THEORY OF REDUCTION FOR ARITHMETICAL EQUIVALENCE. II(1) · 2018-11-16 · THEORY OF REDUCTION FOR ARITHMETICAL EQUIVALENCE. II(1) BY HERMANN WEYL 1. Introduction. Lattices over the

THEORY OF REDUCTION FOR ARITHMETICALEQUIVALENCE. II(1)

BY

HERMANN WEYL

1. Introduction. Lattices over the unit lattice. Given « linearly independ-

ent vectors bi, ■ • • , b„ in an «-dimensional vector space £", the formula

(1) ï = yibi + • ■ • + y»bB

yields all vectors of the space E" or of a lattice 8 in E" if the coordinates y¿

range over all real numbers or all integers, respectively. We take the viewpoint

that the lattice 8 is given but the choice of its basis arbitrary. The several

bases are connected with one another by unimodular transformations. If /(r)

is a gauge function assigning a "length" /(r) to each vector r the problem of

reduction requires normalization of the lattice basis in terms of the given /.

A solution is sought for all possible gauge functions or at least for some im-

portant class. The most significant class is obtained by running f2 over all

positive quadratic forms.

Following in Dirichlet's and Hermite's footsteps, Minkowski developed

such a method of reduction for quadratic forms and established the decisive

facts about it. In Rl I approached the same problem in that geometric way

which Minkowski had initiated but then abandoned for unknown reasons.

The question may be put in a slightly different form. All linear mappings

of En carrying £ into itself carry /(j) into equivalent gauge functions. The

task is to pick out by a universal rule in each class of equivalent gauge func-

tions one particular/(j) which is called the reduced function of its class. Let

'Ro, "R., C m tne future denote the fields of all rational, real and complex num-

bers, respectively. Complex numbers are written in the form £ = Xo+*i2

(xo, Xi real). It is convenient to insert between the full vector space and

the lattice £, the set El of all vectors (1) with rational coefficients yit a set

which we describe as an «-dimensional vector space overeo- Crystallography

has found this advisable in distinguishing between the macroscopic and atom-

istic symmetries of a crystal, and in the theory of algebraic numbers one puts

the field before the ring of its integers.

Let a lattice £ in El be given. With respect to any basis bi, • • • , b„ of El,

formula (1), the function/(j) is represented by a function g(yi, • • • , yn) and

the lattice ? by a "numerical lattice" A whose vectors are «-uples (yu ■ ■ ■ , yn)

Presented to the Society, January 1, 1941; received by the editors December 11, 1940.

(') The first part, which appeared under the same title in these Transactions, vol. 48 (1940),

pp. 126-164, is cited as Rl.

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204 HERMANN WEYL [March

of rational numbers. (Only if bi, • • • , b„ is a true basis of 8 will A be the unit

lattice I whose elements are the w-uples of integers.) Hence/(r) with respect

to 8 is represented by g/A. All representations g/A of/(r)/8 are equivalent,

i.e., they arise from one another by linear transformations of the coordinates

with rational coefficients. In each class of equivalent g/A we are to pick one

individual, the "reduced" g/A. Suppose we have succeeded in doing this by

some universal rule. We then have to select, for each A that may occur in a

reduced g/A, a definite basis b*, • • • , b„* in terms of which 8 is represented

by A. The equation

f*(l) = g(yu ■ ■ ■ . yn) for r = yibi* + • • • + ynb„*

then defines the reduced gauge function /* in its class. By the first step of

reducing g/A no essential progress has been made unless the lattices A which

may occur in a reduced g/A are limited to a finite number of possibilities.

For only then is the selection of a basis b*, • • ■ , b„* for each of these A essen-

tially simpler than the original problem.

The Dirichlet-Hermite-Minkowski method of reduction by admitting only

bases bi, ■ • • , b„ of 8 always represents 8 by the one lattice A = I, the unit

lattice. Thus it provides the ideal solution. Minkowski's construction of con-

secutive shortest distances in the lattice

/(bi) = Mx, ■ ■ ■ , /(b„) = Mn

(for which he obtains the inequality Mx • • • M„V^2") falls under our more

general scheme. That theorem which he describes as indicating a certain

"Oekonomie der Strahldistancen" states exactly that there is only an a priori

limited number of possibilities for A with which to count in a reduced g/A.

In Rl I carried the first method over to those other fields and quasi-fields

which have not more than one infinite prime spot, and I found that it works

only under the hypothesis that the class number for ideals is 1. Simultane-

ously Siegel observed that the rougher second method, by which incidentally

Minkowski had proved that the class number of positive quadratic forms

with integral coefficients and a given discriminant is finite, operates without

this restrictive hypothesis(2). I add the remark that an argument making no

use of the bases of a lattice need not even assume their existence. In an alge-

braic number field y we consider any "order" [j]; in general there are several

classes of lattices belonging to this order. The theory is limited neither to the

principal class nor to the principal order. Following a suggestion by Siegel,

P. Humbert generalized the investigation of quadratic forms to an arbitrary

algebraic number field J with several infinite prime spots(2). No doubt the

whole problem thereby loses much of its simplicity. But once upon this track

one ought to include the quaternions and thus deal also with those noncom-

(2) See P. Humbert, Commentarii Mathematici Helvetici, vol. 12 (1939-1940), pp. 263-306.

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1942] ARITHMETICAL EQUIVALENCE 205

mutative division algebras of finite degree over fr\.0 for which the concept of

infinite prime spots goes through. I resume here the rougher method of reduc-

tion with these further generalizations by the same geometric approach as

in Rl. I am not only interested in the fact that certain numbers are finite;

1 wish to ascertain reasonably low explicit upper bounds for them. The geo-

metric method yields good results in this regard.

Before concluding this introduction I remind the reader of some simple

facts about lattices in El. A vector £ in El is defined as an «-uple (xi, ■ ■ ■ , x„)

of rational numbers. The unit vectors tk=(eik, • • • , enk) are the columns of

the unit matrix ||e<*||. The word lattice means any set of vectors such that

a — b is contained in the set every time a and b are. We assume that the lattice

is «-dimensional, i.e., contains « linearly independent vectors;.and discrete,

i.e., we require that for any given positive integer q there are not more than

a finite number of lattice vectors satisfying the inequalities

| Xi | á q, ■ • • , | xn | ^ q.

From now on the term lattice refers only to discrete lattices which have the

full dimensionality of their vector space. By a familiar argument one shows

that one can find « linearly independent vectors h, • • • , I„ in a given lattice

2 such that every lattice vector

£ = «ill + • • • + M„I„

has integral components Ui. By the same construction one adapts the basis

Ii, ■ • • , I„ of any lattice A containing the unit lattice I to the basis d, ■ • • , en

of I:

ei = cili,

Í2 = C21I1 + C2I2,

tn — Cnl\l + • • • + £„, n-lln-l + Cnln.

The integers ck are positive and the integral skew coefficients Cki (i<k) may

be normalized by

0 ^ Cki < Ci (k = i + 1, • • • , «);

then (íi, ■ ■ ■ , In) is uniquely determined. The index7= [A:I], i.e., the num-

ber of vectors in A which are incongruent modulo I, equals Ci ■ ■ • c„. Let

A(k) denote the part of A lying in the linear subspace Xk+i= ■ ■ • =#„ = 0. The

index jk= [A(4):I(A;)] equals Ci ■ ■ ■ ck. Hence these two lemmas:

Lemma 1. jk is a divisor of jhfor k<h.

Lemma 2. The number h„(j) of different lattices A over I of given index

j= [A : I ] is finite.



Indeed, it equals the sum

Zn—X n—2 0Cx C2 ■ ■ ■ Cn

extended over all factorizations CiC2 • • ■ cn=j of j. (Incidentally, the numbers

AnO) f°ri=l. 2, • • • have as their generating function the Dirichlet series

DO

E hn(j) j- = f(i)f (s - 1) • • • f (s - n + 1)#-i

convergent in the half-plane <Rjs >n.)

2. Vector space and lattice over an algebraic field. Let J be any field of

finite degree / over <r\o. By carefully putting all factors in their proper places

we shall see to it that all arguments and formulas in this and the following two

sections remain valid for any division algebra, whether commutative or not,

of finite degree over 'Rj. We choose a basis ax, ■ • ■ , a ; of J/%s so that each

number £ of J is uniquely represented by

(2) £ = xicri + • * • + Xfo-f (xa rational).

Any w-uple (£i, •••,£„) of numbers £¿ in J is a vector of the w-dimensional

vector space En over J. The fundamental operations are addition of two vec-

tors, r+r', and multiplication 5r of a vector r by a number ô (the numerical

factor always in front of the vector!). Thus we may write

Ï = (li. ••■,£») = £i<!i + ■ • • + £»e„.

A linearly independent vectors bi, ■ ■ ■ , b* span a linear subspace [bi, ■ • ■ , bk]

consisting of all vectors of the form r;ibi+ • • • +r]kbk. Any n linearly inde-

pendent vectors bi, • • • , b„ form a basis of En/J in terms of which each vector

is uniquely expressible as

(3) r = ijibi + ■ • • + Vnbn.

The original coordinates £¿ are connected with the 77» by that nonsingular

linear transformation D,

(4) £< = E Vkôik,k

whose matrix ||5<i|| has for its columns the vectors bk = (oxk, • • ■ , onk).

Expressing each component £i in terms of the basis a of J,

£i = XiXffx + • • • + XijOf,

we identify En/J with the (nf)-dimensional vector space E^r over 'Rj. The ra-

tional numbers x,a are the coordinates of r with respect to the basis a0e¿. One

has to distinguish between linear dependence in J and in 'R.o-

We suppose we are given a lattice 8 in E"¡¡. It will have a basis I„



(p= 1, • ■ • , «/) in terms of which each vector £ of 8,

(5) £ = Z uX

has rational integral components u„. A number ô of J is said to be a multiplier

of 8 if the operation £—>5f carries each lattice vector r into a lattice vector of.

The multipliers of 8 form an order [j]. This assertion is meant to imply the

following four properties (3) :

Io. The number 1 is in [j].

2°. [J] is a ring.3°. Any given number 5 in J may be multiplied by a positive rational in-

teger m such that m ô is in [J].

4°. Each number in [j] is an integer.

Io and 2° are evident. To prove 3° and 4° we write

ôL. = X) d,JL, (dßV rational).V

If ô is any number and m a common denominator of the coefficients d^, then

mô is a multiplier. If ô happens to be a multiplier, then the dß, are rational

integers and ô satisfies the equation

I 5e,ß — dyß I = 0.

In the same manner as for the "principal order" consisting of all integers of J

one proves(4) that any order [j] is a discrete /-dimensional lattice in the

/-dimensional vector space J/1{o, and hence has a basis o~i, • • • , <f/ in terms

of which every number £ of [j] appears in the form (2) with rational integral

coefficients Xi.

The transformation D, (4), maps 8 upon a lattice A: If £ = (£1, • • ■ , £n)

is in 8, then (771, • ■ • , rjn) is in A, and vice versa. We call two lattices equivalent

and admit them to the same class if one is carried into the other by a non-

singular transformation D. The lattices A of one class express a given lattice 8

in terms of different bases (bi, ■ • ■ , b„) of E"/J. Obviously two equivalent

lattices have the same multipliers.

A lattice 8 is said to belong to the order [j] if every number of that order

is a multiplier of 8. (For « = 1 this notion coincides with that of an ideal in [j],

and our classes of lattices with the classes of ideals.) Given an order [j], the

M-uples (£1, •■-,£„) of numbers £¿ in [j] form a lattice I which belongs to

the order [j]; we call it the unit lattice for [J]. The lattices belonging to a

given order [j] are distributed over a number of classes, the class of I being

the principal class.

(3) Notion and name are due to Dedekind. Hubert in his Zahlbericht introduced the word

"ring" for this purpose, but since ring has now acquired a wider meaning I revert, in agreement

with such authorities as Artin and Chevalley, to Dedekind's terminology.

(*) Cf. H. Weyl, Algebraic Theory of Numbers, Princeton, 1940, pp. 145-146.



Let <Ti, ■ • • , af be a basis of [j] and l„ (p= 1, • • • , nf) a basis of A. If A

contains I the vectors aatk, which span I, are linear combinations of the L.

with integral rational coefficients, and their absolute determinant, i.e., the

absolute determinant of the transformation connecting the coordinates uß

with the xka (k — l, ■ ■ ■ , n; a = l, ■ • • ,/) is the index j= [A: I].

Those vectors (£i, ■•-,£„) in A for which ¡¡k+1 = • ■ ■ = £n = 0 form a lat-

tice A(*> in the A-dimensional space Ek/J with the coordinates £i, • • • , £*.

Considering A as a lattice in Ef and using the arrangement

xix, • ■ ■ , xx¡; x2x, • • • , x2¡; • • •

of the coordinates in E% one can apply Lemma 1 to A/ and (A+ 1)/instead of A

and A and thus one derives a corresponding proposition in J instead of i\o:

Lemma 3. In the row of indices

(6) jk=[A^:Vk>] (A=l, •••,«)

eacA number is a divisor of its successor.

The set of vectors (£i, •••,£„) in A outside [ei, • • • , ek-i], i.e., for which

(£t, • • • , £n)^(0, • • • , 0), will be denoted A*. Thus Ak and A(k~l) are comple-

ments in A.

We have seen that the number of different lattices A over a given lattice I

with a given index [A:l]=/ is finite, namely hn/(j). More exactly, one finds

by the same argument that the number H/(jx, • ■ ■ , jn) of different lattices A

over I with given indices (6) has as its generating function the Dirichlet series

of n variables slt ■ ■ ■ , sn:

Z/(si + s2 H-+ sn - nf) ■ Z/ist H-+sn- (n-l)f) ■ ■ ■ Zf(sn - f)

= E H,(jx, •••,/„) jT1-- -jn'"h, • • • , j'n

where

Z/W -f(*+l) •••«*+».

Hence a fortiori:

Lemma 4. We have found upper bounds for the number of lattices A belonging

to a given order [j] which contain the unit vectors tx, ■ ■ ■ , tn and hence the unit

lattice 1 for [j] and which, moreover, have either a given index j= [A:I] or a

given row of indices jx, ■ ■ ■ ,jn.

3. Preliminaries about reduction. Suppose an order [j] and a basis

Ou • • • , ff/ of [J] to be given. We consider a real-valued function

/(£) = /(£i, •■•,£»)

which depends on a variable vector r in £"/7and is positive except for r = 0,

and we assume :



(io) For each positive q one can ascertain a positive q' such that the in-

equality/if) <q implies the «/ inequalities

(7) |*ia|<<7' (*= 1, ■ •• ,n;a= 1, • • • ,/)

for the components x¡a of the £<.

Let 8 be a lattice belonging to the fixed order [J]. n vectors bi, • • • , b„

of 8 which are linearly independent with respect to J constitute a semi-basis

of 8. Because of the discreteness of 8 there is but a finite number of vectors £

in 8 satisfying the inequalities (7). Hence Minkowski's construction of con-

secutive minima of / in 8 is applicable. It yields a semi-basis bi, • • • , b„ of 8

such that

/(£) ^ /(M = Mk

for every vector £ in 8 outside [bi, • • • , b*_i] (reduced semi-basis). Obviously

(8) Mi g Mt á • • • £ Mn.

The mapping

(3) £ = rjibi + • • • + 7?„b„ —> (t)i, ■ ■ • , Vn)

carries/(£) into a function g(rji, ■ ■ ■ , r?„) and 8 into a lattice A which contains

the unit lattice I for [J]. The function g(£i, • • • , £„) is reduced with respect to

A, i.e.,

(9) g(£i, ••■,£„) = g(en, ■■■ , enk)

whenever (£i, ••-,£„) is in Ak.

The Mk are uniquely determined by/and 8; the situation is somewhat less

favorable for bi, • ■ • , bn. Suppose bi , • ■ • , bn' is another set constructed

according to our prescription. If Mk is actually lower than Mk+i then

[bi , • • • , b* ] = [bi, • • ■ , bit]. (Analogues of Theorems 8 and 9 in Rl.)

Being given « real numbers pk,

1 á Pl £ ' • ' è pn,

we say that the semi-basis b/ , • • • , b„' of 8 has the property B(pi, ■ ■ ■ , pn) if

/(£) = -f(b¿)Pk

for any vector £ in 8 outside [b/, • • • , b*_i]. Accordingly we ascribe the

property B(pi, ■ ■ ■ , p„) to a function g(£i, • • • , £n) in conjunction with a lat-

tice A over I if1

g(£i, • • ■ , £n) ^ —g(eik, • • • , enk)

Pk

whenever (£i, • • • , £n) is in A*.



If b* is a semi-basis of 8 with the property 73(pi, • ■ ■ , p„), then

Ml =/(b¿) Ú pkMk.

(Analogue of Theorem 8P.) Indeed, let bi, ■ • ■ , b» be a reduced semi-basis of

8, f(bk) = Mk. At least one of the A linearly independent vectors bi, • • • , b*,

say b¿, lies outside [b/, • • • , b*-i]; hence

f(bi) £—/(W)pk

or

Mi ^ PkMi ^ PkMk.

With the same notations I maintain that [b{, ■ • • , b* ] = [bi, • • • , bk]

provided Mk+i>pkMk- (Analogue of Theorem 9P.) Proof: Suppose that one

of the vectors b{, ■ ■ ■ , b* , say bi, is not in [bi, • • • , b*]. Then

f(b!) ^ Mk+x.

Vice versa, if all A numbers M{, ■ ■ • , Mk are less than Mk+1 then bi , • • ■ , b*

lie in [bi, • • • , b*]. The observation that M' SpiMi^pkMk finishes the proof.

The notation of properly reduced bases depends on a given multiplicative

group U of numbers e in J and deals with functions/which satisfy the further

condition :

(Ho) /(er)=/(r)(ein U).

The semi-basis bi, ■ • • , b„ of 8 is said to be properly reduced provided the

inequality

m > mholds with the sign > for any vector r of 8 outside [bi, • • • , bk-x] except for

the vectors of the special form

r = tbk (e in U).

Accordingly g(£i, • • • , £n) is properly reduced with respect to the lattice A

over I if

g(£i, ■••.£-.)> g(eik, ■ ■ ■ , enk)

for all vectors (£i, •••,£„) in A* except the special vectors

e(exk, ■ • • , enk) (e in U).

With bk the vectors

bk = tkbk (tk in U)

form a reduced semi-basis of 8 under the sole assumption that they lie in 8.



Because £= e^b* satisfies

/(r) =f(i>k), a fortiori /(£) S f(bk),

there is then, according to (i0), only a finite number of possibilities for ek.

We set

IJlbi + • • • + 7J„b„ = £ = 7)1 bi + • • • + 77n'bB'

and denote by A, A' the corresponding images of 8 :

(vu ■ • • ,Vn) in A • <^ • £ in 8 • <=í • (t/i , • • • , 77„') in A'.

The "special transformation"

(10) i)k = nkek (tk in U)

carries A into A'. We count in the same family any two lattices A and A' aris-

ing from each other by such a special transformation. Given the lattice A over

I there is only a finite number of special transformations such that the trans-

formed lattice A' also contains I. In particular, the group {/a} of all special

transformations Ja leaving A invariant is finite. If h is its degree, one has

eî= 1 (k = 1, • • • , «) for each J&; hence the ek are roots of unity in J. The roots

of unity in a field J form a finite cyclic group ; in particular, if J has at least

one real spot, the only such roots are +1. (However, in noncommutative di-

vision algebras the group of the roots of unity is, generally speaking, neither

Abelian nor finite.)

The simple argument in Rl, p. 136, shows:

If bi, • ■ • , b„ is a properly reduced semi-basis and bi, ■ • ■ , b„' any semi-

basis of 8, then the sequence of the values /(bi), • • • , /(b„) is lower than

fW), ■ ■ • ,/(bn )■ If bi , • • • , b„' is reduced and bi, • ■ • , b„ properly reduced,

then

bi = eibi, • • • , b„' = €„bn (a in -U).

4. Extension to the ground field %,. Minkowski's inequality. So far the

function /(r) has been defined merely for the vectors in the space En/J. In

order to introduce geometry we assign to the variables xa in (2) arbitrary real

values :

(2*) £* = xm + • • • + x/oy.

Sticking to the multiplication table of the basic elements <ra, we thus extend

J/%s to a commutative algebra J* over <r\.. But only in the two cases treated

in Rl, where J is 'R.o itself or an imaginary quadratic field over %$, is 7* again

a field. In general it is not. However, any w-uple £* = (£f, • • • , £*) of ele-

ments £f in J* may be considered as a vector in an («/(-dimensional vector

space En¡ over <r\ with the real coordinates #,<,:



£* = XixCx + ■ ■ ■ + XijCj.

We now assume/(r*) to be a gauge function, i.e., a continuous real-valued

function in this space, having the following properties:

(i) /( r*) > 0 except for r* = 0.

(ii) f(t%*) =\t\ -/(r*) for any real factor /.

(¡ii)/(?i + r?)a/(l?)+/(ï?).

The gauge body

K: f(íx*, • • • ,*„*)< 1

and also the solid qK defined by /(£*, • • • ,£*)<<? are bounded; hence postu-

late (i0) of the previous section is fulfilled. Let V* be the volume of K com-

puted in terms of the coordinates x¡0.

Again we fix an order [j] and a basis ax, ■ ■ ■ , a¡ of [j]. Let A be a lat-

tice belonging to this order and containing the unit lattice I for [j] and let

/(£i> • • ■ i £n) be reduced with respect to A. The volume of K in terms of the

coordinates uß as introduced by (5), i.e., measured against the fundamental

parallelepiped of A, equals V*- [A:I]. Hence by the simple argument ex-

plained in Rl, p. 140, Minkowski's second inequality leads to this formula

holding for a gauge function /(£f, ••■,£*) which is reduced with respect to A:

(Mi- • • Mny V*[A:l] ^ 2"'

where

Mk = f(eik, • • ■ , e„k).

5. Splitting. The number of reduced lattices is finite. Up to now every-

thing has worked for a division algebra of degree / over 'Ro just as well as for

a field J. Further progress depends on .the structure of J*. If J is a field,

then J* is isomorphic to the direct sum of a number of components 'R. and Q.

We first study this case.

The decomposition of J* is brought about by conjugation. One knows that

y/'Ro has a determining number 6 whose powers 1, d, ■ ■ ■ , 0/_1 constitute a

basis for J. The number d satisfies an irreducible equation in iRo of degree /.

Let 6" and 9ß, dß (or in one row: 0(1), • ■ ■ , 0(/)) denote its r real and s pairs of

complex conjugate roots. They define the/conjugations

£^£°; £-*£", i->?

each of which projects J isomorphically into 'R or Q. We use the notations

« « J » , . & ,-rß ß . /»,£ = x , £ = xo + ixx (£ = x0 — ixx)

and call the r+s numbers £a, £" the splits, and the/ real numbers x"; Xq, xf



the splitting coordinates of £. The same applies to any element £* of J*. The

product f * = £*77* has the splits

t " = iava, P = 5V-

The arithmetician speaks of the different values of the indices a and ß as

the r real and s imaginary (infinite prime) spots of J; for the sake of brevity

we often drop the adjectives in parentheses. If a definite arrangement is de-

sired, we write a = oci, ■ ■ ■ , <xr; ß — ßi, ■ • • , ß,.

The splitting coordinates x"; Xq, x\ are connected with the components

Xi, ■ ■ ■ , Xf of £* by the linear substitution

(11) S = j|o-i, • • • , (r/H

where in the symbol on the right side each term stands for the column of its

splitting coordinates (in a definite arrangement). The splitting of J* into r

components %, and s components Q is established as soon as it is certain that

the absolute determinant

A = abs. | ai, ■ ■ • , o-f |

of the matrix 2 is different from zero. For the particular basis 1, 6, ■ ■ ■ , 6f~l

one sees that ( — 2i)s -A is the Vandermonde determinant of 0(1), • • • , 0(/), and

hence indeed AÔ. This fact carries over to any basis ai, ■ ■ ■ , crf of J.

The number in J* with the splitting coordinates x"; x$, — icf is denoted

by £*. As absolute value ] £* | we introduce the greatest of the r-\-s numbers

|£"|, |£"|. One could agree on other definitions, but this one is most con-

venient for our future applications. What usually is called a unit in J is a

number of J which is a unity at all finite prime spots. None but the infinite

prime spots matter for our investigation; hence we take the liberty of using

the term "unit" for those numbers e of J which are unities at all infinite prime

spots, i.e., for which the r+s equations | e"| = 1, | eß\ = 1 hold.

For any element 5* of J* one introduces the real matrix ||¿a6|| of the linear

operation £*—->-£*ô* in J*:

X„ —> 2-1 dabXb ( 0-at>* = 2_, dbaVb jb \ b /

and its characteristic polynomial

| te„b - dab | ■» ¥ — dit'-1 + • ■ • ± df.

di and d¡ are called trace (tr) and norm (Nm), respectively. In terms of the

splitting coordinates our operation of multiplication splits into the trans-

formations

xa—> xad"; £" —>• £"5",



each corresponding to a real or imaginary spot a or ß. Of course, £"—>£ßoß

stands for

Hence

ß ßß ßß ß ßß , ßßXo —> Xod0 — Xxdx, Xx —► Xoö! + Xiao.

tr(í*) = E¿°+2E^-ß

Nm(¿¡*) = Ud"-Il{(dSo)2 + (d3x)2}.ß

If ô* = 5 is in J the dab are rational numbers. For a unit e in J our formulas show

that the determinant Nm(e) of the transformation £*—>£*e is of absolute

value 1 and hence as a rational number equal to +1.

Considering the trace tr(f) one readily verifies that (2äA)2 is rational for any basis

o-i, • • • , 07 and especially a rational integer for a basis of an order [j].

The transformation (4) in E"f,

(4) £f = E V*Sik (Sik numbers in J)k

splits into the components

£l = ¿^ VkSik, £¿ = 2-4 Vkàik,k k

each «-component involving n, each /3-component 2n real variables:

a a 8 ß ß

£/b = xk; £*; = Xko + ixicx-

How closely can one approximate an element £* of J* by a number y of our

order [j] with the basis ax, ■ ■ ■ , a/? For an appropriate y in [j] the real

components xá of £*— y,

£* — 7 = x/ o-i + ■ • • + x/cf,

will satisfy the inequalities |x„' | ^2, and thus

I £* - 7 I =s pwhere

i a i i a , i ¡S i i ß i

p = 5 max ( \ ax \ + ■ ■ ■ + \ a/ |, \ax\ + • - - +\a/\).a.ß

The "circles" of radius p around all numbers y of [j] cover the whole J*.

(Such a radius was denoted by the letter r in Rl, which now serves a different

purpose.)

Let us now return to the situation explained at the end of the previous

section and let V be the volume of K computed in terms of the splitting

coordinates of £j*, ■ ■ ■ , £*. Then V = V*/An. Moreover we observe that



/(£*. •••>£*. 0> • • ■ . 0) is reduced with respect to the lattice A(*\ and de-

noting by Vk the volume of the solid

/(£f, •••,£**, 0, • • -, 0) < 1

in Ek/ computed in terms of the splitting coordinates of £f, • • • , £*, we ob-

tain these fundamental inequalities for Mk =f(eik, ■ • • , enk) :

Theorem I. For a reduced f(û ■ ■ ■ , £n)/A one has

(12) (Mi-- ■ Mn)'V [All] ^(2'A)",

more generally

(12,) (Mi--- M*)/-F*[A<*>:I<*>] = (2'A)*.

At this point we introduce the further assumption:

(ii*) /(r*£*) g|r*| •/(£*) (t* any element of J*),

and henceforth the term "gauge function" is to be taken in this restricted

sense. Following Minkowski's own argument, we then prove

Theorem II. For a reduced/(£i, • • ■ , £„)/A one always has

(13) /=[A:l]^(«/)!(±)n3.(^)"

and more generally

(13k) yfc- [A«>:l«»]i <*/) ! (-¿)\j) (A-1, •••.»).

Hence in any class of lattices belonging to the order [J] there is always a

lattice A which contains I and satisfies (13) and (13k). Together with Lemma 2

this proves(6) :

Theorem III. The number of classes of lattices belonging to a given order

[j] is finite.

(6) This theorem is well known. We are concerned only with those lattices A over I which

are in the class of 2, but as our bounds (13) or the sharper bounds (35) depend on the order

rather than on the special class it seemed worth while to mention Theorem III in passing. For

a commutative field J and its principal order [7] E. Steinitz, Mathematische Annalen, vol. 71

(1912), pp. 328-354, and vol. 72 (1912), pp. 297-345, proved that the number for classes of any n

is the same as for n = l, namely equal to the number of classes of ideals. See also I. Schur,

Mathematische Annalen, vol. 71 (1912), pp. 355-367; W. Franz, Journal für die reine und ange-

wandte Mathematik, vol. 171 (1934), pp. 149-161; C. Chevalley, L'Arithmétique dans les

Algebres de Matrices, Actualités Scientifiques et Industrielles, no. 323, 1936, in particular Theo-

rems 3, 7 and 8.



(The proposition implies the corresponding one about classes of ideals.)

Any gauge function will do for the proof, for instance

/(ÍÍ, ••-,£»*) = |£í| + ---+|£n*|.

We shall soon see that much better upper bounds for the number of classes are

obtained by using for/2 the trace of a positive Hermitian form. However, our

present Theorem II goes far beyond Theorem III because it deals with any

gauge function/ in conjunction with a lattice rather than with lattices alone.

Proof. Observe that the "octahedron"

| £l* | + • • • + | £n* | < 1

contains no vector of A except the zero vector. Hence owing to Minkowski's

chief inequality we find this upper bound for its volume W:

Wj g (2'A)".

Let (£1, •■-,£„) be a vector in A and £4 be the last nonvanishing one among

its coordinates £,-. Then by the definition of reduction

(14) /(£i, ■■•.ÜiJÍ»- f(exk, ■■■ , enk).

On the other hand the assumptions (iii) and (ii*) imply

(15) /(*iei + " ' + Un) - Ml I ?11 + ' ' ' +Mn I *n I

= Mx | £i | + • • • + Mk | £* |.

Because of (8) the relations (14) and (15) are incompatible unless

|£i| + ••• +|£n| = |£i| + • • • +|ft| à 1.

We base our computation of W upon the following general remark about

gauge functions/ in an «-dimensional vector space over <R. If V is the volume

of the gauge body 7sT:/(r) < 1, then the integral / of e~f over the whole space

equals n\V. One simply evaluates the integral by decomposing the space into

the infinitely thin shells

q ̂ M) <q + dq

and thus finds

/-/;e~g-nq"~1dq = nlV.

Applying this remark to the gauge function |£f| + • • ■ +|£*| in our («/)-

dimensional vector space and to the gauge function | £*| in the/-dimensional

space J*, one gets this double value for /:

(nf)\W = (f\wY,



w being the volume of the "cylinder" defined by

| £* | < 1, or by | x" \ < 1, (xl)* + (*?)* < 1.

Therefore w = 2r7r*.

(ii*) entails the property (ii0) of §3, provided U is the group of units in

our sense. From now on we shall abide by this convention and interpret the

term "properly reduced" accordingly. Then the transformation £*—»£*•«

(e in U) and hence every special transformation (10) has the determinant

±1 and thus the indices jk for two lattices A and A' over I which are in the

same family coincide : jk =jk for A = l, • • • , «.

The values y* of a Hermitian form in J*,

(16). 7*(£*) = Z £*7«£** (yt = y*k)i.k

are totally real in the sense that 7* = 7*, or that even the /3-splits 7" = g%+ig\ = g*

of 7* are real. What such a Hermitian form does is to associate a quadratic

form {gjj with each real spot a and a Hermitian form {y^} with every imag-

inary spot ß. The splits of 7*(£*) are

a TT-, a o o ß ■r-i ß ß -ß

(17) g = 2-, *< g'kXk, g = ¿^£>7.A¿*i,k i,k

where *? = £? and ff are the splits of £f. The form 7*(£*) is said to be positive

if each of the r quadratic forms {gJt} and each of the 5 Hermitian forms {7ft}

is positive definite.

We now apply our theory to the gauge function / introduced by

(18) P = tr (7*(£*)).

In terms of the splits (17) one has

(i9) /2 = Zr + 2Zsfl-a ß

The properties (i) to (iii) of §4 are readily verified ; (ii*) is also fulfilled because

of

/Vf*) = Z |r"|V + 2 D I t* IY.ß

6. Quaternion algebra of totally positive norm over a totally real field.

Turning to noncommutative division algebras, we denote by SI the quasi-

field of quaternions

a = ao + aiii + a2i2 + a3i3

whose components cto, ii, «2, a3 are arbitrary real numbers, and use the nota-

tions ä and \a\ in the customary manner:



II2 2 i 2 2 2aa = | a \ = a0 + ax + a2 + a3.

For which of the noncommutative division algebras of finite degree over

iRo does the concept of infinite prime spots work in a way similar to that in

the previous section for fields ? 1 am going to describe one such situation with-

out discussing the question whether or not it is the only one (though, as a

matter of fact, it is).

Suppose we are given a field £ of degree e over fR0 and two numbers coi, a>2

in £. We put w3 = wi«2 and form the quaternion algebra J over £ whose ele-

ments £ are quadruples (£0, £1, £2, £3) of numbers in £,

(20) £ = £0 + £iu + £2'2 + £3i3,

with this multiplication table for the unities ti, i2, i3:

222

n = — wi, 12 = — o)2, 1» — — o>3;

Lli2 = — l2ll = IS, 13'1 = — tl'3 = <<Ht2, l2l3 = — l3l2 = 0)2'1.

The conjugate £ is £0 — £iii — £212 — £313 and

— 2 2 2 2

££ = £0 + «)i£i + o)2£2 + o>3£3.

If the equation2222

(21) £0 + coi£i + o)2£2 + co3£3 = 0

has no solution (£0, £1, £2, £3) in £ except (0, 0, 0, 0), then J is a division algebra

of degree 4 over £ and of degree/ = 4e over <R0- We assume £ to be totally real

(to have no imaginary infinite prime spot) and a>i, co2 to be totally positive

numbers in £ (i.e., cheir e conjugates w", co£ are all positive). Then the quad-

ratic form of the variables £0, £1, £2, £3 at the left of (21) is positive definite

in each conjugate £" of £ and hence (21) has no solution except 0. Denoting

as before by t" the conjugate of any number t in £ corresponding to the spot a

of £, we map (20) upon the element

a a a, a 1/2 a, or. 1/2 or, a 1/2

(22) £ =£o+£i(cci) -tx + £2(a)2) -Í2-T-£3(0*3) •*,

in §,. This "conjugation" is an isomorphic mapping and defines the "infinite

quaternion prime spot" a of J. (22) are the splits, and the 4e=/ real numbers

or a a a or. 1/2 a a a 1/2 or or or 1/2

Xo = £o, Xi = £l(o)!) , X2 = £2(«2) , X3 = £3(0)3)

are the "splitting coordinates," of £. Application to the elements £*, equation

(2*), of J* is immediate.

There is only one thing to settle : The splitting coordinates, xô, x", x2, x"

arise from the components xa by the substitution (11), each aa standing for



the column of its splitting coordinates. Is its determinant, whose absolute

value will again be denoted by A, different from zero? To answer the question,

let (ti, • ■ ■ , t«) be a basis of £ and set A0 = abs. |ti, ■ • ■ , t,\ . From it we

obtain the following basis of J:

Ta, TaLl, Tai2, Tal3 (ö = 1, ■ • • , e).

The A of this particular basis is given by

A = YL (ü,iü,2) 'Ao = Nm o)3-Ao.a

Thus A?íO for this and consequently for any basis.

Incidentally A is a rational number for any basis of 7 and 4"A a rational integer if

a\, • • • , oy is a basis of [j]. The characteristic equation of the multiplication £*—>£*• 5 consid-

ered as a linear operation in J* is the square of a polynomial (of degree 2e), and so is the char-

acteristic polynomial of the linear substitution (4) in E"1'.

The notion of unit and the absolute value |£*| of any element £* of J*

are introduced as before. The constant on the right side of (13^) is to be

changed into

/32YYAVW)!fc) (jj ■As gauge functions / we employ in particular those whose square equals

I tr (y*) = E Ta

where y* is any positive Hermitian form (16) in J*.

7. The theorems of finiteness for quadratic forms. After so many pre-

liminaries which stake out the ground covered by our investigation, I now

come to the core of the matter, which may be explained fairly completely by

the simplest example 7 = £Ro- Here we have only one order [j] consisting of

the ordinary integers 0, ±1, +2, • • • and only one class of lattices. For any

given lattice A over I and any positive quadratic form

/2(ï) = E gikXiXk (gki = gik)

the conditions of reduction read :

PO) ^ gkk whenever r = (xi, • • • , x„) is in A*.

Each of them is a linear inequality for the coefficients gi¡.

With the notation used in Rl we carry out Jacobi's transformation:

2 2 2

/ = çizi + • • • + qnz„.

The volume V of the ellipsoid/2<1 is given by



2 2

V = COn/çi •••?„,

w„ being the volume of the «-dimensional sphere. Hence the inequality (12),

(23) Mi ■ ■ ■ MnV[A:l] ^ 2",

turns into

(24) gn---gnn[A:l]2 = (2V«.)'-ii • • • ?..

As Minkowski observed, (23) may be proved much more easily for quadratic

forms than for an arbitrary gauge function. By an argument similar to the

one employed in proving Theorem II we see that the ellipsoid

t'2 ?1 2 _i_ i ?n 2 / i/ =-zi + • • • H-z„ < 1

M2 M21 n

contains no lattice vector except zero. Hence its volume V satisfies the in-

equality

F'[A:l] ^ 2», and V - Mt • • • MnV.

If k„ is a number such that the part of space covered by impenetrable «-di-

mensional spheres in any lattice arrangement may never exceed the propor-

tion k„:1 then we can even write Kn2n instead of 2n and thus replace co„ in

(24) by 7r„=w„//c„. The most primitive choice is «„ = 1 ; however, according to

Blichfeldt's ingenious device(6),

Kn = (« + 2)-2-1-"/2

is a permissible and better value.

Making use of the inequalities

ga ^ ?v

on the left side of (24) we get for the index _/ = [A:I] this upper bound

(25) j á 2»/7r„

which is a considerable improvement over (13),j^w! For w = l, 2, 3 it yields

the result j= 1, to which the theory of reduction for binary and ternary forms

owes its comparative simplicity.

For similar reasons

2 ¡fc 2(24*) gn ■ - - gkk-jk = (2 /ick) qi- - ■ qk,

(25*) j* = 2*/»* (* = 1. • • • - n).

Unless the lattice A satisfies the « inequalities (25*) for its indicesj* = [A(t) : I(t)]

there can be no A-reduced forms.

(») H. F. Blichfeldt, Mathematische Annalen, vol. 101 (1929), pp. 605-608.



Dividing (24*) by

gll • • ■ gk-l,k-l è ?i • • • qk-i

we find that our reduced form satisfies the fundamental relations

(26) qk ^ \kgkk

where

(27) X, = (jk*k/2ky.

This lower bound for qk is much better than the corresponding one holding for

the method of reduction studied in Rl.

The first theorem of finiteness deals with the subset A¡fc( = ) of Ak to which

a vector r in A* belongs if there exists a A-reduced positive quadratic form f2

satisfying the equation /2(r) =gkk. The set Ak( = ) is finite. The proof is as in Rl,

but the upper bounds arrived at are a good deal lower. The first part of the

proof yields the bounds

2X,z¿ ̂ 1 (for i = A, A + 1, • • • , n)

where the X, are now defined by (27). In the second part one replaces the vec-

tor r in A*( = ) by r— rA where rh is any vector in A(A) (A<A) and observes that

f(ï - rÄ) ^ gkk.

This is true in particular if r* is in Iw, and as in Rl one thus derives the rela-

tions

X**î ̂ p2A (p = 1/2; A = 1, • • • , A - 1).

Once the discrete lattice A is given, the resulting universal upper bounds for

\xn\, • ■ ■ , \xx\ leave only a limited number of possibilities for a vector

i=(xx, • ■ ■ , x„) in A.

The second theorem of finiteness shall be restated in a more natural and

slightly more general form. Let p^l and wÔ be given. With respect to the

lattice A over I the positive quadratic form /2 will be said to have the prop-

erty B (p, w) provided

(28) p(%) ^ --f2(tk)P

for any vector r in A*, and

(28') p(tk - n) è f(tk) - w-f(th)

for A<A and any vector r„ in A(w. Again, each of these conditions is a linear

inequality for the coefficients gtJ- of/2. We maintain :

Given two lattices A and A' over I, there is only a limited number of linear



transformations carrying A into A' and at the same time capable of carrying an

unspecified A-reduced form f2 into an unspecified f'2 which has the property

B(p, w) with respect to A'.

We write the transformation as

£ = Z x¿< = Z yùi'-i i

if (xi, ■ ■ • , x„) is in A, then (yu ■ ■ ■ , yn) is in A', and vice versa. In particular,

the bi, • ■ • , bn are vectors in A. (p, e¿, b¿ were denoted in Rl by p2, b,-, 8,-.)

More explicitly as has been done in Rl, we divide the row of indices 1, ■ • • , «

into a number of sections by means of the subspaces

Ek = [d, • • • , e*], Ek' = [bi, ■ ■ • , b*] (A = 0, 1, • • • , »).

We pick out those k = l0,li, ■ ■ ■ , lv,

0 = lo < h < • • • < lv-i < lv = «

for which Ek = Ek , and divide the range of k into the v sections

lu-l < k ^ lu (u = 1, ■ • • , v).

We then study the possibilities for transformations (bi, • • ■ , b„) with given

h, " ' ' , 4-1-By the analogues of Theorems 8P and 9P we have

(29) g'kk = pgkk (A - 1, • • • , n)

and moreover

(30) gi+i,i+i ^ pgu

whenever i and ¿ + 1 are in the same section. Consider a b* of the last section

(lv_i<kSn). The first part of the proof in Rl yields for {= b* the simple upper

bounds

2 U-äI + 1XaZa Ú p

if h also belongs to the last section, [k] denoting 0 or k according as ^^0

or &>0. The second part requires a slight modification. Suppose h lies in the

«th section (u<v), and set for the moment /„=/. Since Ei = E[, the vectors in

Au) are obtained from the expression yibi+ ■ ■ • +y¡b¡ by running (y¡, ■ ■ ■ , y¡,

0, • • • , 0) over A'Ci). Hence, according to the postulate (28'):

r'2 r >\ ^ I 'f (tk - t> ) ^ gkh - Wgu

for any vector t¡' in A'(n, or

/ (b* - f') è g'kk - wgu



for any vector r' in Ac", a fortiori for any vector in ACÄ), a fortiori for any vec-

tor r' in Icw. Following the same argument as in Rl, one gets the inequality

I 2 2wgu + p hghh ^ \hghhZh (p = 1/2).

But because A and I are in the same section, (30) and (29) lead to

gu ^ Pl~hghh, g'u ^ pgu ^ pl'h+lghh

and thus finally

XfcZ/, S hp + w-p (lu-x < A è h; u = 1, • • ■ , v — 1).

It is clear how the same procedure applies to a A in the lower sections. De-

noting the values of the variables zit ■ ■ ■ , zn for r = b* by zlk, ■ ■ • , z„k ,one

finds:

Zhk = 0 if A is in a higher section than A;

i^hztk is p\k-h)+x ¿f ^ ancj £ are ¡n ^jje same section;

XaZa* is hp2+wp'~h+1 if A is in a lower section than A which ends with /.

8. Modifications in arbitrary fields and quasi-fields. Our next concern is

to examine whether any serious modifications of the procedure just described

arise in the two general cases of a field and a quaternion algebra over a field.

Take the case of the field first. With a positive Hermitian form 7* in J* we

combine its trace/2:

(31) / (£1, • • • , £„) = E E gik Xi xk + 2 E E 7i*£; Í* (£/£ = xk0 + ixki).a i,k ß i,k

y* is called reduced with respect to A if the gauge function / is, i.e., if

(32) /(£!, •••,£„) £ tr (y*kk) = m\

for any vector (£1, ••-,£„) in Ak. Each part is subjected to its Jacobi trans-

formation :

Zor a a ^-y a a 2gik Xi Xk = 2-, °i (s." ) >

i,k i

Zß ß-ß ^ ß i ß |2yik £¿ £i = 2jÎi| s i \ ■i,k i

Besides

tr (qi) = E 1° + 2 E ??. Nm (<?,) = ü ?."• II (??)*or (3 a /S

we introduce the mean value (qi) by

/•(?i) = tr(ai).



In terms of the coordinates»:*; x^, x{i the volume F of the ellipsoid /2(f*) <1

is

u„f divided by 2"' I JJ Nm q{ J .

Instead of applying Minkowski's second inequality to the present gauge func-

tion we again consider the ellipsoid

/'*(£*) = tr ( Z !^) < 1

which contains no lattice vector except zero, and thus establish the inequality

(33) II(tr7,*)/-[A:l]2= a -ÜNrng,i 7Tn/ i

for any reduced 7*/A.

Now enters the only new feature: Making use of the inequality between

arithmetic and geometric means in the form

Nm qi ^ (qiy

we infer from

ß ^ ßqi è gu,

the relation

qi ^ ga, qi ^ 7i

<7«y è (qiY è Nm q{,

and then (33) yields the following upper bound for j = [A : I ] :

Tn/-/n//2

;" i£ 1/jUn with the abbreviation pn =-•(2»+«A)»

For the same reasons

* ' *

(34) II (yfi)f(Wk)2 = II Nm qi»•—1 <-l

and hence

(35) pkjk = 1.

These estimates are an improved substitute for Theorem II. Combining (34)

with

n <7* y ^ n (qiY è n Nm ?i



one gets

(36) Nm?*^ (m*/*)2-<7**)/-

Not only does this inequality establish a lower bound for the trace of qk,

tr qk S: (pkjk)2l/tr yXk,

but it also shows that the geometric mean of the conjugates of qk is not much

smaller than their arithmetic mean. Therefore none of the conjugates can be

much smaller than their arithmetic mean. We have a special case of the situa-

tion dealt with by the following

Lemma 5. Let fi, ■ ■ ■ , fm be positive integers and «i, ■ • ■ , um; Vi, ■ ■ ■ , v„

two rows of positive numbers. We set

/' (M) = /l«l + • ■ • + fmUm,

-T /l /m

Nm u = «i • • • um .

If ua^va (a = l, ■ ■ ■ , m) and

(37) Nm u ^ p-(v)f

with some constant pî, then

ua =" Xo-(»)

where X„ depends on p but not on the u and v.

(In our case r among the weights fa are 1 and J of them equal 2.)

Proof. In the trivial case m = 1 one determines X by

(38) X' = p.

If m>\ we set Ui=\-(v) and assume X=T. Then

Nm u = X (v) -u2 • • • um ^ X (v) (u)i

Here (u)i denotes the arithmetic mean of u2, ■ ■ ■ , um formed with the weights

ft, ■ • ■ , fm oí sum f-fi:

(/ _ /O • (M)l ■ /a«2 + • • • + fmUm = /• («) - flUl

-/•<«>-/*<?> S (f-fi\)(v).

Therefore

Nm u ^ p-{v)f

where



ff - /ixv_/i(39) " = K^f)As its logarithmic derivative shows,

dp fx(l - X) fd\(40)

P / - /iX X

this function /x(X) maps the interval 0 :SX :£ 1 monotonically upon 0 iS/x ̂ 1 and

thus will assume the given value p (^1) for a certain X=Xi (^1). Thus we

cannot have Mi<Xi-(z>) under the condition (37).

(We wish to obtain the best value for the constant Xi. If one is content with a little less, one

may choose r\¡ according to the equation

<(yh)'-*or even, as

,(/-/i)//i

(-A) < e ( = basis of natural logarithms),

(*ie)'i = m.

Incidentally, formula (40) holds good also for the function (38) which rules the trivial case

m = \, /,=/.)

In this way we ascertain constants \k, \¿ such that

each qk ^ \k-{ykk) and each qk è X* • (y**).

In case there is only one infinite prime spot, Xi and \¡¡ are determined by the

relation

! /2

(pkjk) = Xt = X*

in case of several spots by the equations

(»kjk)2 = xk(j-T) =^(yrrj

together with Xj¡ ̂ 1 and X* ;S 1.

Similarly for the other case studied in §6, that of a quaternion algebra J

with totally positive relative norm over a totally real field. The constants

Pk, i\k in the inequalities

Pkjk ts 1 and qk ^ \k- (ykk)

are then given by

Pk = irike A-k(e/4yk°,



(e - XA«-1(Pkjk)lt2 = \k or \kl _ J (XtSl),

according as e = J/is 1 or is greater than 1.

After this the proofs for the first and second theorems of finiteness roll

along as before.

9. The pattern of equivalent cells. The Hermitian forms {y**} constitute

a linear space of

N = f-\n(n - 1) + (r + s)n = f-\n(n + 1) - sn (field J)

or

AT = en(2n - 1) (quasi-field J)

dimensions, the positive ones a convex cone G in that space. G is an open set;

we operate within G throughout. "Form" means any positive Hermitian form.

Let A be a lattice over I. A A-reduced form 7* has been characterized by

the inequalities

(32) P(tx, • ■ • , U è P(exk, ■■■ , enk)

holding for/2 = tr (7*) whenever (£1, ■•■,£„) is in A*. For a given vector

(£1, ••■,£„) the equality sign in (32) will hold identically for all Hermitian

forms 7* only if

(£1, • • ■ , £n) = e(elk, ■ ■ ■ , enk) (e a unit),

as follows at once from the expression (31). For any other vector (£1, •••,£„)

the equation determines a (TV — l)-dimensional hyperplane in our TV-dimen-

sional linear space of forms. This remark justifies our definition of "properly

reduced" in terms of the group U of units.

The forms 7* which are reduced with respect to A make up a convex cone

G a in G. The properly reduced forms are the inner points of Ga; see Rl, p. 150.

Ga may be empty; indeed it will be so unless the indices jk of A satisfy the in-

equalities (35). Even if it is not empty it may be without inner points. Theo-

rem 10 in Rl, together with the first theorem of finiteness, proves:

Theorem IV. If Ga has inner points, then Ga is a convex pyramid defined

within G by a limited number of linear inequalities.

A linear mapping j—*r' of En/J upon itself is one satisfying the conditions

(ïi+Ï2)'= £1' + f2 and (5r)'=5r' for any number 5 in J. We also require

that r = 0 is the only vector whose image r' equals 0. If bi, • • • , b„ is any basis

of En/J the mapping S may be defined by giving the images b,' = biS of the b,.

The mapping 5 carries a form 7* into a form 7/ according to the equation

7s*(r-5) =7*(î). An order [j] in J and a lattice 8 belonging to the order [j]

are supposed to be given. The linear mappings 5 which leave 8 invariant are



said to form the modular group(7). In terms of a basis bi, • • ■ , b„ of En/J the

lattice 8 is represented by A :

£ = Thbl + • • ■ + 7?„b„ in 8 • <=± • (vi, ■ ■ ■ , Vn) «» A,

and a form 7*(f) is represented by a form T*(vi, • • • , t;„) :

(41) 7*Ml + • • • + T,,b«) = r*(l,„ • • • , 77n).

The linear mapping defined by b¿—*bí carries 7?ibi+ • ■ • + 77„bn into

lib/ + • ■ • +77„b„' ; hence it leaves 8 invariant and thus belongs to the

modular group if and only if A=A', A and A' being the representations of

8 in terms of bi and b '. For a vector b in 8 there are not more than a finite

number of units e such that eb also is in 8. Indeed, for the splits of £=eb

one finds

|£a|=l<H> M=l5"l> a fortiori | £<" | ^ | Ô" \, \ £" | = | 5" |,

which in view of the discrete nature of 8 proves the point.

We want to divide G without gaps and overlappings into domains which

are mutually equivalent under the modular group. We shall introduce these

cells first as entities which have nothing to do with Hermitian forms, adopting

a criterion of identity other than the set-theoretic one. The systematic place

for this introduction would have been at the end of §3. Only afterwards shall

we explain the meaning of the phrase "a form lies in a cell." Here are the defi-

nitions:

A semi-basis bi, • • • , b„ of 8 determines a cell Z(bi, • ■ • , bn) ; the semi-

basis bi, • ■ • , b„ is said to determine the same cell if

(42) bi = tibi (a units).

Let S be an operation of the modular group. The image Zs of Z = Z(bi, • • • , b„)

is defined as Z(b{, • • • , b„') where b' = b¿5. (Notice that if Z is written as

Z(bi, ■ ■ ■ , b„), bi = e¿b„ then Z(b[, ■ ■ ■ , b/ ) is the same Zs because bi = e.b/ ;

thusZs is independent of the fixation of the defining semi-basis bi, • • • , b„.)

Those S of the modular group for which Zs = Z shall be denoted by Jz; they

form a finite group {Jz} ■ Indeed, for such an S = Jz one must have

(43) b/ = biS = o-ibi (o-i a unit),

and the Jz are those mappings of the special form (43) which leave 8 invari-

ant. (In terms of another defining semi-basis b¡= e¿b¡ the same Jz is expressed

by bi = eiffif^1 ■ bi.) Any operation 5 of the modular group has the same effect

upon Z as JzS.

(7) If one feels that this term ought to be reserved for the group which is fundamental in

the theory of the modules of the theta functions then a new word, say "lattice group," is indi-

cated for our purpose.



In terms of (bi, • • • , b„) the lattice 8 is represented by an admissible lat-

tice A, i.e., by a lattice A over I which is equivalent to 8. Hence to the cell

Z = Z(bi, • • • , b„) there corresponds a family of admissible lattices A, and the

same family to each equivalent cell Zs- We have a one-to-one correspondence

between the classes of equivalent cells on the one hand, and the families of

admissible lattices A on the other. We distinguish them by different colors.

The operations Jz are represented by the operations J\ in terms of the basis

(bi, • • • , b„).Now we come to the realization of cells as point sets in G. A form 7*

is said to lie in Z = Z(bx, ■ • • , b„) if (bi, ■ • • , bn) is reduced with respect to 7*,

i.e., if for /2 = tr(7*) one has f2(l)^f2(bk) whenever | is in 8 and outside

[bi, • • • , b*-i]. Because (42) implies

f(bk) = f(bk), [bx, ■■■ , b*_i] = [bx, ■■■ , bk.x],

the definition is independent of the fixation of the defining semi-basis

bi, • • • , b„. If 7* lies in Z = Z(bi, ■ ■ • , b„) then the transform V* introduced

by (41) lies in Ga, and 7s* lies in Zs.

The fact that there always exists a reduced semi-basis for a given 7* and

the concluding sentence of §3 can now be stated thus:

(a) Every point 7* lies in at least one cell Z.

(b) An inner point of a cell Z cannot lie in a cell Z' unless Z' is the same

as Z (or briefly: different cells have no inner points in common).

The fact (a) will of course not be altered by suppressing all empty cells and

their colors. Thus we have to look only for those admissible A whose indices

satisfy the conditions (35) ; and this brings the colors down to a limited num-

ber. Will (a) still prevail after suppressing all cells without inner points and

their colors? The answer is affirmative because there is no inner clustering

of cells in G. This is a consequence of the second theorem of finiteness, which

now takes on the following form. Let Oi, • • • , a„ be a semi-basis of 8, p 2:1

and wSïO. The form 7* is said to lie in Z(oi, • • • , an\p, w) if

f(ï) è — f(ak)p

whenever r is in 8 and outside [01, ■ • • , ak-x], and if, moreover,

f2(ak - &) è f(ak) - wf(ah)

whenever A<A and rh is in 8 and [oi, • • • , 0*].

Theorem V. 7"Aere is only a finite number of operations S of the modular

group such that the image Zs into which a given cell Z is thrown by S will have

points in common with the domain Z((Xx, ■ ■ ■ , an\ p, w).

Application to p = l, w = 0 proves in particular that a cell borders on not

more than a finite number of other cells. And since Z(ai, • • • , a„| p, w) sweeps



over the whole G if p and w increase to infinity we are sure that the cells

cluster around no point in the interior of G ("the modular group is properly

discontinuous in G"). We therefore definitely admit only those colors whose

cells are TV-dimensional solids, i.e., have inner points. In our summary we

talk of them as point sets in G.

Theorem VI. (a) G is divided into a pattern of cells, each cell bearing a

color out of a finite palette of colors. The cells cover G without gaps and overlap-

pings. Each cell is a solid convex pyramid (in G). The mappings of the modular

group leave this design, including its coloring, invariant. Any two cells of the

same color can be carried one into the other by an operation of the modular group.

(b) Given a point in G and a cell Z one can assign a neighborhood 9Î to the

point such that there is only a limited number of operations S of the modular

group for which the image Zs penetrates into 3Í.

(c) The operations of the modular group which carry a cell into itself form a

finite subgroup. This group of linear operations in the vector space En/J is

equivalent (in J) to a group whose elements are of the special form

£1 -» fm, ••-,$«-* Lin («< units).

(Of course, in view of statement (c) the statement (b) could have been re-

placed by the simpler one that only a finite number of cells penetrate into 5t.)

We form a nucleus by selecting one cell Zc of each color c. All cells adjacent

to the nuclear cells form a wreath around the nucleus. Here the word "ad-

jacent" may be interpreted either in the wide sense of "having a point in com-

mon," or in the narrower sense of "having a wall of N—Í dimensions in

common."

Theorem VII. Determine for each cell Z'c of color c in the wreath an opera-

tion S'c of the modular group which maps the nuclear cell Zc of color c into Z'c .

The SI thus selected, together with the operations Jc of the modular group which

carry Zc into itself, generate the whole group if all colors c are taken into account.

Were it not for the groups {j¿.} the nucleus would form a fundamental

domain. As it is, one has first to replace in our construction each G\ by a

part G\ which in G\ is a fundamental domain for the finite group of special

transformations

-7a: £*-*£*«* (e* a unit)

carrying GA into itself. The effect of J\ upon the coefficient y*K is described by

* *.yn —► e.'7ae*-

If in one split a the transformation of the variable y%c = 'E,

a a a °\~I

a—>ei A€* = e<E(e*) ,



is the identity, then the same is true of every split. Hence it is sufficient to

consider one split a only, and after choosing it we write simply 7^ = 7«*,

e" = e¿. If the transformation E—»tiEer1 is the identity, one must have ex = e2

as the specialization 2—1 shows, and E—>e2Ser1 is also the identity. More-

over, if E—»eiHer1 and E—>e2aei~l are identities, then E—»eiEer1 is. Conse-

quently we may well limit ourselves first to the coefficients y,k (i<k) on one

side of the diagonal, and then more particularly to

El = 7l2, Ä2 — 723) ' -• • , û«-l = yn-i,n-

Let us at once consider the most disagreeable case, that of a quaternion quasi-

field J as described in §6.

The group { Ja} induces a group of transformations of the type

( I ex I = I £2 I = 1)

for 7i2 = Ei = E. This is a finite group of orthogonal transformations J in the

space of the four components Xo, Xx, X2, X3 of the variable quaternion E.

Denote by aJ the transform of E by J. The simplest way of ascertaining a

fundamental domain for this group {j} is as follows: One chooses a point

E = A which differs from all its transforms AJ (JVidentity). The fundamental

domain consists of all points E which are nearer to the center A than to the

other equivalent centers AJ and is thus characterized by the inequalities

E-A - A/+ (A - A/)-È~è 0.

Fortunately these are linear inequalities, namely of the form

a0X0 + aiXi + a2X2 + a3X3 ^ 0

(a0, ai, a2, a3 being the components of A—AJ). After having done this we limit

ourselves to those operations Ja which leave Ei unchanged. They form a sub-

group and we study its influence upon E2, • • • , En-i- The next step would

consist in singling out E2. By induction we thus obtain a finite number of

subsidiary linear inequalities each concerned with the four components of one

of the variables 712, • • • , 7n-i,n only, and by them we define the fundamental

domain Ga' in Ga for the group { Ja} •

I set little store by this whittling down of Ga to Ga' . It seems less artificial

to operate with the whole cells Z; in doing so one has to keep in mind that the

modular group in its influence upon Z matters only modulo {Jz\ ■

Institute for Advanced Study,

Princeton, N. J.

a —>• ex a e2


THEORY OF REDUCTION FOR ARITHMETICAL EQUIVALENCE. II(1) · 2018-11-16 · THEORY OF REDUCTION FOR ARITHMETICAL EQUIVALENCE. II(1) BY HERMANN WEYL 1. Introduction. Lattices over the

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