Hazlett, Olive C.

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OLIVE C. HAZLETTOn formal modular invariantsJournal de mathématiques pures et appliquées 9e série, tome 9 (1930), p. 327-332.<http://www.numdam.org/item?id=JMPA_1930_9_9__327_0>Article numérisé dans le cadre du programmeGallica de la Bibliothèque nationale de Francehttp:// gallica.bnf.fr/et catalogué par Mathdocdans le cadre du pôle associé BnF/Mathdochttp:// www.numdam.org/ journals/ JMPAON FORMAL MODULAR INVARIANTS. 327 On formal modular invariants ; BY OLIVE C. HAZLETT. I. Summary and relation ίο the literature. — This note gives ari extension of the principal results containd in a paper (') by VV. L.-G. Williams in this journal, in Part I, he proves that p . 1 a(, J V 4- ( ^ J «, i- . . . 4- a,,] i n and p _ 1 Κ ™; at'-* ^α:υ /'/ -+- ̂  jo, e/'"1 -i- . .. + a,f j {>.= ι . ). 1=0 are formal modular invariants, modulo />, of the binary <f — ic form aD. xq 4-( χ'' ■+- ■ - ■ ^ a'ifr· where ρ and q are such that none of the binomial coefficients is divisible by p. He also proved that( if ρ is odd and λ is an integer such that ~A(p — ι)Jq is integer, then I. ~ 4- β0 p _ 1 '14- ( ^ 1 <Z| 4-. . . 4- α η I , ι=ο is a formal modular invariant of the above q — ic. These he calls, iri (') On the formal modular invariants of binary forms {Journal \de Mathéma-tiques. 9e série, t. 4. ig^ô. p. îtig-ig?.). Journ. de Math., tome IX. — Faso. IV, ig3o. 42 3?. 8 OLIVE C. HAZLETT. order, Theorems I, Il and III. The fust two are due to Hurwitz (') and the third is a generalization of the second. In part III, he proves four theorems (2) each of wicli asserts that, if /(.*, y ) — aa::t -+- ?) bar y -f- 'λ rx.y- -+- rly'\ then certain special polynomials in the values of f(x, y) when .r, y arc marks of the Galois field, GF|p],of order ρ — where ρ is a prime — are formal modular invariants of J\oL\y'). These theorems are not only very special in that they are stated for special invariants, hut they are special in another sense, since they are stated and proved only for the cases where no one of the bino-mial coeflicients is congruent to zero modulo p. This last is more than a slight restriction. For, if the study of formal modulai· invariants ever approach completion, we shall want to study the invariants of the quadratic, cubic, quartic, etc. modulo 2 and similarly modulo/>where/> is any fixed prime. But, when/; = 2, we can not use binomial coefficients for the binary η — ic when rj = 2, 4> 5, G, 8, 9, 10, etc., when ρ — 3, we can not use binomial coefficients when η — 3, /j, G, 7, 9, ίο, etc.; nor if ρ = 5, when </ = 5, 6, 7, <8, 10, etc.; and similarly larger primes. In none of the cases just indicated, do Lhe proofs in Mr. Williams1 paper hold. Hence it would seem desirable to have proofs that hold without exception. The present note not only gives such proofs, but also generalizes the invariants to which the theorems apply. A.s will he seen, the proofs are exceedingly simple and use no technical knowledge of the subject. 2. Definitions. — Consider the polynomial (I) /( .R,. J-., ) — -+· ave'!," 1 .v., + ο,,χ'Ι,. in wich the coeflicients //,, ... are indéterminates and subject the ("1 ) Hôhere Corigruenzen (Archiv tier ΛίαίΙι. u. I'hysik. 3e série, vol. 5, 190'i, p. 17-27)· (2) Each of the first two theorems also asserts that, no invariant whose leading term is a power of d is essentially different, from those given by the formula in the theorem. ON FORMAL MODULAR INVARIANTS, 329 variables to any transformation' (2) xr1 = E1. x'1 -4 7, ,.r'2, x2 = Ec x. -4 r/2, in which the coefficients ξ,, £a, rn, rJ2 are integers reduced modulo/>, a prime, subject to the restriction that the determinant of the trans-formation is not congruent to zero, modulo p. Any polynomial in the 7/'s which is unaltered in form, modulo p, under the total group of all such transformations is called a forma! modular invariant of (i), modulo p. For simplicity, we shall use the fact that this group is generated by the transformations t.'h α· , = .·/, 4- ./2. ,v.,~ .74, and ( A) x1 = x'2, x2 = - -F,, of determinant unity and the transformation ( Γ) ) x1 = x'1 x2 = Q x'2 ( ρ ;ι primitive root of llie field ). Similary we hawe the notion of a formal modular invariant of (i) with respect to the Galois field, GF[ p" |, of order ρ". 5. The two theorems. — First we prove a generalization of Prof, Williams theorems of his Part I. Since the transformation (3) permutes the pairs (l, ι)(ί = ο, ι, , .., ρ— ι) modulo p, any symmetric func-tion of all the values of/(.r,, x2) as (#,, χφ ranges over the above pairs and the pair (ο, ι ) is invariant modulo ρ under (3). hnder (4), /( ι, o) goes into/(ο, ι ),/(<>, i) goes into /(— ι, o) = (— ι>'/(ι, o) and when t φ ο, /'(ι, ι) goes into ]\— ι, t) — f j\t', i)where i'=—i/1. Finally, under (5), /(ι, o) is invariant and /(/, i) goes into f(t, ρ) = ff(f, i) where t" = — tj p. Hence if λ be an integer such that /.· = ~L(p — ι )/ </ is an integer, then the k' th powers of the values of f(x,, a?3) where (xx, xf range over the above set-are merely permuted among themselves bv any transformation ( 3), f4>, (5) and hence by any transformation (2), Thus we have. 33o OLIVE C. HAZLETT. THEOREM 1. — Let f(x,, x2 ) == «0x'\"1 A?3 +. . . + Β*/#* ^ any binary q — ic and let k be any integer of the form \(p — i)Jq where λ is an integer. Then any symmetric function of the k'th powers of f(\,o) and f(i, i)(i = o, .··,/?— \) is a formal modular invariant of f with respect to the Galois field, GF[p], of order ρ if there is made no lowering of the exponents is made by application of Fermât1 s theorem. Special cases of this are the three theorems of Williams1 Part I of wich the first two are due to Hurwitz. In essentially the same is proved the COROLLARY. — This is Theorem 1 stated for the general Galois field, GF[/;" ], of order ρ". In his paper, he used essentially the symbolic notation of classic invariant theory which can not, however, be used for formal modular invariants of a binary q — ic in case any of the binomial coefficients is congruent to zero, modulo p. 13ul the above theorem and corollary can be proved equally well for the general binary q— ic by using the symbolic notation, for formal modular invariants (■). In this latter notation /(«·*··,, ··'-'·.) 2-i J£( «,./·, H- X,( a -h I ( β I-!- £,.7;.,').. . α where the </ symbolic linear factors are symbolically distinct. Since (α,, a2) are subjected to a linear transformation when the x's are subjected to a linear transformation (2), then L(«) = Π<«·'· + «·'.) I is a formal modular invariant of /(χ,, x..) when (i,, z2) range over the set τ : (ι, o), (/, J) (t — ο, 1, . . ., ρ — ι). Similarly with Ρ(β), L(y), etc. Hence II L(oc) =/*(*,, (...) is a formal modular invariant α of f(x 1, x.f) and thus we have Theorem 1 in part 1, Similarly, we prove Theorem Ί and 3 of Part II. (') HAZLETT. A symbolic theory of format modular invariants (Transactions of the Amer. Math. Soc.. vol. 24. ig-i?.. p. ·>86-3ιι). ON FORMAL MODULAR INVARIANTS. 331 in the same manner as above, we may also prove the four theorems in Part III. We do not make any actual change in his proof, but merely note that he does not actually use the binomial coefficients in his proof in any way. His proofs hold with complete generality although that was not true of his proofs of the theorems in his Part 1. Moreover, they are special cases of a more general theorem which we shall now proceed to prove. LetF, be any polynomial in /(i,o)andlhe/U, ι)(/=ο, ι — i). Then F, is a polynomial in the a's in which we understand that no reduction in the exponents of the a's has been made by applying Fer-rnafs theorem. If we apply any particular transformation (2), F, is replaced by another polynomial in the a's in which, also, we unders-tand thai there has been made no reduction in the exponents of the rt's. If these two polynomials are always identical (moduloyj) — that is, if F, be formally unaltered under all transformation — it is a formal modular invariant of J\xx, x-2). If, however, F, be altered by some transformation σ,, let F2 be the transform of F, under σ,, let F:, be the transform of F2 under σ,, etc. If Φ,, denote any symmetric function of those conjugates of F, under the group S, generated by <7, which are incongruent modulo ρ, then Φ, is formally unaltered under the group S,. Now let Φ3 be the conjugate of Φ, under some transforma-tion σ2 of the group (2) not in S,, and similarly Φ, has conjugates under the various transformations of the group generated by σ2. If Φ, be unaltered under σ2, then Φ, is an invariant under the group S2 of transformations generated by cr, and σ2. If Φ, be note unaltered under σ2, let us form the same symmetric function Ψ, of Φ,, Φ2, . .., that is of F,, F2, .... Then Ψ, is invariant under the group S2. Continue in this way. Since there is only a finite number of conjugate of F,, incongruent modulo p, under the group (2) this simple process is feasible in the modular case at it would not be in the classic case. AnoLher natural process for forming the formal modular invariants obtained in the above manner is the following (1 ). Let F, be as above. Under any particular transformation (2), F, is replaced by the pro-(') Strictly speaking, tlie invariant obtained by the first method is a constant mul-tiple of the one obtained by the second method. 132 OLIVE G. HAZLÉTT. — ON FORMAL MODULAR INVARIANTS. duct of a non-zéro o',/: — vvliere /.' is I,lie degree of F, in the a's, — and F,·, where Ft is here obtained from F, by changing the order of tjie pairs τ. Consider the y[ th powers, P', of F, where */ is an integer of the form \(p— OAA' i'1 which λ is a fixed positive integer-. Then any symmetric function of the P, is a formal 1 nodular invariant. In a similar manner we prove THEOREM 2. — Let f (χ,, χ., ) be any homogeneous polynomial in two variables of order η and let F, be any homogeneous polynomial in J\i, o) and the f\t, 1) t ranges over the marks of the Galois field f « F| ρ" | of order ρ". Let F, range over all the conjugates of K, under the permutations of the pairs τ which are incongruent in the Galois field, GF| yj" |, of order p". Ify=~/fp — 1 )[(/ be an integer {where is some fixed integer), then any symmetric function of the yj th powers of the l\ is a formal modtdar invariant under the group (2). A. More than two variables. — Now let /"(or,, x.,, . . ., xn) he the general homogeneous polynomial in n variables with coefficients denoted by a with suitable subscripts. We subject the variables to the transformations of the group (.6) Iciij.r'j ( /',./ = 1, where the a's are marhs of the Galois field, GF| //"], of order pm such that their determinant is not zero in the field. Instead of the former set τ of pairs (a.·,, xf) we have the set σ : (ι, ο, . . ., ο), ( ρ,, ο, .... ο), (ρ,, ρ2, ι, ο, . .., ο), . .(ρ,, ρ,, . .., ρ„....,, ι). A proof closely paral-lel to the one above for the binary case holds here although the sym-bolic proofs do not seem to hold. Thus we have THEOREM .*1. — Let f(xK, . .., x,f) be any homogeneous polynomial in n variables of order q and let F, be any homogeneous polynomial in the values of f as the (xt, .. ., x,j) range over the set a. Let F, range overall these conjugates of F, under the trans formations of the group (6), which are incongrucnt in the field. If 'y = λ (p'n— i)jq be an integer (where λ is some fixed positive integer), then any symétrie function of the y' th powers of the F,: is a formal modular invariant of J' under the group (6) with respect to the field GF [ ρ"' |. 

On formal modular invariants

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On formal modular invariants

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