English translation of Frobenius' and Stickelberger's "On the theory of elliptic functions"

§ 12. English tr anslation of F r ob enius’ and Stickelb er ger’s ON THE THEOR Y OF ELLIPTIC FUNCTIONS Journal für die r eine und angewandte Mathematik 83, 175–179 (1877) F erdinand Georg F rob enius with Ludwig Stick elb erger Züric h Marc h 10, 1877 A remark able form ula comm unicated b y Mr. Hermite in a recen tly published note on elliptic functions (this Journal V ol. 82, p. 343) prompts us to p oin t out the connections b et ween sev eral related formulas. T o pass from the general equation with which w e b egin to the more sp ecialised form ula of Mr. Hermite , and from there to the further sp ecialised one that Mr. Kiep ert 1 has used as the basis of his solution to the m ultiplication problem, w e emplo y a limiting pro cess which w e first present in its general form. Let f 0 ( u ) , f 1 ( u ) , . . . f n ( u ) b e n + 1 conv ergen t series of p ositiv e p ow ers of u , and let u 0 , u 1 , . . . u n b e arbitrary v alues within their common region of conv ergence. Then the determinan t of ( n + 1) th degree | f α ( u β ) | = F ( u 0 , u 1 , . . . u n ) can b e expanded in to a series of p ow ers of u 0 , u 1 , . . . u n and this, as an alternating function, can b e brough t to the form F = G ( u 0 , u 1 , . . . u n )Π( u α − u β ) where the symmetric function G is likewise a series of p ositive p o wers of u 0 , u 1 , . . . u n . (In the difference pro duct, α and β are to run through the pairs of num b ers 0 , 1 , . . . n here and in the follo wing suc h that α > β .) Then the v alue whic h the function G ( u 0 , u 1 , . . . u n ) assumes when all n +1 v ariables are set equal to u can easily b e expressed as a determinan t. F or simplicity , let h denote a small quantit y and set u β = u + β h, ( β = 0 , 1 , . . . n ) 1 Kiep ert , A ctual execution of the in teger multiplication of elliptic functions, this Journal V ol. 76, p. 21. There one finds the definition of the functions σ ( u ) and ℘ ( u ) whic h Mr. W eierstr ass has in tro duced in to the theory of elliptic functions, as well as a brief compilation of their most imp ortant prop erties. Regarding the formulas and theorems from this theory which we will use in the following, we refer to that treatise. 1 and in tro duce the notation ∆ f ( u ) = f ( u + h ) − f ( u ) , then b y a kno wn determinan t theorem | f α ( u β ) | = | ∆ β f α ( u ) | and consequen tly G = F Π( u α − u β ) = 1 Π( α − β )     ∆ β f α ( u ) h β     . F rom this one obtains, b y letting h approach the limit 0 , the sought relation lim F ( u 0 , u 1 , . . . u n ) Π( u α − u β ) = 1 Π( α − β ) | f ( β ) α ( u ) | . (1) On the left side, the quotien t is first to be expanded into a series of p ositiv e p ow ers of u 0 , u 1 , . . . u n , and then the arguments are all to b e set equal to u . W e now apply a similar pro cedure to the determinan t R =         0 1 . . . 1 1 ψ ( u 0 + v 0 ) . . . ψ ( u 0 + v n ) . . . . . . 1 ψ ( u n + v 0 ) . . . ψ ( u n + v n )         , where ψ ( u ) = d log σ ( u ) du . The difference ψ ( u + v ) − ψ ( u ) is a doubly perio dic function of u . By m ultiplying the elemen ts of the first row of R with ψ ( u 0 ) and subtracting them from those of the second ro w, one recognises that this determinan t is a doubly perio dic function of u 0 . Since the same conclusion applies to the other v ariables en tering into R , this function of 2 n + 2 argumen ts also has the remark able prop ert y of b eing doubly p erio dic with resp ect to eac h of them. The function ψ ( u ) b ecomes infinite only for u = 0 (and congruen t v alues), and its expansion in ascending p o w ers of u b egins with 1 u . Considered as a function of u 0 , R therefore b ecomes infinite only at the n + 1 v alues u 0 = − v 0 , − v 1 , . . . − v n and congruent v alues, with simple p oles at eac h. On the other hand, R ob viously v anishes for u 0 = u 1 , . . . u n . No w ho wev er an elliptic function 2 has as man y zeros as infinities, and (after Ab el’s theorem) the sum of the v alues for whic h it v anishes is congruen t to the sum of the v alues for which it b ecomes infinite. ( Briot et Bouquet , F onctions elliptiques, I I. éd., p. 241, 2 F ollo wing Mr. W eierstr ass , we call a doubly perio dic function elliptic if it has the character of a rational function ev erywhere in the finite plane, i.e. if it can b e expanded in the neigh b orho o d of ev ery finite v alue a as a series in integer p ow ers of u − a containing only finitely man y negativ e p o wers. 2 Théor. I I I; p. 242, Théor. V. Kiep ert , l. c. p. 24 and 25.) Consequen tly R m ust also v anish for an ( n + 1) th v alue of u 0 , whic h is to b e calculated from the equation u 0 + v 0 + · · · + u n + v n = 0 By a kno wn theorem ( Briot et Bouquet , p. 242, Théor. IV; p. 243, Théor. VI. Kiep ert , l. c.) that determinant is therefore, up to a factor indep endent of u 0 , equal to σ ( u 0 + v 0 + · · · + u n + v n ) σ ( u 1 − u 0 ) σ ( u 2 − u 0 ) . . . σ ( u n − u 0 ) σ ( u 0 + v 0 ) σ ( u 0 + v 1 ) . . . σ ( u 0 + v n ) . If one in v estigates in a similar wa y its dep endence on the remaining 2 n + 1 argumen ts, one finds 3 , up to a constant factor, that, −          0 1 . . . 1 1 σ ′ ( u 0 + v 0 ) σ ( u 0 + v 0 ) . . . σ ′ ( u 0 + v n ) σ ( u 0 + v n ) . . . . . . 1 σ ′ ( u n + v 0 ) σ ( u n + v 0 ) . . . σ ′ ( u n + v n ) σ ( u n + v n )          = σ ( u 0 + v 0 + · · · + u n + v n )Π σ ( u α − u β )Π σ ( v α − v β ) Π σ ( u α + v β ) . (2) In the denominator of the right side the pro duct is to b e extended o ver all pairs of n um b ers from 0 to n , and in the numerator only ov er those for which α > β . T o v erify that the constan t factor in equation (2.) is correctly given, one observ es that it holds immediately for n = 0 , and in general follows easily b y induction from n to n + 1 . Indeed, if one m ultiplies the elemen ts of the last row on the left side by u n + v n and then sets u n = − v n , they all v anish except the last, which b ecomes 1 . Therefore R reduces to the determinan t formed analogously from the n argumen t pairs u 0 , v 0 , . . . u n − 1 , v n − 1 . In the expression on the right side of equation (2.), when u n = − v n w e ha v e lim u n + v n σ ( u n + v n ) = 1 , σ ( u n − u β ) σ ( v n − v β ) σ ( v n + u β ) σ ( u n + v β ) = 1 ( β = 0 , 1 , . . . n − 1) . Th us the right side likewise reduces to the expression formed analogously from the n argumen t pairs u 0 , v 0 , . . . u n − 1 , v n − 1 . In equation (2.) we now set v β = β h ( β = 0 , 1 , . . . n ) and transform the left side b y the metho d already applied ab o ve in to a determinan t in whic h the ( α + 2) th ro w con tains the elements 1 , ψ ( u α ) , ∆ ψ ( u α ) , ∆ 2 ψ ( u α ) , . . . ∆ n ψ ( u α ) . Since in this transformation the elements of the first row b ecome 0 , 1 , 0 , 0 , . . . 0 3 F or n = 1 this form ula essentially coincides with that whic h Jac obi has giv en in V ol. 15 of this Journal (p. 204, 13). 3 the left side reduces to a determinant of ( n + 1) th degree, which we denote in an easily understandable manner with − R = | 1 , ∆ ψ ( u α ) , ∆ 2 ψ ( u α ) , . . . ∆ n ψ ( u α ) | . Consequen tly     1 , ∆ ψ ( u α ) h , ∆ 2 ψ ( u α ) h 2 , . . . ∆ n ψ ( u α ) h n     = σ ( u 0 + v 0 + · · · + u n + v n )Π σ ( u α − u β ) Π σ ( u α + v β ) Π σ ( v α − v β ) h , therefore, in the limit that h approaches zero, | 1 , ψ ′ ( u α ) , ψ ′′ ( u α ) , . . . ψ ( n ) ( u α ) | = σ ( u 0 + · · · + u n )Π σ ( u α − u β )Π( α − β ) (Π σ ( u α )) n +1 . Setting according to Mr. W eierstr ass − ψ ′ ( u ) = − d 2 log σ ( u ) du 2 = ℘ ( u ) , this form ula reads         1 ℘ ( u 0 ) ℘ ′ ( u 0 ) . . . ℘ ( n − 1) ( u 0 ) 1 ℘ ( u 1 ) ℘ ′ ( u 1 ) . . . ℘ ( n − 1) ( u 1 ) . . . . . . . 1 ℘ ( u n ) ℘ ′ ( u n ) . . . ℘ ( n − 1) ( u n )         = ( − 1) n Π( α − β ) σ ( u 0 + · · · + u n )Π σ ( u α − u β ) (Π σ ( u α )) n +1 . (3) This is essen tially the equation given by Mr. Hermite in the ab ov e cited letter. W e no w apply the limiting pro cess from the b eginning once more, choosing in formula (1.) f 0 ( u ) = 1 , f 1 ( u ) = ℘ ( u ) , . . . f n ( u ) = ℘ ( n − 1) ( u ) . Then in the determinan t | f ( β ) α ( u ) | the elemen ts of the first column v anish except for the first, and it reduces to a determinant of n th degree. One thus arriv es at the form ula of Mr. Kiep ert (l. c. p. 31, formula ( 29 a .))         ℘ ′ ( u ) ℘ ′′ ( u ) . . . ℘ ( n ) ( u ) ℘ ′′ ( u ) ℘ ′′′ ( u ) . . . ℘ ( n +1) ( u ) . . . . . . ℘ ( n ) ( u ) ℘ ( n +1) ( u ) . . . ℘ (2 n − 1) ( u )         = ( − 1) n (Π( α − β )) 2 σ (( n + 1) u ) σ ( u ) ( n +1)( n +1) . (4) • ⋄ • 4 T ranslator’s Notes Original paper. F rob enius, F. G., & Stic kelberger, L. (1877). Zur Theorie der elliptis- c hen F unctionen. Journal für die r eine und angewandte Mathematik , 83, 175–179. DOI: https://doi.org/10.1515/crll.1877.83.175 Wikimedia scans: https://en.wikipedia.org/wiki/File:Zur_Theorie_der_elliptischen_Functionen.djvu T ranslated by K. Khazanzheiev and G. Hesk eth, March 2026. 5