev - * (mx) (my) eX - e-,* (nx) (ny)] -2 (ro r,)2 (r X) (r, y) =

(ev- ek) [- V/e,y, -eV . (lx) (ly) + V'e,-e.* (mx) (my) -Vek-e,. *(nx) (ny)], - 2 (ror,)2 (r, x) (r, y)

(eA -e,.) [-Ve-eV. (lx) (y) -Ve,-e.* (ma) (my) + _V eA-e,. . (nx) (ny)],

2(r,, rV) (r rA) (rA r,) *(xy) - 2 Ve,,-e, e. - * Ve- ef-,. .(xy);

whence, by an easy reduction, (22) becomes

-2 (dx) (Sy)(Ve, - e)3. (li) (ly) + (/ve -e)3 _ (mx) (My) 1 23 + (V/eA-e,_.)3. (nx) (ny)-V/eg -e, . VeV-eA . V, eA- e,, . (xy). (2)

The equianharmonic case is characterized by the vanishing of the invariant

g2. The cubic resolvent becomes then

4e3- g3 =0 and its roots,

ex= SlIL3 ea,, = Ee 93 e,= E3 93 (24)

* This transformation will hold rO unaltered if (rO rx)4 + (rO r,)4 + (rO r,)4 = 0; or by (14), if (e _-e)2

+ (e,- ex)2 + (ex - e6,)2 = O. In the equianharmonic case [see (24)], this condition becomes (62 -C3)2 + (63 _ 1)2

+ (el- _)2 = 0, and this is a numerical identity.



36 WMSON: Automorphic Transformations of the Binary Quartic.

where El, E2, c3 are the cube roots of unity, named arbitrarily. The radical expressions in (23),

(^/e,-r)3 (ev - x)3 le, -y3V-e.v^ A e S (25)

take now, except for sign, the same value, and indeed each is equal to i A (V - 3)g3 or each is equal to V (V - 3)3g3. From the trans-

formation (23) we may omit then a common factor, the common value of the expressions (25). We shall have different transformations according to the choice of signs of the radicals in (14), which was arbitrary; there will in fact be exactly eight which are essentially different, each of which will transform the quartic into a multiple of itself. These may also be derived by (3); we have in fact, putting

e = -(Ix) (ly) - (mx) (my) - (nc) (ny) + (xy),

G12: to = (XY), tl = (lX) (Iy), t2 = (mX) (my), t3 =(nx) (ny), 'r to = - (lx) (ly) - (mx) (my) - (nx) (ny) + (xy),

= tl = (lx) (ly) - (mx) (my) + (nx) (ny) + (xy), er t2 = (lx) (ly) + (mx) (my) - (nx) (ny) + (xy), rt3 = - (lx) (ly) + (mx) (my) + (nx) (ny) + (xy), (26)

- T2 to = (lx) (ly) + (mx) (my) + (nx) (ny) + (xy), -r2 t, = _ (lX) (ly) - (mX) (my) + (nx) (ny) + (xy), A T2 t2 = (1x) (ly) - (mx) (my) - (nx) (ny) + (xy),

-

=t3 - (lx) (ly) + (mX) (my) - (nx) (ny) + (xy).

The respective invariants of the last eight of these transformations have the same value. Writing

(dx) (3y) = i (lx) (ly) i (mx) (my) i (nx) (ny) + (xy), we have

i (d3) = 2, j = g (dd') (R(')

= W { i (dl) (61) + (dim) (3m) iz (dn) (3n) + (d3) = 4 ~(1l)2 + (mml)2 + (nn' )2 + 2 } = 4.

Hence, in every instance i2 j= 0; that is, the transformations rtr? are all of period 3.

HAVERFORD COLLEGE



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The Automorphic Transformations of the Binary Quartic