ON THE ORDER INVARIANTS OF INTEGRAL
QUADRATIC FORMS
By GORDON PALL (Montreal)
[Received 27 March 1934]
and Minkowskif laid the foundations of an arithmetical
theory of quadratic forms in their definitions of orders and genera
of quadratic forms. In this paper we are concerned with those
invariants of a quadratic form which characterize its order. We shall
see that of Minkowski's invariants (olt..., on_i, av..., a,^) the latter
»—1 are quite superfluous, after a modification in the definition of
the former. In Smith's terminology, it is no longer necessary to
distinguish specifically between properly and improperly primitive
forms.
For purposes of arithmetical theory the use of Kronecker's binary
forms ax2
-\-bxy-\-cy2
and discriminants b2
—4ac has conduced to
greater simplicity than the restriction to Gauss's forms ax2
-\-2bxy+cy2
and determinants ac—bz
or b2
—ac. The following developments seem
to lead to a corresponding advancement in the theory of quadratic
forms in several variables.
1. The matrix and discriminant of a quadratic form. Any
quadratic form f in a variables with integral coefficients may be
written as
where the coefficients aH and 2a{j (i ^ j) are integers. The matrix of
/• is the matrix (a{j); the determinant \ati\ of / will be assumed
(throughout this paper) to be not zero. The discriminant d of / is,
by definition, the determinant multiplied by
(_)»/22« if s i s even, (_)(»-iV!2»-i if 8 is odd. (2)
That a discriminant is an integer if s is odd, and is an integer
congruent to 0 or 1 (mod 4) if s is even, is a corollary of the following
lemma.
t The contents of this paper are cognate with the following: H. J. S. Smith,
Collected MathematicalPapers, i. 412-15, 510-12, and ii. 623-36; H. Minkowski,
OesammeUe Abhandlungen, i. 4-8, 8-33, 72, 76-9.
ON INTEGRAL QUADRATIC FORMS 31
LEMMA 1. Let k ^ 1; let etj (i,j = l,...,k) be integers such that ei{
is even and eti = eit for ail values of i and j . Then the determinant
|e,j| is always even if k is odd, and is congruent to 0 or (— 1)*" (mod4)
if k is even.
For consider the expansion of |ey| = ]F ( i ) ^ - - ^ . With each term
T = (±)el4...eto. let us associate the transpose term T* = (±)cgl...ert
which is equal to T. NOW T may be its own transpose; but, if Jfc is
odd, this happens only if T contains some factor eit whence T is
even; and, if k is even, T will contain an even number of such factors.
tHence, if k is even,
|e,y| = (-)t/2
{elge34...et_1>t+...}* (mod 4), (3)
where the expression in braces is the sum of all algebraically distinct
terms ew...e,r such that all the indices p, q,..., t, r are unequal.
2. Notation. With certain exceptions small Latin letters will
connote integers. The exceptions are: /, g represent forms; aijt by
denote halves of integers if i ^ j . Otherwise the role of the various
letters will be defined.
3. Classes of forms. Index. If the transformation
Z y J j (* =
with the matrix T = (ttj) carries f(xl,...,xB) into
gtov->y.) = 2b
nyiVi ib
a =fy<;*.i= i,-.«), (5)
then, representing {2ai}) and (26^) by A and B, we have
B = T*AT, (6)
where T* denotes the transpose of matrix T. We can then say that
'/contains g'.
Let T be unitary, i.e. have determinant 1. Then T~l
is unitary,
and (6) implies A = (T-^'BT-1
. Thus g contains /. If / is transformable
into g by a transformation of determinant 1, we say that
/ and g are equivalent, and write f' ~ g. The relation of equivalence
is reflexive, symmetric, and transitive. All forms equivalent to a
given one are equivalent to one another, and constitute a class of
forms.
The discriminant is an invariant of a class. Another invariant is
the index (to be denoted by I), defined as follows. Since d =£ 0, / can
be expressed in the form
32 GORDON PALL
where the o^ are rational non-zero numbers, and the Xi are linear
combinations with rational coefficients of the xit the determinant of
the Xt being not zero. The number / of negative coefficients c^ in
every expression of this type for /is the same, and is called the index
of/. The signature of /, defined to be 8—21, is frequently taken to
replace / as an invariant of the -class.
4. The g.c.d. of order k. We shall conveniently employ the letter
a to connote a subsequence of k (1 ^ k ^ s) elements of (1, 2,...,«),
that is, a sequence of the type (it, it,..., ik) (1 < »\ < it < ... < ik ^ s).
The minor determinant of a matrix C formed by the elements at the
intersections of rows 4, it,..., ik and columnsjvjt,---,jk will be denoted
by Cfaa,], where ax = (ilt...,ik) and o2 = {jlt...,jk).
From the equation (6) we have, by a simple property of deterUfa
aj = 2 AWFlaoJTlo'oJ (7)
&
summed for all subsequences a, a'. Since C[aa'] = C[a'ff] (C = A
or fi), the g.c.d. of all the .4[crcr] and 2^4[aa'] is a divisor of every
B[oo] and 2J3|W]. (8)
The g.c.d. of order k of / is defined as follows, and is denoted by
dk {k = 1,...,«). Let A = (2a{j), and, with Lemma 1 in mind, write
/it = 1 or 2 according as k is even or odd. Then filcdk is the g.c.d.
of all the principal minors and doubles of the secondary minors of
order k in A:
fj.kdk is the g.c.d. of all the -<4[cra] and 2.4[a<r']. (9)
For example, d^ is the g.c.d. of the actual coefficients aH, 2aif of /.
Since d, > 0, the discriminant is equal to
d=(_)wa-/rf,. (10)
For future expediency we shall define
d^=0, do =l, ds+1 = 0. (11)
llf~g, A and B may be interchanged in (8). The g.c.d.'s dk are
invariants of a class.
We call dj the divisor of/ or of its class. If dx = 1, the form and
class are called primitive; if d^ is prime to N, they are primitive to
modulus N. The form //c^ is primitive, and / is said to be derived
therefrom.
ON INTEGRAL QUADRATIC FORMS 33
5. The o-invariants of /. Definition of order. For odd primes
p it is plain that if pn
\dk_1 then pn
\dk. This is proved in § 8 for
p = 2. Hence ^ ^
^ ^ (i < * <
The following two theorems are also established in § 8.
THEOREM 1. Each of the numbers ok defined by
. = ^ i W i ^ - i = ilM-i^k+1dk_t (fc = Q> s)
(Mt»t) d\
is an integer. Further:
(k = 0,...,«). (13)
/ / any ok (1 < k < a—1) is odd, then ok_1 = ok+1 = 0 (mod 16).
Thus o0 = 0 = o,, ox = djj/df. As here defined, o^..., ogHl are
positive.
These ot, together with dv will be chosen to replace the dk as
invariants of a class, and may be called the o-invariants. All forms
or classes in s variables with the same index /, the same divisor dv
and the same system of invariants o1,...,os_1 constitute an order.
The g.c.d.'s dk are given in terms of the ok by the equations
^*+i _. o
i°8~1
---o
fc (u — i o i>_n
d1 being an arbitrary positive integer.
The greatest common divisor of a set of numbers Aa< (t = \,...,n),
where A is a real number differing from zero and the a{ are integers,
may naturally and without ambiguity be defined to be |A|Z>, where
D is the g.c.d. of the a{. Thus the g.c.d. of order k of the form A/
is ±\k
dk. Observing with a view to (12) that
(A*+1
At
-1
)/(Ai
)2
= 1,
we see that: the invariants ok ofXf are the same as those off.
5 a. Definitions of even, odd, and classical forms. An integral
quadratic form / is called even or odd'f according as the primitive
form f\dx from which / is derived has all its cross-product coefficients
even or has at least one of them odd. A form is called classical if all
its cross-product coefficients are even. The determinant of a classical
form is an integer.
•f It is appropriate virtually to reverse Smith's use of these terms.
3MB.«
34 GORDON PALL
6. The comitant forms fk and Fk. Let/ be a form (1). Employ
the notations of § 4. The form fk in the fik variables £o defined by
HkM£) = lA[oo']{Jo, (k = 1,...,8-1) (15)
a,a'
(jj.k = 1 if k is even, fik = 2 if k is odd)
is called the kth comitant of/. The divisor of fk is dfc: cf. (9). Also
ft = /. By § 8, we have
THEOREM 2. Theformfk is even or odd according asokis even or odd.
Let us write
(—)„ = 1 or —1 according as the sum of the elements in a
is even or odd. (16)
Replacing every £„ in/fc(£) by (—)o£a yields a new form fk differing
from fk only in the signs of certain secondary coefficients. The forms
Fk=fjdk (k=l,...,8-l) (17)
are called the primitive comitanU of/. The (s—l)th primitive comitant
Fa_t is called the reciprocal otf/dv
6 a. Reciprocal orders. (§ 6a is not used in proving Theorems
1 and 2.)
The form <f> = JB_X = d,^ F8-1 is the contravariant of /. We have
where Atj denotes the cofactor of 2aij in (2aiy). Write E = (A{i).
Let T and T' denote subsequences of a—k elements of ^1,2,...,«), and
a and a' the conjugate subsequences consisting of the remaining k
elements (k = l,...,s— 1). Then by a simple property of determinants
E[oa'] = | 2 a y | ' - ' ( - ) T ( - ) ^ H '
Hence, if « is even, the kth comitant of ^ is seen to be
^ = (-)«*-1V2
rf*-VU (A)
where d is the discriminant of/, i.e. d = (—Y1
*\2ayl. If a is odd, the
kth comitant offaf>reduces similarly to
From (A) and (B) we can easily write down the divisors of <f>k and
($<j>)k. Let o'k (temporarily) have the same signi^iance for <f> (and
hence, by the end of § 5, for \<f>) as ok has for /. Substituting for the
various g.c.d.'s in (12) we immediately find thao
o'k = o,_k (k =!,...,«-!). (18)
ON INTEGRAL QUADRATIC FORMS 35
If a form / has index / its reciprocal has index /', where
/ ' = / if s is even, /' = «—/ if a is odd. (18')
If / belongs to the order {a\ = 1; /; olt..., o,^), its reciprocal belongs
to the order (1; /'; o^,...^^. By (A) and (B) with k = s - 1 , f/d^
is the reciprocal of F3_x. The primitive orders (1; /; ol!...,o,_l) and
(1; /'; £>,_!,....Oi) are called reciprocal orders. The itth primitive comitant
of Fs^iBf3_k/d8_k (k = 1,...,«-1).
7. Canonical forms of / to modulus p1
. In studying the properties
of the minors of A, to modulus N, it is expedient to transform
/ into a simple equivalent form, to modulus N. Two forms / and g
of type (1) are said to be equivalent, to modulus N, if there exists in
the class of / a form whose coefficients are congruent, to modulus N,
to the corresponding coefficients of g. Equivalence, to modulus N, is
reflexive, symmetric, and transitive.
T/KMATA 2. Let 8 ^ 2, f being a form (\). Let t be positive, p an odd
prime. Then f is equivalent, to modulus p1
, to a form g of the type
(0 < « ! < « , < . . . < «i), (19)
the act being integers, and the mi prime to p.
LEMMA 3. Let s > 2, t > 0. Then f is equivalent, to modulus 2',
to a form g of the type
y^), (20)
where (a) the /3t and Yj are non-negative integers, the mi and mU)
are
odd, and s = u+2v, u ^ 0, v ^ 0;
(6) the m<f> may be taken to be arbitrary odd integers, and
74, Tig,..., n^-i to be odd;
(c) for no i and j is a /?(+1 equal to a yy.
In proving these lemmas we shall assume without loss of generality
that / is primitive to modulus p (greater than or equal to 2 respectively),
i.e. at least one of the integers aH and 2ai} is prime to p.
In the case p = 2 and / even, at least one of the au is prime to
p. In the case p > 2 and every au divisible by p, some 2ajk is prime
to p; we apply then the unitary transformation
- V . xk = yi+yk, x, = y, (I # *), (21)
36 GORDON PALL
which replaces a}i by ajj-\-2ajk-{-akk, which is prime to p. I n view
of the unitary transformation Plt, where P^ is
p
ki- *k = Vi> *i = —yk> *i = Vi (J # ». *), (22)
we may assume an prime to p.
Suppose in (1) that a n is prime to p. The transformation
x1 = yl+h2y2+...+hey,, xl = yl (1^2), (23)
carries / into g with au as the coefficient of y\ and with
bu= 20^+20^,
as the coefficients of ylyl (I = 2,...,s). The Aj can be chosen to make
each bu divisible by p1
, except when p = 2 and / is odd. Thus
y(yi,-,y,) = anyl+pa
g'(y ,ys) (mod^), (24)
where g' is primitive to modulus p, and a is an integer ^ 0.
Even if / is odd and p = 2, / may be equivalent to a form of
type (24). At any rate in view of transformation (22) we may assume
2a12 odd. We apply then the unitary transformation
x2 = y2+k3y3+...+kayg, x, = yt (I > 3). (25)
This yields a form g in which the coefficient of y^t is 2a11A2+2o12
and is odd; and the coefficients of ytyt and y2yt {I ^ 3) are respectively
^ = 2
and bn= buh%-\-2al2hl-\-2ai2kl-\-2atl.
Now the congruences
+2a1Mk,^-2au ) •
2an?il-\-2a2tkl = —2an
are solvable simultaneously for A, and kh the determinant
4anai2—(2ali)i
being odd. Thus
9(yi,-,y,) = n1y\-\-myly2+n2y\+2'i
g"(y3,...,y,) (mod^),
where m is odd and the notations are self-explanatory.
It is clear how Lemmas 2 and 3 (a) follow by repeated applications
of these results. Lemma 3 (b) is a corollary of the following result.
LEMMA 4. Let m be odd, n^, n2 be integral, and t be positive. Then
»1 if+ma;1 x2 +n2 a^ is equivalent to a like form in which 7^ is odd and
m has any desired odd residue, to modulus 2*.f
•f It should be noted that then n, is odd or even according as the discriminant
mt
— 4n1 n, is congruent to 6 or 1 (mod 8).
ON INTEGRAL QUADRATIC FORMS 37
If % is not odd, but nz is odd, we employ P12; if n2 is even also,
we employ S12. Suppose n^ to be odd. By the unitary transformation
^I =
J/i+%2. X
2 =
J/2. m
Soea m
*o W = m+2Are1, and A may be
chosen to give m' any desired odd residue, to modulus 2f.
To prove Lemma 3 (c) we have
LEMMA 5. Let m, m' be odd, nlt n2 be integral, and t be positive.
The form mx\-\-2(n1x\-\-m'' x2x3-\-n2x\) is equivalent, to modulus 2f, to
afOnU
rn1yt+m2yl+m3yl (26)
in which m1; m2, m^ are odd.
Replacing x2 by y2+yv x1 by ylt x3 by y3, we obtain
mx yl+Zn^ yx yi+2m'y1 y3+ 1nx y\+2m'y2 y3+2n2 y\,
where m1 = m+2?^ is odd. Now write
yx = ZJ^+AJZJJ+ASZS, y2 = z2, y3 = %.
The coefficients 2mlh2-\-2n1 and 2m1h3-\-2m' of zxz2 and zxz^ can be
made divisible by 2? by choice of an odd h^ and an integral h2. The
new form is congruent, to modulus 2f, to m1zl-\-S, where 8 is a binary
form in z2, z3 in which the coefficient of z\ is odd and that of z2z3
is even. i
After arranging the s numbers
0i+l>&!+l»-,/3u+1
> y1.y1.y2.y2.--.yr.Vr. (27
)
where no ft+1 is equal to any yp in order of magnitude, we denote
them by o^, ^.....a,. Thus at < a2 < ... < <xa, and each c^ is either
a /?+1 or a y. Let the number of distinct values among the <xi be
q, and arrange them into q sets 2r (r = l,...,g) of elements of equal
value, thus:
Sj = (a^aj,,...,^), 2 2 = (ag+1,...,aSi),..., £8 = (a^.^i,...,«,). (28)
We may write sQ = 0, aq = s.
The variables yi in (20) may be rearranged accordingly: thus
with each Sr there is associated a form 2*'^ in the variables yk
(k = 8,^+1,..., ar), where er +l is the constant value of the elements
of 2r. If these elements are of type
where the mk are odd. But if the elements are of type y, sr—sr_1 is
necessarily even, say sr—8r_l = 2h, and
38 GORDON PALL
where the fa are odd binary quadratic forms of the type in Lemma 4,
in the successive pairs of variables yk. By Lemma 3, every / is equivalent,
to modulus 2*, to a canonical form of the type
4> = 2*01+2*08+... + 2 H r (29)
These results should be expressed in a form analogous to Lemma 2:
LEMMA 6. Let t > 0. Any classical, integral quadratic form in 8
variables is equivalent, to modulus 'if, to a form of the type (29), where
ifil,...,\fiq are classical, integral quadratic forms, each in variables different
from those of the remaining forms, and each of odd determinant;
and the er are integers (0 < e1 < e2 < ... < eq).
LEMMA 7. Let t > 0. Let >p be a classical, integral quadratic form
in v variables, of odd determinant. Then, if ifj is even, ifi is equivalent,
to modulus 2?, to a form of the type
m1x{+msa%+...+mva*, (30')
where the mt are odd integers. If <fi is odd, then v is even, say v = 2r,
and ifi is equivalent, to modulus 2f, to a form of the type
2(m1xs
1+mWx1xi+n1x*)+... + 2(mrx*v_1+m<r
h:yL1xy+nrxl), (30')
where the m^ are odd integers, the ni are integers, and the mw
are arbitrary
odd integers.
A canonical form, (19) if p > 2, and (29) if p = 2, of / is called
a principal residue of f to modulus p1
.
In connexion with (29) it is useful to define
dk = 0, if af c isaj3+l \
1, if ak is an initial y >. (31)
— 1, if ak is a terminal y )
An initial y denotes a term ak of type y occupying an odd place of
its set 2 r in (28), that is, for which sT_x < k ^ sr and k—8r_x is odd;
a terminal y occupies an even place. An initial y — <xk and its succeeding
terminal y = ak+l may be called twins.
With each pair of twin terms a^, ak+1 of type y is associated a
matrix , ,v
a
% (32)
\6 cJ
where y = <xk = at+1, a = Zr+hn, b = 2?m', c = 2r+1
n, m and m'
being odd, n integral. If ^ is a principal residue of /, to modulus 2*,
the matrix of 2<f> consists of a series of single terms 2P+1
mk (ak = £+1)
and binary matrices (32) situated along and symmetric with the
ON INTEGRAL QUADRATIC FORMS 39
principal diagonal, all remaining elements being zero. Further,
alt...,as are in order of magnitude, and any <xk of type y is unequal
to the nearest preceding and succeeding <x's of type j9+l. It should
be noted that ac—b2
= 2a
*+a
*+iJlf, where M — imn—m'* is odd.
8. Proof of Theorems 1 and 2. With Minkowski let us denote
the exponent of the power of p dividing dk by dk = dk(p). By (11),
d0 = 0. Forms which are equivalent, to modulus p1
, for a sufficiently
large t have the same values dk. It will suffice to have t > <xs in
(19) and (29).
(i) p > 2. By (19) we have
Hence 8k ^ 8k_1. The numbers a>k defined, when p > 2, by
«* = (3*+ i-3*)-K-3*-i) (* = I,-,*-1), (33)
are also positive or zero, since in fact
w
k = a
*+i—<**• (34)
Plainly, ojt is the exponent of the power of p dividing ok in (12).
(ii) 2> = 2. The exponent of the power of 2 dividing ok is
<"* = dk+1-28k+8k_1+2{l + (-l)"} (k = 1,...,«-1). (35)
Let us write
Pt = M-aj+.-.+ofc, (36)
efc = 0, if ak is a /?+1 or a terminal y,
= 1, if ock is an initial y. (37)
Hence, by (31), 0t = ek—ek_v
B}r
the sequel to (32), the exponent of the power of 2 dividing the
leading principal minor determinant of order k (in the matrix of 2<j>)
is pk+ek, and this is the greatest power of 2 dividing all the principal
minor determinants of order k. Further, the exponent of the greatest
power of 2 dividing the doubles of all the secondary minors of order
k is never less than pk+l, and is pk+l if <xk is an initial y. Consequently
we have /*fc28
* = 2*+«i> (38)
fk is odd or even according as ek = 1 or ek — 0. (39)
Hence, at once, by (9), (38), (36), and (37),
0*+i-3* = «*+i-(-l)fc
+0*+i (fc = 0,l>...,«-l). (40)
Except possibly when ak+1 = 0 or 1 and 6k+l = — 1, it is now obvious
that dk+l— dk ^ 0. This is true even in that case, for then aA.+1 is
of type y and equaPto <xlt whence k is odd.
40 GORDON PALL
Finally, by (35) and (40),
«* = «*+i-«*+2+0*+1-0*Since
at+1 ^ <xk, and, by (31),
oifc = 0, if and only if ak is an initial y, (42)
i.e. if and only if efc = 1,
i.e. if and only if fk is odd.
This completes Theorem 2. Also, for k = 1,...,«—1, (41) yields
oifc = 0 or o)t ^ 2;
then a>ft_j > 4 (A; > 2), wt+1 > 4 (fc < «-2). '
These facts involve Theorem 1.
Corresponding to (31) we now have the following values for 6k:
0 A = - 1 , if 0 , ^ = 0, \
= 1, Xwk = 0, ( 4 = l , . . . , « - l ) . (44)
= 0 otherwise J
We can thereby determine a^,..., <x8 and the type /3+1 or y of each a{,
uniquely in terms of 8t and co^..., tUg^. For we have
ock = <*!+...+ak_1-2k-0k+3+d1, (45)
ak is of type y, if and only if cok = 0 or tot_1 = 0. (46)
8 a. The correspondence with the Minkowski-Smith invariants.
By (41), ak+1 = ak, if and only if
<vk = 2; or cut = 0; or wk_1 = 0 , u>k = 4, a>i+1 = 0. (47)
Thus <xk+1 is the first element of its set Sr in (28), only for the values
k not satisfying (47).
It is to be observed that the classical Minkowski's ok and Smith'?
Ik, which are identical and which we denote temporarily by Smith's
notation Ik, are related to ours by the equation
°k = ^ t - i I
k a
k+i!<*k (48)
where ak is 1 or 2 according as ok (or fk) is even or odd. It is easily
verified by use of (13) that the cases (47) are precisely those in which
Ik is odd.
9. A special condition if ov o3,..., o8_1 are odd, s even. Let
ov o3,..., <>,_! be odd, s even, whence, in (20), 8 = 2v, u = 0. We may
suppose dl= 1. Then (29) is of the type
g = &+2<V,+ 2»i06+... + 2<*«0gB_1> (49)
ON INTEGRAL QUADRATIC FORMS 41
where each ifti is an odd, primitive binary quadratic form in the
variables yit yi+1. By (10) and (14) the determinant of 2g is of the
f 0 r m
{Yo<ho
where A and M are non-negative integers. But the product of the
determinants of the %\ii is the product of \s numbers of the form
4n— 1. Since the modulus 2? can be taken arbitrarily large, we
must have (-jo-JW^o,... Og_1 = i (mod4). (50)
THEOREM 3. The condition (50) is satisfied by the signature 8—21
of any order offorms (1) in which s is even and ov o3,..., o,_1 are odd.
10. Necessary and sufficient conditions on (1, /,• o1,...,ol_1) for
an order to exist. Smith's enumeration! of the further relations,
in addition to (13) and Theorem 3, which the invariants
defining an order must satisfy in order that corresponding forms may
exist, is incomplete; his specification of certain relations to be satisfied
by the generic characters probably covers this omission, but is,
of course, complicated to apply. Minkowski's discussion, covering
the generic characters, is rather intricate.% The relations in question
take a distinctive form in our notations. In view of these circumstances
we shall now present a direct investigation for the simpler
case of an order. The extension to a genus is then in fact more
perspicuous.
Analogously to § 4, let
p = (l,...,i-l), r=(l,...,k+l) (0<k<s).
Then we have the identity
A[ao]A[a'a']-A[aa']A[a'a] = A[PP]A[TT], (51)
for any square matrix A of order s, a well-known relation for a
second-order minor in the adjoint of the determinant -4[TT].
Let A be the matrix of 2/, whence ^4[CT<T'] = ^4[CT'CT]. The leading
principal minor determinant of order k, namely, ^4[aa], is of the form
fikdklk, where lk is an integer. In particular, by (10) and (11),
lo=l, l8 - ( - 1 ) ' . (52)
The sequence of numbers lo,lu...,ls (53)
forms a. reduced leading chain of minor determinants of f; lk is the
leading coefficient of Fk (k = l,...,s— 1).
t Loc. cit. 512-13. X Loc. cit. 78-9.
42 GORDON PALL
We require the following theorem of Minkowski and Smith:
LEMMA 8. In the class of forms equivalent to f there are forms in
whose reduced chain (53)
lk is prime to 2llc_llk+1o1o2...os^1 (k = l,...,s— 1). (54)
The proof, which is worth our while to review, is based by
Minkowski (whose proof, pp. 21-2, is faulty) on the lemma that if
<f>v...,(f)r are any r forms in the class of /, and Nlt..., Nr are any r nonzero
integers which are relatively prime in pairs, then there exists
a form <j> in the class of / such that <f> = fa (modiV,) (i = l,...,r).
Smithf gives a satisfactory proof of a more fundamental lemma,
from which Minkowski's follows: if a determinant \t{i\ = 1 (modi^),
we can alter the elements tti by multiples of N to secure a determinant
actually equal to 1; the extension to moduli Nv...,Nr
relatively prime in pairs is obvious.
Consequently, by Lemmas 2 and 3, there exists in the class of /
a form <f> which is, to modulus p'p, a principal residue of / to the
same modulus for any number of powers of different primes. Such
a form is called a principal representativeX of / to modulus JJ pf). We
shall include among the pi all the primes dividing 2o1...oa_1, and
shall always suppose n^ > as(pj). The latter convention ensures that
the leading principal minor determinant of order k in the matrix of
2+ is of the form
where nij is prime to pp for each j ; hence it is of the form fikdklk,
where lk is prime to all the Pj.
It remains to secure that lk be prime to lk_v Minkowski's treatment
at this point (p. 72) seems to be not quite complete, but is
supplemented by Bachmann.§ We have I, = ± 1 . If A; is the largest
integer for which lk is not prime to lk_v consider
i/i(xu...,xk) = ^(x1,...,a;t,0)...,0).
Then d1,...,dk_l are the same for iji as for <f>, but dk(ip) = ±dklk. If
we apply t o / any unitary transformation T leaving xk+1,...,xs unaltered,
the determinants of </r and of <j>(z1,...,x{,0,...,0) (k < t ^ s)
are unchanged, whence 74., lkil,..., ls are unchanged. We employ such
a transformation T which carries >p into a principal representative
+ Op. cit. ii. 635-6.
J The determination of a principal representative in a finite number of
ste])S is discussed by Minkowski (pp. 33-5).
§ P. Bachniann, Die Arithmetik der qiutdrntischeii Formen, i. 452-3.
ON INTEGRAL QUADRATIC FORMS 43
of itself for the primes dividing lk as well as 2o1o2...oa_1. Then the
new lk_t will be prime to lk, and we have reduced the problem to
a lower value of k.
To proceed: by (9), we may write (51) in the form
where z and l'k are integers; and hence by (12) we have
—oklk-ilk+i = zi
— Ukl'k. (55)
Consequently Lemma 8 implies
LEMMA 9. If the order (.1; I;o1,...,og_1) actually contains forms, there
exist integers l0 = 1, ll,...,lt-1,la = (— I)7
satisfying (54) and such that
the congruences - o j ^ l ^ = z\ (mod4Z4) (56)
are solvable in integers zk (k = 1,...,«—1).
The index / of/is equal to the number of consecutive sign-changes
in a chain of principal minor determinants and hence in the sequence
(53). We shall hereafter assume (54), so that none of the lk are zero.
We write ek = + 1 or —1 according as lk is positive or negative,
whence e o = l , c3 = I, = ( — I)7
, tklk > 0 (k = 0,...,s).
Since (54) holds, (56) implies both of
U = (-°*I*-i**+iM*) = 1 (* = I,-,*-1) (57)
and —OfcZfc..!^! = 1 (mod 4), if ok is odd. (58)
It is easily verified that
(li+i\e<h)(-l<\e<+ik+i) = *tK (• = 0,...,a-l), (59)
where
(60)
and it is plain that Ci ••• £s-i i8
^ e
product of the s left-hand members
of (59) by the s—1 numbers
v*=(o*|e*y (*=1> ...,«-1). (61)
Consequently, by (57), (56) requires £i...£g_i = 1, that is,
«o-ViV-ViI
'i-Vi= L
(62
)
Now / is the number of consecutive sign-changes in (eo,...,ee). (63)
To each change from -f- to — corresponds a factor A^ = — 1. Hence
and (62) reduces to
44 " GORDON PALL
We shall now show, that, if there exist integers l0 = 1, lv..., ls-u
l3 = (— I)1
(the sign ±1 of lk being represented by ek) satisfying
(54), (58), (63), and (64), (P)
then there exist other integers lk having the same properties and
signs and also satisfying (56); and hence we shall be able to construct
a form in the order (l;I;ol,...,oa_1).
For set m0 = l0 = 1. Without affecting the validity of (P) we can
replace lt by mv where e1m1 is a positive prime (or 1) and ml = lx
(modSoj). Next, by (54) and (58), we can replace l2 by m2, where
e2m2 is a positive prime, m2 = l2 (mod8o2), and the congruence
—o1Tn0m2 = z\ (modkm^)
is solvable for an integer zx; the truth of (P) being retained. We can
proceed in this fashion until we have chosen e,_im
»-i to be a positive
prime (or unity) satisfying
—os_2ms_3TOs_1 = zf_2 (mod4mg_2), ma_1 = l,^ (mod80^^;
while the system Z0->-l, Z2->•»»!, ..., la-i-*• m^^ I, -*• ms = (— l)z
satisfies (P). Further, with the m4 substituted for the l{,
whence (64) shows that £s_x = 1, and hence that the last of the
congruences
~°km
k-im
k+i = z
l (mod4mt) (k = l,...,a— 1) (65)
is solvable for an integral zg_v
We shall now try to construct a form <f> in the order (1; /; ov...,oa_1)
with a matrix for 2<f> of the type
" 2
^o w
i
Wl 2vx w2
Wo 2v9 ws
2t>,
(66)
with zeros elsewhere. Let Mk denote the leading principal minor
determinant of order k. We wish to identify Mk with \ik dk mk. Thus
3t0 = 1, Mx = 2v0 = 2d1m1, whence v0 = mlt and generally, if
0 < k- •< s, ^ „ TUT ,,,2 iw
(67)
ON INTEGRAL QUADRATIC FORMS 45
Now, since by (12) or (14)
4*~V*<** = 2o1o2...o4_1/xk_1dt_i> (68)
we must have (for k = I,...,*—1)
42
-2i
of...o|_1ofcmi+1+mk_1«^ = 4*-k
o1...ok_1vkmk. (69)
We shall therefore, in view of (65), writef
Okm
k+l+ m
k-lZ
l = 4m
t<
*' V
0 = m
l>
wk=^-"o1...ok_1zk, vk = 41
-"o1...ok_1tk (k = l,...,a-l). (70)
By this construction the leading minor Mk has the value fikdkmk,
(k — 0,...,«). Hence the index, determined by the signs of the mk, is /.
Since mk is prime to ^k+\dk+1mk+x = Mk+1, we have merely to see
that all the remaining principal minors and twice all the secondary
minors of (66), of order k, are divisible by iikdk, to ensure that dk is
the g.c.d. of order k. Every such minor M' of order k is obtained by
bordering some Mr (0 < r < t < «) with k—r rows and columns,
the first row being at least the (r+2)th, the first column at least the
(r+l)th.
On bordering Mk_x with the (A+2)th row and column, we obtain
a determinant having the value 2vk+1nk_1dk_1mk_1. The quotient of
this by nkdk is ^m*^ok tk+1, by (68) and (70), and must be integral. By
(70), if ok+1 is even, 4|zi+1 if and only if 4|<t+1; hence we need
4|zfc+1 and 4|<t+1 for each k such that ok is odd (0 < k < s— 1). (71)
To satisfy this condition we replace zk+1 by 2\mk+1\—zk+1, which is
also a solution of (65), if zk+1 = 2 (mod4). Condition (71) is finally
seen to be sufficient.
For, generally, the minor M' is equal to /xr dr mT multiplied by zero,
or by a sum of terms of a type ±ux... uk_r characterized as follows:
let r + 2 <C ht < h2 < ... < hk_r ^ s; u{ is one of the numbers
W
A.-1. 2
«A,-1. w
h,
chosen from the A4th row, no two factors u^,...,uk_r belonging to the
same column in (66). The terms of a secondary minor M' are distinguished
by containing at least one factor u{ = wh such that neither
ui+1 nor w ^ is a.lso wh; in such a case we can prefix a factor 2 to
each term. '
t Minkowski (loc. cit., 77) introduces an extra factor on in his definitions
corresponding to v* and wk, in order to simplify his discussion. But this
weakens the analogy with the best treatment for s = 2 and 3, and hampers
an extension to the case mk not prime to ok.
46 GORDON PALL
By (68), pkdk = nrdr 2£r+12fr+8... 2&, where
iH = ox...oh.yl^-^ (h=l,...,s). (72)
Hence we have merely to prove the integrality of
W(2£+i)}-K-r/(2f*)}> (73)
or of its double, if M' is secondary. We abbreviate
Sit = W£-+< = <W< -. Or^-l/**-* (0 < » < J < »~r). (74)
By (70), wfc = £kzk, vk = £ktk, and the tth factor of (73) is of type
2vr+j/(2$r+1) = ^tT+i or wr+j/(2£r+1) = tf{jzr+i. (75)
Now, by (13), £y is an integer unless
or+i, or+t+2,..., o,^.! are odd (j—i odd); (76)
and in this case 4^u is a n
integer, and hence £ijtr+i, £i)Zr+i are still
integral by (71). If then there is only one factor of type ££^zr+y in
(73), M' is secondary and the prefixed factor 2 ensures the integrality.
Finally, if there are two or more such factors, consider any of
them other than the last, say $£{izr+i. If or+i is even, zr+j is even
by (70), and £ti is an integer save in case (76); if or+j is odd, \Q\oT+j^
and hence, by (13), £y is a multiple of 4 unless"
or+{, or+M, ...,or+j_2 are odd (j—i even), (76')
in which case £{i is^still an integer. Thus £|^2r+j may be half an
integer only in case
or+i,or+i+l, ...,or+i_K (K = 0OT 1) are odd. (77)
The succeeding factor in (73) may be (i) $£i+ijZr+i (i <j < s—r),
(ii) Ui+i,j+hz
r+i+h (*<.? <j+h^s—r), or (iii) £i+1J+htr+j+h. In
case (i), (77) implies that 16|or+i+1 whence either £i+ij or zr+i is a
multiple of 4 and both are integral; in case (ii) or (iii) similarly,
(77) and (71) compel the factor to be a multiple of 4.
What, finally, are the conditions on / and the ok, beyond (13) and
Theorem 3, that there shall exist integers l0 = 1, lv-.., la-v la = (— I)1
(with signs eO)...,ea), satisfying (P)?
We can choose infinitely many sets of integers l^...,^-!, with
arbitrary signs ek and arbitrary odd residues ± 1 (mod 4), to satisfy
(54). Condition (58) limits the possible residues, to modulus 4, but
leaves the choice of signs unrestricted except when o1,o3,...,ot_1 are
odd and s is even; then (58) requires that
{-Y**o1oa...o^1l0l, = 1 (mod 4),
where l0 = 1 and ls = (—1)^, which is condition (50).
ON INTEGRAL QUADRATIC FORMS 47
With this one restriction on / satisfied, we consider the effect on
vk of replacing lk by Zt+4n, where n is an integer of the same sign
as lk and (54) still holds. If ok is not a square there are infinitely
many values n for which vk = 1 and infinitely many for which
vk = —1. Consequently (64) can be satisfied except when
all of ovo2,...,o8_1 are perfect squares. (78)
When (78) holds, v1 = ... = vB_x = 1 and (58), (64) become
lk+1 ^ Zt_1 (mod 4) whenever ok is odd, (79)
*o"i-*»-i = (-1F+1
"2 1
(*< = (—l)d+W/+i-iV4). (80)
Evidently ^ = — 1, if and only if Z4 = 1 and l{+1 = 3 (mod 4). We
call such a consecutive pair l{, li+1 a (1, 3)-change. Then (80) gives
the number of (l,3)-changes in lo,...,ls is congruent, to
modulus 2, to [(/+l)/2]. (81)
Given an index / and square invariants o1,...,og_1, the remaining
question is: can we choose odd lk, whose signs ek agree with (63), to
satisfy (79) and (81)? We shall now. see that such a choice can be
made, if there are three different values of k (0 ^ k ^. s— 1) such
that ok = ok+1 = 0 (mod 4); and that, when there are not more than
two such values of k, the choice can be made if and only if
5—2/^3,4,5 (mod8). (82)
For the only restrictions on the residues of the lt to modulus 4 are
(79) and the values l0 = 1 and ls = ( — I)7
. We are free to assign to
any particular lt either residue ± 1 (mod 4), unless either ovo3,...,oi_1
are odd, in which case l( = (—I)''2
, or ol+1,oi+3,...,og_1 are odd,
whence Zj = (— 1)'-K«-O/2
.
To expedite the counting of (1, 3)-changes consider the case of
°t+2>°(+d>—>°j-i< oj+2,oj+4,...,ol._lodd (83)
(whence ot and oi+1 are even), where — l^Li<j<k^.s, a,ndj—i
and k—j are odd. The number of (I, 3)-changes in /,-+i,^+2,--A- is
independent of j , and depends only on i, k, and the choices of ll+1
and lk; it is therefore the same as in the case
°i+2.0f+4>-"> °k-2 odd, ok_l and ok even, (8+)
with the same choices of ll+1 and lk. For the residues of ll+v ll+3,....lj
and of lic,lic-2>—>lj+i a r e
alternately 1 and 3, or 3 and 1, depending
on li+1 and lk; the values of li+2>h+*>-~>h-2 &n(
^ h+3'h+&>—>h-2 a r e
48 GORDON PALL
therefore immaterial and may be disregarded in counting (1,3)change8.
If now j < k— 1 and we replace j by j-\-2, the number of
(1, 3)-changes is unaltered: e.g.
131, 313 -• 1313,13; 131,1313 -• 1313,313.
Suppose then that oh and oh+1 are even for the three values
h = i,j,k at least, where O^i<j<k^. a—I. Employing the
preceding transformation we can suppose that there are four consecutive
even ok, say ok_i, ok_lt ok, ok+1. Since the residues 1 or 3 of
Zj.^ and lk are unconstrained, we can satisfy (81), whatever be the
values of lk_2 and lk+1, as is plain from the following schemes:
1111 or 1331; 3113 or 3333; 1133 or 1313; 3131 or 3311.
Next let there be only two values h for which oh = oh+1 = 0
(mod4), say i and j (0 ^ i <j ^. 8— 1). Then ov o3,...,oi_1, oi+i,
oi+v...,oj_1,oj+t,oi+A,...,oll_1 are odd; i— 1, j , 8—j are odd, 8 is even.
By the above transformation we can suppose (without loss) that
ovo3,...,o8_3 are odd; og_2, os_lt and o, even. Then ls_s = (—I)*'-2
"2
,
ls = ( — I)1
. If %(8— 2) = / (mod2) the choice l,^ = 1 or 3 gives
at will an even or an odd number of (l,3)-changes. If, however,
8—21 = 0 (mod4) the number of (l,3)-changes is J{«-f-l — ( — I)2
},
whence (81) excludes only a—21 = 4 (mod 8).
If ol,o$,...,os_l are odd, 8—21 = 0 (mod4) by (50), and, as the
number of (1, 3)-changes is again J{«+1 — (— I)7
}, 8—21 = 4 (mod8)
is the only excluded possibility.
Finally, if oh and oh+1 are even for only one value of h, we can
transform it to be h = 8—1. Then s is odd, and the number of
(l,3)-changes is J{s+2—(—1)'}, and (81) thereby rejects only
s— 21= ± 3 (mod 8).
THEOREM 4. Let 0 < / < s, let o1,...,o,_1 be positive integers, and
let o0 = o,= 0. Then a form exists having these invariants, if and
only if
(i) ofc^2(mod4) (0 < k < a);
(ii) if ok is odd, ok_l = ok+1 = 0 (mod 16);
(iii) if o1o3...og_1 is odd, s even, then (—)1(
*~2i)
o1o3...o4_1 = 1
(mod 4);
(iv) if all the ok are squares, and if ok = ok+1 = 0 (mod 4) for not
more than two values of k (0 < k < «), then 8—21 ^ 3,4,5
(mod 8).
ON INTEGRAL QUADRATIC FORMS 49
It is worth while to remark the special result:
If 8 = 2, 3, or 4, and / = 0, and if all the ok are squares,
then none of the ok can be odd. (85)
11. The o-invariants of the comitants of the comitants.
Smith, who was apparently the first to recognize the intermediate
concomitants [comitants] ft,...,f,^2, left an interesting point unsettled.
This we now propose to elucidate.
If s > 3, there are, besides the fundamental concomitants/i,.-.,/,-i,
an infinite number of others, namely the concomitants of the concomitants,
and so on indefinitely. Their invariants being also invariants
of/, Smith remarksj" that 'it is important to know whether,
in order to obtain the distribution into orders, it is, or is not, necessary
to consider these other concomitants'. He states that 'it can be
shown that it is unnecessary to consider any concomitants other than
the fundamental ones, as regards the primary divisors' [the g.c.d.'s
of all the minor determinants of any given order]. 'It is probable
(but it seems difficult to prove) that the same thing is true for the
secondary divisors' [the g.c.d.'s of all the principal and doubles of
the secondary minors of any given order].
By our results (Theorem 2) the primary divisors are completely
determined by the secondary divisors, and it is required only to show
precisely how to find the o-invariants of fk from those of fv
The simplest relationship among these invariants, namely,
°i(A) = o*(/i) (k=l,...,a-l),. (86)
is of some importance and easily proved otherwise. The index /' of
fk is readily expressed by the following formula in terms of the index
To proceed, denote by p°" the power of p dividing oi(/1)
(» = 1,...,«— 1), and by pwu
the power of p dividing oh(fk), where
0 <k <8 and 0 < h < A (A = sCk).
In (19) or (29), there are A sums of the numbers a1,...;<x3 taken
k at a time. Arranged in ascending order of magnitude they may be
denoted by Sj,..., 8^; thus 8X = aj+.-. + at, 82 = a14-...+at_1-(-at+1,...
(0 < 8X < 82 < ... < 8A).
First, suppose p > 2. If we take the kth comitant of the canonical
•(• Loc. cit., 415.
3695-6
50 GORDON PALL
form (19) of / we obtain a canonical form x of like pattern iov fk, the
modulus p1
being sufficiently large. It is plain from x a
P<i (34) that
• «** = S*+i-8* (A=1,,.,A-1). ' (88)
In particular as regards (86), o>u = ak+1—ak = <ak.
Secondly, suppose p = 2 and t sufficiently large. The kth comitant
of (29) is not in general in the pattern of (29), but we shall-see how
to put it into that form.
Each sum 8^ is of the form
«il+ail+...+«fc (1 < h < »2 < ... < tfc < a). (89)
With the sum 8^ we associate the subsequence (*i,...,tjt), and denote
this subsequence by oy The A subsequences of (1,...,«) of k elements
are, in a certain order, al,...,O)l.
There is an indeterminacy in the arrangement of equal sums (89).
Each ah of type y is associated with its twin, which is aA+1 or cc1l_1
according as ah is an initial or a terminal y. If r (^ 0) terms ah
(h == &!,...,&,., say) occur in a sum (83) without their twins, that sum
forms part of a system of 2r
twin sums, obtained by replacing some
or all of the terms aA (A = h^,...,^) by their twins. The sums 8^ may
" be so ordered that a system of twin sums occurs consecutively.
Let M denote the matrix of (29). The matrix of the jfcth comitant
of (29) has as its element in the ith row and jth column the value
of the determinant M[at aJ. Now the elements of one row of M[at a^]
are all zero unless 8{ and 8y belong to the same system of twin sums.
Consequently the matrix of the kth comitant consists of a series of
square matrices situated one after the other down and symmetric
with the principal diagonal, with zeros everywhere else; one
square matrix of order 21
" corresponding to each system of twin
sums (89).
Consider such a system of 21
" twin sums, and denote the corresponding
matrix of order 21
" by R. We may suppose r ^ 2. Necessarily
r ^ k. Let ^1,..,^_r be the indices t of those c^ which are
common to all the twin sums of the system; and let ah (h = hv..., hr)
be the initial y's of the system, so that ah (h = Ax-f 1,..., hr-{-1) are
the terminal y's of the system. Then all the elements of R have
a common factor of the type
2T
m (r = ail+ait+...+oilt_r),
where m is an odd integer. Write R' for the matrix R/iFm) obtained
ON INTEGRAL, QUADRATIC FORMS 61
by removing this factor. Then R' may be described as follows. We
may set la h\
(< j (90)
where a< = 2r<+1
oi<, bt = 2wm», c, = 2w+1
tti (»' = l,...,r), the m, and
m(f)
being certain odd integers, the n{ certain integers. The elements
of R' are the 4r
possible products of r elements, one chosen from each
of the matrices v<. Thus the first row consists of all products in which
one element is taken from the first rows of each of v1,..., vr; the second
row employs similarly the first rows of vv...,vr_1 and the second row
of vr\ and so on, the selection of rows being parallel with the selection
of columns:
a^i...ar a^...aT_xbT a1...of._26f._1or . . . 6162...6r
In the notations (90), if we set 77 = y1+...+yr, then T+TJ is the
common value of the sums 8^ of the system. The factor 2V can now
be removed from every row of R'. The matrix left behind is evidently
the matrix of a classical, odd form, since the diagonal elements are
all even while at least one secondary element (6X ... 6r/21
>) is odd. The
form is, in fact, also of odd determinant, since, reducing the elements
to modulus 2, we substitute 0 for at and ct, 1 for bt in (90) (» = 1,..., r),
and obtain in place of R' a matrix in which a unique element in
each row and column is 1, the rest zero.I Thus Lemma 7 applies
and shows that the form of matrix R is equivalent, to modulus 2',
to a form of the type (30"), multiplied by 2T
+7
'.
Among the sums 8^ in (89) certain ones may have no twin sums;
these correspond in R to an isolated term 2S
J+L
m^ (m^ odd) as the
jth element on the principal diagonal. If any such isolated term
exists with 8y+l equal to the value T+77 of a system like that considered
above, that system together with the isolated term can be
brought as in Lemma 5 to a diagonal form.
The Jfcth comitant of (29) is thereby transformed into a form of
the same kind as (29), in a manner which determines uniquely the
powers of 2 dividing the various quantities oh(Jk) in terms of the
powers of 2 dividing o^-.^o,^.
t It can indeed be shown that the determinant of R' is equal to the product
cf the determinants 4mini—m^1
each raised to the (ST^Jth power.