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NOTE ON MODULAR INVARIANTS.

By Dr. L. E. DICKSON.

sem elementary standpoint, the functions. -། or more varies which are invariant under rus cuate, modulo p (a prime), of deter

sar is de case of a binary form

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..... we get c=(-1)'c.. Hence c=(-1) c exists no invariant binary form. Varg ne transormation (+ey, y) to (1), we get

A

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we videor the transformation with e=1. Ten C=cm=0 shows tha

ewus *\____]+p>2* C, ..., C2, give

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To prove by induction that, if p>3, Cp., ...,

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Cap-3 give

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CH=0, C 0, Cap-3 = 0,

p+2

Cp+1=0,..., Cpp-3=0 for 1 <p<p-1. Then C., gives

(5)

let c,

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The second and third binomial coefficients are evidently multiples of p; also the first is a multiple of p since m is a multiple of p and pp is not. Hence

We shall prove by a two-fold induction that

(6) C(-1)p11=0, C(j-1)p+2=0, •••9 Cjp-j-1=0 (j=1, 2, ..., m/p). We assume that these relations hold for j=1,..., s and prove that they hold also for j=s+1. They are true for j=1 and j=2 by (3) and (4). By our hypothesis c, (isp) is zero unless has one of the values

(7)

0, jp-j, jp-j+1, ..., jp (j=1, ..., 8).

We proceed to prove that Cp. C

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Cl- give
= 0,

viz., relations (6) for j=8+1. We may set p>8+2, since in the contrary case the statement is vacuous.

proof by induction, let

To make this

Cap=0, Cp=0,..., Cp.P-=0 (1<p<p-s).

Hence in C, for r=sp+p, the only c, not known to be zero is that with sp+p-1 and those for which i has one of the values (7). The coefficient (") of c, is a multiple of

numbers (7), other than 0, are

i=jp-k (0 ≤k<j≤8).

Since m is of the form pu, the coefficient of c, in C, is

which is a multiple of

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p. The

p. Indeed, (4) is a multiple* of p iť

a = £a,p',_b = 2b,p* (0 ≤a,<p, 0 ≤b;<p),

* Dickson, Annals of Mathematics, vol. xi. (1896), p 75.

DERIVED RELATIONS INVOLVING B, En,

§§ 18-23.

V-formulæ

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..., §§ 18-27. Relations between Bernoullian numbers, derived from the

94

S$ 24-27.

Relations involving En, In, Jn, derived from the V-formulæ

99

§§ 28-33. §§ 31-35.

GENERAL AND SPECIAL U-FORMULE (NOT PROCEEDING BY POWERS), §§ 28-35. General formulæ relating to Un (x)

102

Special cases of the formulæ relating to Un(x)

105

$36. $$ 37-38.

DERIVED RELATIONS INVOLVING Bn, En, $$ 36-41.
Special values of Un (x)

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106

Relations involving Bernoullian numbers derived from the

U-formulæ

106

§§ 39-41.

Relations involving En, Hn, Pn, Qa, Tn derived from the

U-formulæ

107

GENERAL AND SPECIAL - AND U-FORMULE PROCEEDING
BY POWERS OF 2, §§ 42-44.

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GENERAL AND SPECIAL V- AND U-FORMULE PROCEEDING
BY POWERS OF r, §§ 52-62.

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RELATIONS INVOLVING Ba, En, AND PROCEEDING
BY POWERS OF 1, §§ 63-70.

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SPECIAL RELATIONS INVOLVING Bn, En, CORRESPONDING TO 1, 2, 3, .

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p. 43, 1. 5, for (i) read (iii).

p. 48, 1. 4 and 1. 8, and p. 49, 1. 4, for he read han

p. 49, 1. 15, for 3 read =33.

p. 62, 1. 15, for Pa-2 read Pn-1

p. 75, 1. 6, for 32′′ – 2 read 32′′ – 3.

p. 89, 1. 4, for § 147 read § 145.

p. 94, 1. 17, for n=2n read n=2m.

p. 102, 1. 1, for § 178 read § 177.

p. 102. The formula in §194 is not true except for c=1, and this section should be omitted. The error arises from writing - instead of eac-1 in the denominator in 1. 7 of the section. The formula in the case of c=1 is at once derivable from the equation on p. 102, 1. 2, by putting æ+1, x+2, x+-1 for x and adding.

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NOTE ON MODULAR INVARIANTS.

By Dr. L. E. DICKSON.

WE consider, from an elementary standpoint, the functions

of two or more variables which are invariant under every linear transformation, modulo p (a prime), of determinant unity.

Consider first the case of a binary form

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The conditions for invariance of B under (y, x) are

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c=(-1) c Cm-i (i=0, 1, ..., m).

cm-i

Replacing i by m-i, we get c=(-1)'c,. Hence c=(-1)" c.. If m is odd and p>2, there exists no invariant binary form. Applying the transformation (x+ey, y) to (1), we get

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which is identical with (1) if, and only if,

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For the present we shall employ the transformation with e=1. We first assume that c0. Then C=cm=0 shows tha m is a multiple of the modulus p. If p>2, C,..., C, give

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To give a proof by induction, we assume that, for 1<r<p, c=0,..., C=0. Then C, gives

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The first binomial coefficient is a multiple of p and the second is not. Hence c,, = 0. Next, C, gives

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P+1

Now C is satisfied if p>2, and is considered under C-1 if p = 2.

*The case p = 2 is not exceptional; (3) is then a true but vacuous statement. Similarly in (5) if p 3.

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