Sigma functions for a space curve of type (3, 4, 5)

Sigma functions for a space curv e of t yp e (3 , 4 , 5) SHIGEKI MA TSUT ANI AND JIR YO K O MEDA Abstract. In this article, a gener alized Kleinian s igma function fo r an affine (3 , 4 , 5) space curve of genus 2 w a s constructed a s the simplest ex ample of the s ig ma function for an affine space curve, and in terms of the sigma function, the Jacobi inv ers ion formulae for the curve are obtained. An int eresting r e lation b etw een a spa c e curve with a semigroup generated by (6 , 13 , 14 , 15 , 1 6) and Nor ton n um ber asso ciated with Monster gro up is also men tioned with a n App endix by Komeda. sigma function, space curv e, Jacobi in v ersion form ula MSC 2010: 14H05, 14H42, 14H50, 14H55, 20M07 1. Intr oduction Recen tly the Kleinian sigma function fo r hyperelliptic curv es, a natural generalization of the W eierstrass sigma function, is re-ev aluated b ecause in terms of the sigma functions, it is more con ve nien t to in ves tigate the prop erties of the ab elian functions and their in teresting prop erties are rev ealed naturally [2, 17, 6 ]. F urther in [5], Enolskii, Eilb ec k, and Leykin disco v ered a construction whic h generalizes the K leinian sigma function a sso ciated with h yp erelliptic curv es to one for a n affine ( r , s ) plane curv e, where r and s ( r < s ) are coprime p o sitiv e in tegers g = ( r − 1)( s − 1) / 2. In [5], they , firstly , constructed the f undamental diff erential of the second kind o v er a n affine ( r , s ) plane curv e and using it, obtained the Legendre relation as the symplectic structure o v er the curv e. Using the Legendre relation, they define d the generalized Kleinian sigma function o v er the image of the ab elian ma p C g . They also found the natura l Jacobi in v er- sion formulae in terms of their sigma f unction. W e call the construction EEL c o nstruction in this article. Using the EEL construction, w e hav e some in teresting results [20, 21]. In this article, we consider a generalized Kleinian sigma function for an affine (3 , 4 , 5) space curv e of genus 2, which is the simplest affine space curv e. Our purp ose of this article is to sho w that the sigma function is also defined for an affine space curv e as w e can do for plane curv es. F ollo wing the EEL-construction, w e define the fundamen ta l differen tial of the second kind ov er it and obtain the Legendre relation as the symplectic structure o ver it. With the ab elian map to C 2 , w e sho w that the symplectic structure determines the sigma function. Date : 1 2 SHIGEKI MA TSUT ANI A ND JIR YO KOMED A F urther using the sigma function, w e obtain the Jacobi inv ersion form ulae for the curv e and the Jacobian fo llowing the previous w o r k [20, 21]. It means that the gene ralization of the s igma functions fo r the affine plane curv es to ones for the space curv es is basically p ossible and is useful. Recen tly , Korotkin with Shramc henk o defined a sigma function for a compact Riemann surface [15] but it is not directly associated with an algebraic curv e. F urther Ay ano in tro duced sigma functions for space curv es of special class [1 ], whic h are called telescopic curv es, but the class do es not include this (3,4 ,5) curv e. In Remark 4.8, w e also show a problem of a space curv e asso ciated with the semigroup generated by (6 , 13 , 14 , 15 , 16) with an App endix b y Komeda. The semigroup might b e related to Nor ton num b er asso ciated with t he Monster group, the simple largest sp oradic finite group [22]. 2. Preliminar y 2.1. Numerical semigroup. Here w e g iv e a short o v erview of recen t study of the n u- merical semigroups as sub-semigroups of non-negative integers N 0 related to algebraic curv es. W e call an additive semigroup in N 0 numeric al se m igr oup if its complemen t in N 0 is finite. F or a numerical semigroup H = H ( M ) generated by M , the n umber of elemen ts of L ( H ) := N 0 \ H is called genus and L ( H ) is called gap sequence. F or example, we hav e semigroups H 2 , H 4 , H 12 generated by M 2 := h 3 , 4 , 5 i , M 4 := h 3 , 7 , 8 i , M 12 := h 6 , 13 , 14 , 15 , 16 i respectiv ely whose genera are g ( H g ) for g = 2 , 4 , 12 due to L ( H 2 ) = { 1 , 2 } , L ( H 4 ) = { 1 , 2 , 4 , 5 } , L ( H 12 ) = { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 1 1 , 17 , 23 } . F o r a complete non-singular irreducible curv e C of gen us g o v er a n algebraically closed field k of c haracteristic 0, the field of its rational functions k ( C ), and a p oint P ∈ C , w e define (2.1) H ( P ) := { n ∈ N 0 | there exis ts f ∈ k ( C ) suc h that ( f ) ∞ = nP } whic h is called the W eierstrass semigroup of the p oin t P . If L ( H ( P )) := N 0 \ H ( P ) differs from the set { 1 , 2 , · · · , g } , w e call P W eierstrass p o in t of C . A nu merical semigroup H is said t o b e W eierstrass if there exists a p ointed algebraic curv e ( C , P ) suc h that H = H ( P ). Hurwitz conside red whether ev ery n umerical se mi- group H is W eierstrass. This w as a long-standing problem but Buc h w eitz finally sho we d that eve ry H is not W eierstrass. His first coun terexample is the semigroup H B generated b y 13, 14 , 15, 16, 1 7, 18, 20, 2 2 and 23, whose gen us is 16. Thus in general, it is not so trivial whether a g iven semigroup is W eierstrass o r not. Komeda has b een inv estigated this problem with Ohbuc hi and Kim [10, 11, 12 , 14]. 2.2. Comm utative Algebra. Here w e review a normal ring and normalization in com- m utat ive ring [16]. W e a ssume that ev ery ring is a comm utative ring with unit. B is a ring and A is a subring of B . B is said to b e extension of A . An elemen t b of B is said to b e inte gr al o v er A if b satisfies a monic p olynomial ov er A , i.e., there exist n and { a i } i =1 ,...,n ∈ A suc h that b n + a 1 b n − 1 + · · · a n = 0 . 3 W e say that B is inte gr al ov er A , or B is an inte gr al ring ov er A , or B is an inte gr al extension of A if eve ry eleme n t b of B is in tegral o v er A . An inte gr al closur e in B o v er A is defined b y ˜ A := { b ∈ B | b is integral ov er A } . If A = ˜ A , A is inte gr al close d in B . Definition 2.1. A is a ring and Q ( A ) is a quotient ring of A . We assume that A is an inte gr al domain. A is nor ma l if A i s inte gr al close d i n Q ( A ) , i.e., fo r ˜ A := { q ∈ Q ( A ) | ther e exist n and a i ∈ A s uch that q n + a 1 q n − 1 + · · · a n = 0 } , A = ˜ A . We define the minimum extension ˆ A of A in Q ( A ) so that ˆ A is inte g r al close d in Q ( A ) . We say that ˆ A is normalization of A o r the normalized ring o f A . Through the corresp ondence b etw een an algebraic v ariet y and a comm uta tiv e ring, w e ha ve the w ell-kno wn normalization theorem [9, p.5, p.68]: Theorem 2.2. F or any irr e ducible algebr aic curve X ⊂ P 2 C , ther e exists a c omp a ct Riemann surfac e ˜ X and a holomorphic mapping s : ˜ X → P 2 C such that s ( ˜ X ) = X and s is inje ctive on the inverse image of the set of smo oth p o ints of X . F urther the Riemann surfac e is uni q ue up to its isomorphism; if ther e a r e two Riemann surfac es ˜ X and ˜ X ′ given by normalizations of X , ther e is a biholomorphic fr om ˜ X to ˜ X ′ . As examples of Theorem 2.2 , we giv e three examples. Example 2.3. ( x 3 − y 2 ): R := C [ X , Y ] / ( X 3 − Y 2 ) is not no rmal b ecause Y X ∈ ˜ R \ R ⊂ Q ( R ) due to  Y X  2 − X = 0 . Since R ≈ C [ t 2 , t 3 ], the normalized ring is ˆ R = C [ t ]. Example 2.4. ( y 3 = x 5 − 1 and w 3 = z − z 6 ): F o llowing The orem 2.2, w e con- sider the cov ering of a curv e of f ( x, y ) := y 3 − x 5 + 1. Let us consider a homogeneous p olynomial F ( X, Y , Z ) := Y 3 Z 2 − X 5 + Z 5 ∈ C [ X, Y , Z ]. Around Z 6 = 0, w e ha v e F ( X , Y , Z ) = Z 5  Y 3 Z 3 − X 5 Z 5 + 1  and thus by regarding that x = X/ Z and y = Y / Z , w e ha v e F ( X , Y , Z ) = Z 5 f ( X/ Z , Y / Z ). R 0 := C [ x, y ] / ( f ( x, y )) is a normal ring. On the o ther hand, around Z = 0 and X 6 = 0, w e ha ve F ( X , Y , Z ) = X 5  Y 3 Z 2 X 5 − 1 + Z 5 X 5  , and then we obtain a polynomial, g ( w , z ) = w 3 z 2 − 1 + z 5 b y rega r ding w = Y /X and z = Z /X . Ho wev er R ∞ := C [ w , z ] / ( g ( w , z )) is not a normal ring. A s a v ector space, R ∞ is C 1 + C z + C z 2 + · · · + C w + C w z + C w z 2 + · · · + C w 2 + C w 2 z + C w 2 z 2 + · · · + C w 3 + C w 3 z . W e sho w that q ∈ Q ( R ∞ ) \ R ∞ exists suc h that q n + a 1 q n − 1 + · · · a n = 0 for c ertain a i ∈ R ∞ . Noting 1 1 − z g ( w , z ) = w 3 z 2 1 − z + 1 + z + z 2 + z 3 + z 4 = 0 ∈ Q ( R ∞ ) , w e consider q := w 3 1 − z + 1+ z z 2 ∈ Q ( R ∞ ) \ R ∞ , which is in tegral o v er R ∞ . By normal- ization, w e define ˆ w := w z = y /x 2 . ˆ R ∞ := C [ ˆ w , z ] / ( ˆ g ( w , z )) is a normal r ing, where ˆ g ( ˆ w , z ) := ˆ w 3 − z + z 6 . The minimal condition is obv ious. Example 2.5. (a space curve; y 3 = x 2 ( x 2 − 1) and w 3 = x ( x 2 − 1) 2 ): Let us consider a p olynomial f ( x, y ) = y 3 − x 2 ( x 2 − 1) and show that R 0 := C [ x, y ] / ( f ( x, y )) is not a normal 4 SHIGEKI MA TSUT ANI A ND JIR YO KOMED A ring. As a v ector space, R 0 is C 1 + C x + C x 2 + · · · + C y + C y x + C y x 2 + · · · + C y 2 + C y 2 x + C y 2 x 2 + · · · . W e sho w t ha t w ∈ Q ( R 0 ) \ R 0 exists such that w n + a 1 w n − 1 + · · · a n = 0 for certain a i ’s of R 0 . In other w ords, noting that y ∼ 3 p x 2 ( x 2 − 1) and y 2 ∼ x 3 p x ( x 2 − 1), one of w is that w := y 2 x whic h is integral ov er R 0 b ecause w 3 = y 6 x 3 = x ( x 2 − 1) 2 or w 3 − x ( x 2 − 1) 2 = 0 ∈ R 0 . Let g ( x, w ) = x ( x 2 − 1) 2 . Noting the relations that w = y 2 x , y = w 2 x 2 − 1 , and w y = ( x 2 − 1) x , w e hav e ˆ R 0 := C [ x, y , w ] / ( f 1 ( x, y , z ) , f 2 ( x, y , z ) f 3 ( x, y , z )) , as the normalized ring of R 0 , where f 1 ( x, y , w ) = y 2 − w x , f 2 ( x, y , w ) = w y − ( x 2 − 1) x , and f 3 ( x, y , w ) = w 2 − y ( x 2 − 1). The minimal condition is also obv ious. This example corresp onds t o the sp ecial case of the affine ( 3 , 4 , 5) space curv e in this article. Due to Theorem 2.2 , the corr espo nding Riemann surface uniquely exists up to an isomorphism. 3. A Cur ve (3,4,5) Since H 2 generated b y h 3 , 4 , 5 i is W eierstrass and is the simplest semigroup whose cardinalit y of the generators is greater t ha n 2, we consider a curv e C ( H 2 ) ex plicitly in order to construct the sigma functions for C ( H 2 ) fo llo wing the EEL construction. F o llowing The orem 2.2, in order to construct a non-singular curv e X 2 = C ( H 2 ), w e consider tw o singular curv es X 3 and X 4 generated by ∞ p oin ts and the zero es of f 3 , 12 ( x, y 4 ) := y 3 4 − k 4 ( x ) , f 4 , 15 ( x, y 5 ) := y 3 5 − k 5 ( x ) where k 4 ( x ) := k 2 ( x ) k 1 ( x ) 2 , k 5 ( x ) := k 2 ( x ) 2 k 1 ( x ), k 2 ( x ) := ( x − b 1 )( x − b 2 ) = x 2 + λ (2) 1 x + λ (2) 2 , a nd k 1 ( x ) := ( x − b 0 ) = x + λ (1) 1 for finite b a ∈ C ( a = 1 , 2 , 3) which is distinct from eac h other. Let us consider comm utativ e rings R 3 := C [ x, y 4 ] / ( f 3 , 12 ( x, y 4 )) and R 4 := C [ x, y 5 ] / ( f 4 , 15 ( x, y 5 )) related to X 3 and X 4 resp ectiv ely . These genera o f the semigroups asso ciated with their W eierstrass non-gap sequences at ∞ -p oin ts are three and four respectiv ely , though the geometric genera a r e not. F ollowing the normalization in section 2, w e normalize R 3 and R 4 . Since in terms of the language of the commutativ e algebra [1 6], y 2 4 ( x − b 0 ) is in tegral ov er R 3 in Q ( R 3 ) and y 2 5 ( x − b 0 )( x − b 2 ) is integral ov er R 4 in Q ( R 4 ), R 3 and R 4 are not normal ring s. Th us w e will nor ma lise them in C [ x, y 4 , y 5 ] in the meaning of the comm utative a lgebra [16] (see Example 2 .5 in § 2.2). F o r the zero es o f f 3 , 12 ( x, y 4 ) and f 3 , 15 ( x, y 4 ), w e could ha ve the relations, (3.1) y 4 y 5 = k 2 ( x ) k 1 ( x ) , y 5 = y 2 4 ( x − b 0 ) , y 4 = y 2 5 ( x − b 1 )( x − b 2 ) · Here for the primitive ro ot ζ 3 ( ζ 3 3 = 1 , ζ 3 6 = 1), ζ 3 acts o n X 3 and X 4 resp ectiv ely . The first relation is ch osen in the p ossibilities y 4 y 5 = ζ i 3 k 2 ( x ) k 1 ( x ) i = 0 , 1 , 2. As a normalization of these singular curves , w e ha v e the comm uta t ive r ing , R 2 ≡ R := C [ x, y 4 , y 5 ] / ( f 8 , f 9 , f 10 ) and X 2 := Sp ec R . Here w e define f 8 , f 9 , f 10 ∈ C [ x, y 4 , y 5 ] by f 8 = y 2 4 − y 5 k 1 ( x ) , f 9 = y 4 y 5 − k 2 ( x ) k 1 ( x ) , f 10 = y 2 5 − y 4 k 2 ( x ) 5 whic h are a lso regarded as the 2 × 2 minors of     k 2 ( x ) y 4 y 5 y 4 y 5 k 3 ( x )     . Here ζ 3 acts on X 2 b y ˆ ζ 3 ( x, y 4 , y 5 ) = ( x, ζ 3 y 4 , ζ 2 3 y 5 ). Let X b e the Riemann surface whic h is naturally obtained as an extension of X 2 as men tioned in Theorem 2.2, i.e., X = X 2 ∪ {∞} as a set. It is not ed that when x div erges, y 4 and y 5 also div erge vise v ersa. Thus the infinity p oin t ∞ uniquely exists in X . G m acts on R b y setting g − 3 m x , g − a m y a for x , y a , g m ∈ G m and a = 4 , 5. By Nagata’s Jacobi-metho d [16], it can b e pro ved that X is non- singular . Though they do not explicitly app ear, we may a lso implicitly consider parametrizations of y 4 and y 5 b y y 4 = w 2 w 2 1 , and y 5 = w 2 2 w 1 , where w 3 1 = k 1 and w 3 2 = k 2 . When w e consider ˜ R := C [ x, w 1 , w 2 ] / ( w 3 1 − k 1 ( x ) , w 3 2 − k 2 ( x )), it is related to a natura l co vering of X . 3.1. The W eierstrass gap and holomorphic one forms. The W eierstrass gap se- quences at ∞ are giv en in T a ble 1. F or the lo cal para meter t ∞ at ∞ , w e ha v e x = 1 t 3 ∞ , y 4 = 1 t 4 ∞ (1 + d ≥ ( t ∞ )) , y 5 = 1 t 5 ∞ (1 + d ≥ ( t ∞ )) . (3.2) Here for a give n lo cal parameter t at some P in X , the series o f t , whose orders o f zero at P are greater than ℓ or equal to ℓ , is denoted by d ≥ ( t ℓ ). H ( ∞ ) in (2.1) is H (3 , 4 , 5) as Pinkham considered (3 , 4 , 5) curv e as the simplest example of the n umerical semigroup H (3 , 4 , 5) [23, Sec.14]. Its monomial curv e is defined by , Z 2 4 = Z 3 Z 5 , Z 4 Z 5 = Z 5 3 , Z 2 5 = Z 3 3 Z 4 , or the 2 × 2 minor of     Z 3 Z 4 Z 5 Z 4 Z 5 Z 2 3     . Z 3 , Z 4 and Z 5 corresp ond to 1 x 1 y 4 and 1 y 5 resp ectiv ely and t hese relations corresp ond to ( 3 .1). T able 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 8 9 10 X 3 1 - - x y 4 - x 2 xy 4 y 2 4 x 3 x 2 y 4 xy 2 4 x 4 x 3 y 4 x 2 y 2 4 x 5 x 4 y 4 X 4 1 - - x - y 5 x 2 - xy 5 x 3 y 2 5 x 2 y 5 x 4 x 2 y 5 xy 2 5 x 5 x 2 y 2 5 X 2 1 - - x y 4 y 5 x 2 xy 4 xy 5 y 4 y 5 x 2 y 4 x 2 y 5 x 4 x 3 y 4 x 3 y 5 x 5 x 4 y 4 There w e define φ ( g ) i as a non-gap monomial in R g for g = 2 , 3 , 4 and e.g. , φ (2) 0 = 1, φ (2) 1 = x , φ (2) 2 = y 4 , φ (2) 3 = y 5 , φ (2) 4 = x 2 , · · · and φ (3) 0 = 1, φ (3) 1 = x , φ (3) 2 = y 4 , φ (3) 3 = x 2 , φ (3) 4 = xy 4 , · · · . W e in tro duce the w eigh t N ( g ) ( n ) b y letting N ( g ) ( n ) := − wt( φ ( g ) n ), where wt() is the degree o f divisor at ∞ of eac h curv e X ’s. It is no t ed that H 2 is iden tical to { N (2) ( n ) | n = 0 , 1 , 2 , . . . } . F or later con v enience, w e also introduce φ H 1 i ∈ R ( i = 1 , 2 , 3 , · · · ) b y φ H 1 0 := y 4 , φ H 1 1 := y 5 , φ H 1 2 := xy 4 , φ H 1 3 := xy 5 , f or i > 3, φ H 1 i :=    x ( i − 4) / 3 y 4 y 5 i ≡ 1 mo d 3 , x ( i +1) / 3 y 4 i ≡ 2 mo d 3 , x i/ 3 y 5 i ≡ 0 mo d 3 . 6 SHIGEKI MA TSUT ANI A ND JIR YO KOMED A W e also define the w eight N H 1 ( n ) b y N H 1 ( n ) := − wt ( φ H 1 n ); N H 1 (0) = 4, N H 1 (1) = 5, and N H 1 ( n ) = n + 5 for n ≥ 2. By letting Λ (2) i := N H 1 (2) − N H 1 ( i − 1) + i − 3 , Λ ( g ) i := N ( g ) ( g ) − N ( g ) ( i − 1) − g + i − 1 , ( g = 3 , 4) the related Y oung diagrams, Λ ≡ Λ (2) := (Λ 1 , Λ 2 ) = (1 , 1), Λ (3) := (Λ (3) 1 , Λ (3) 2 , Λ (3) 3 ) = (3 , 1 , 1) and Λ (4) := (Λ (4) 1 , Λ (4) 2 , Λ (4) 3 , Λ (4) 4 ) = ( 4 , 2 , 1 , 1) a re given b y resp ectiv ely: , , . The Y oung diagram Λ is not symmetric, whereas t Λ (3) = Λ (3) and t Λ (4) = Λ (4) . Then the follo wing prop ositions are ob vious: Prop osition 3.1. Bases of the holomorphic one forms over X ar e exp r esse d by ν I 1 = d x 3 y 5 and ν I 2 = d x 3 y 4 or ν I i := φ H 1 i − 1 d x 3 y 4 y 5 , ( i = 1 , 2) . W e note their divisors and linear equivlanece; for B a := ( b a , 0 , 0) ( a = 0 , 1 , 2), ( ν I 1 ) = ∞ + B 0 ∼ (d x/y 2 5 ) = 2(3 ∞ − B 1 − B 2 ) and ( ν I 1 ) ∼ ( ν I 2 ) = B 1 + B 2 ∼ (d x/y 2 4 ) = 2(2 ∞ − B 0 ) = 2( ∞ + ( ∞ − B 0 )). Prop osition 3.2. P n i =0 a i ˜ ν i b elongs to H 1 ( X \ ∞ , O X ) , wher e ˜ ν i := φ H 1 i d x 3 y 4 y 5 and the or der of the singularity of ( ˜ ν i ) at ∞ is given by N H 1 ( n ) − 5 . Lemma 3.3. a 0 d x y 4 y 5 + a 1 x d x y 4 y 5 + a 2 x 2 d x y 4 y 5 is not h o lomorphic one form over X if a i do es not vanish. Pr o of. F or n < 3, eve ry P n i =0 a i x i d x y 4 y 5 has singularities at p oints in X \ ∞ .  W e c ho o se the bases α i , β j (1 ≦ i, j ≦ 2 ) of H 1 ( X , Z ) suc h that their inters ection n umbers are α i · α j = β i · β j = 0 and α i · β j = δ ij , and w e denote the p erio d matrices b y [ ω ′ ω ′′ ] = 1 2  Z α i ν I j Z β i ν I j  i,j =1 , 2 . L et Π 2 b e a lattice generated b y ω ′ and ω ′′ . F or a p oin t P ∈ X , the a b elian map ˆ u o : X → C 2 is defined b y ˆ u o ( P ) = Z P ∞ ν I ∈ C 2 and for a p o int ( P 1 , · · · , P k ) ∈ S k X , i.e., the k -th symmetric pro duct o f X , the shifted ab elian map ˆ u : S k X → C 2 b y ˆ u ( P 1 , · · · , P k ) := ˆ u o ( P 1 , · · · , P k ) + ˆ u o ( B 0 ) 7 where ˆ u o ( P 1 , · · · , P k ) := P k i =1 ˆ u o ( P i ). Then w e define the Jacobian J 2 and its sub v ariet y W k ( k = 0 , 1 , 2 ) b y κ : C 2 → J 2 = C 2 / Π 2 = W 2 , W k := κ ˆ u ( S k X ) resp ectiv ely . F urther the singular lo cus of S 2 X is denoted by S 2 1 X as in [20 ]. F o r a p oint ( P 1 , P 2 ) ∈ S 2 X around the infinit y po in t, b y letting their lo cal pa r a meters t ∞ , 1 and t ∞ , 2 , u ≡ t ( u 1 , u 2 ) := ˆ u o ( P 1 , P 2 ) is given b y u 1 = 1 2 ( t 2 ∞ , 1 + t 2 ∞ , 2 )(1 + d > 0 ( t ∞ , 1 , t ∞ , 2 )) , u 2 = ( t ∞ , 1 + t ∞ , 2 )(1 + d > 0 ( t ∞ , 1 , t ∞ , 2 )) , where d ≥ ( t 1 , t 2 ) is a natural extension of d ≥ ( t ). 3.2. Differen tials of the second and the third kinds. F o llowing the EEL-construction [5] for a ( n, s ) c urv e, w e giv e an algebraic represen tation of a differen tia l form w hic h is equal to the fundamen tal normalized differen tial of t he second kind in [7, Corollary 2.6], up to a tensor of holomorphic o ne forms: Definition 3.4. A two-form Ω ( P 1 , P 2 ) on X × X is c al le d a fundamental differ ential of the se c ond kind if it is symmetric, Ω( P 1 , P 2 ) = Ω ( P 2 , P 1 ) , it h a s its only p ol e (of se c ond or der) along the diagonal of X × X , and in the vici n ity of e ach p oint ( P 1 , P 2 ) is ex p ande d in p ower series as (3.3) Ω ( P 1 , P 2 ) =  1 ( t P 1 − t ′ P 2 ) 2 + d ≥ (1)  d t P 1 ⊗ d t P 2 (as P 1 → P 2 ) wher e t P is a lo c al c o or dinate at a p oint P ∈ X . Here w e use the conv en tion that for P a ∈ X , P a is r epresen ted b y ( x a , y 4 ,a , y 5 ,a ) or ( x P a , y 4 ,P a , y 5 ,P a ) a nd for P ∈ X , P is expressed by ( x, y 4 , y 5 ). Then the follo wing prop o- sitions holds. Prop osition 3.5. B y letting Σ  P , Q  := y 4 ,P y 5 ,P + y 4 ,P y 5 ,Q + y 4 ,Q y 5 ,P ( x P − x Q )3 y 4 ,P y 5 ,P d x P Σ ( P , Q ) has the pr op e rties: 1) Σ ( P , Q ) as a function of P is singular at Q = ( x Q , y 4 ,Q , y 5 ,Q ) and ∞ , and vanishes at ˆ ζ ℓ 3 ( Q ) = ( x Q , ζ ℓ 3 y 4 ,Q , ζ 2 ℓ 3 y 5 ,Q ) , ( ℓ = 1 , 2) , and 2) Σ ( P , Q ) as a function of Q is singular at P and at ∞ . Pr o of. Direct computations lead the results.  Prop osition 3.6. Ther e exist differ entials ν I I j = ν I I j ( x, y 4 , y 5 ) ( j = 1 , 2) of the se c ond kind such that they have their only p ole at ∞ and satisfy the r elation, (3.4) d Q Σ  P , Q  − d P Σ  Q, P  = 2 X i =1  ν I i ( Q ) ⊗ ν I I i ( P ) − ν I i ( P ) ⊗ ν I I i ( Q )  wher e d Q Σ  P , Q  := d x P ⊗ d x Q ∂ ∂ x Q y 4 ,P y 5 ,P + y 4 ,P y 5 ,Q + y 4 ,Q y 5 ,P ( x P − x Q )3 y 4 ,P y 5 ,P . 8 SHIGEKI MA TSUT ANI A ND JIR YO KOMED A The differ entials { ν I I 1 , ν I I 2 } ar e determine d mo d ulo the C -l i n e ar sp ac e sp an ne d by h ν I j i j = 1 , 2 ; we fix  ν I I 1 , ν I I 2  =    −  2 x + λ (2) 1  d x 3 y 4 , − x d x 3 y 5    as their r epr esentative. Pr o of. ∂ ∂ x Q y 4 ,P y 5 ,P + y 4 ,P y 5 ,Q + y 4 ,Q y 5 ,P ( x P − x Q )3 y 4 ,P y 5 ,P d x P is equal to 1 ( x P − x Q )9 y 4 ,P y 5 ,P y 4 ,Q y 5 ,Q h 3( y 4 ,P y 5 ,P + y 4 ,P y 5 ,Q + y 4 ,Q y 5 ,P ) y 4 ,Q y 5 ,Q ( x P − x Q ) +  y 4 ,P y 4 ,Q y 5 ,Q (2 k 2 ,Q k ′ 2 ,Q k 1 ,Q + k 2 2 ,Q k ′ 1 ,Q ) + y 5 ,P y 5 ,Q y 4 ,Q (2 k 2 ,Q k 1 ,Q k ′ 1 ,Q + k ′ 2 ,Q k 2 1 ,Q )) i . Here k a,P = k a ( x P ) a nd k ′ a,P = d k a ( x P ) / d x P . W e hav e ∂ ∂ x Q y 4 ,P y 5 ,P + y 4 ,P y 5 ,Q + y 4 ,Q y 5 ,P ( x P − x Q )3 y 4 ,P y 5 ,P − ∂ ∂ x P y 4 ,Q y 5 ,Q + y 4 ,Q y 5 ,P + y 4 ,P y 5 ,Q ( x Q − x P )3 y 4 ,Q y 5 ,Q = 1 ( x P − x Q )9 y 4 ,P y 5 ,P y 4 ,Q y 5 ,Q ( B 2 ( P , Q ) − B 2 ( Q, P )) where B 2 ( P , Q ) = y 4 ,P y 5 ,Q  2 x Q + λ (2) 1 − x P  . Then w e obtain the statemen ts.  Corollary 3.7. 1) The one form, Π P 2 P 1 ( P ) := Σ ( P , P 1 ) − Σ ( P , P 2 ) , is a diff er ential of the thir d kind, whose only (firs t-or de r) p oles ar e P = P 1 and P = P 2 , and r es idues +1 and − 1 r esp e ctively. 2) Ω ( P 1 , P 2 ) is define d by d P 2 Σ ( P 1 , P 2 ) + 2 X i =1 ν I i ( P 1 ) ⊗ ν I I i ( P 2 ) Ω ( P 1 , P 2 ) = F ( P 1 , P 2 )d x 1 ⊗ d x 2 ( x P 1 − x P 2 ) 2 9 y 4 ,P 1 y 5 ,P 1 y 4 ,P 2 y 5 ,P 2 wher e F is an element of R ⊗ R . Pr o of. Direct computations giv e the claims.  Lemma 3.8. We have lim P 1 →∞ F ( P 1 , P 2 ) φ H 1 1 ( P 1 )( x P 1 − x P 2 ) 2 = φ H 1 2 ( P 2 ) = x P 2 y 4 ,P 2 . Pr o of. B 2 in the pro of of Prop osition 3.6 leads the r esult.  F o r later con venie nce w e in tro duce the quan tity , Ω P 1 ,P 2 Q 1 ,Q 2 := Z P 1 P 2 Z Q 1 Q 2 Ω ( P , Q ), (3.5) Ω P 1 ,P 2 Q 1 ,Q 2 = Z P 1 P 2 (Σ ( P , Q 1 ) − Σ ( P , Q 2 )) + 4 X i =1 Z P 1 P 2 ν I i ( P ) Z Q 1 Q 2 ν I I i ( P ) . 9 4. The sigma function for (3 , 4 , 5) c ur ve 4.1. Generalized Legendre relation. Corresp onding to the complete in tegral of the first kind, w e define the complete in tegral of the second kind, [ η ′ η ′′ ] := 1 2 " Z α i ν I I j Z β i ν I I j # i,j =1 , 2 . Let τ Q 1 ,Q 2 b e the nor malized differen tial of the third kind suc h that τ Q 1 ,Q 2 has residues +1 and − 1 at Q 1 and Q 2 resp ectiv ely , is regular ev erywhere else, and is normalized, R α i τ P ,Q = 0 for i = 1 , 2 [7, p.4]. The following Lemma corresp onding to Corollar y 2.6 (ii) in [7] holds: Lemma 4.1. By letting γ = ω ′ − 1 η ′ , we have Ω P 1 ,P 2 Q 1 ,Q 2 = Z P 1 P 2 τ Q 1 ,Q 2 + 2 X i,j =1 γ ij Z P 1 P 2 ν I i Z Q 1 Q 2 ν I j . Pr o of. The same as [20, I: Lemma 4.1].  The following Prop osition provide s a symplec tic structure in the Jacobian J 2 , kno wn as gene r alize d L e ge ndr e r elation [3, 4, 20]: Prop osition 4.2. M  − 1 1  t M = 2 π √ − 1  − 1 1  for M :=  2 ω ′ 2 ω ′′ 2 η ′ 2 η ′′  . Pr o of. The same as [20, I: Prop ositon 4.2].  4.2. The σ function. Due to the Riemann relations [7], Im ( ω ′ − 1 ω ′′ ) is p ositiv e definite. Theorem 1.1 in [7] giv es δ :=  δ ′′ δ ′  ∈  Z 2  4 b e the theta c haracteristic whic h is equal to the R iemann constan t ξ R and the p erio d matrix [ 2 ω ′ 2 ω ′′ ]. W e note that ξ R = ˆ u ( P R ) for a p oint P R ∈ X satisfying 2 P R + 2 B 0 − 4 ∞ ∼ 0. W e define an en tire function o f (a column-v ector) u = t ( u 1 , u 2 ) ∈ C 2 , σ ( u ) = c e − 1 2 t uη ′ ω ′ − 1 u X n ∈ Z 2 e  π √ − 1  t ( n + δ ′′ ) ω ′ − 1 ω ′′ ( n + δ ′′ )+ t ( n + δ ′′ )( ω ′ − 1 u + δ ′ )  where c is a certain constan t as in (4.1). F o r a giv en u ∈ C 2 , we in tro duce u ′ and u ′′ in R 2 so that u = 2 ω ′ u ′ + 2 ω ′′ u ′′ . Prop osition 4.3. F or u , v ∈ C 2 , and ℓ (= 2 ω ′ ℓ ′ + 2 ω ′′ ℓ ′′ ) ∈ Π 2 , by letting L ( u, v ) := 2 t u ( η ′ v ′ + η ′′ v ′′ ) , χ ( ℓ ) := exp[ π √ − 1  2( t ℓ ′′ δ ′ − t ℓ ′ δ ′′ ) + t ℓ ′ ℓ ′′  ] , we have a tr anslation a l r elation, σ ( u + ℓ ) = σ ( u ) exp( L ( u + 1 2 ℓ, ℓ )) χ ( ℓ ) . Pr o of. The same as [20, I: Prop.4.3].  10 SHIGEKI MA TSUT ANI A ND JIR YO KOMED A The v anishing lo cus of σ is simply giv en b y Θ 1 := ( W 1 ∪ [ − 1] W 1 ) = W 1 . 4.3. The Riemann fundamen tal relation. As in [2 0, I: Prop4.4], we ha ve the Riemann fundamen tal relation: Prop osition 4.4. F or ( P , Q, P i , P ′ i ) ∈ X 2 × ( S 2 ( X ) \ S 2 1 ( X )) × ( S 2 ( X ) \ S 2 1 ( X )) , exp 2 X i,j =1 Ω P ,Q P i ,P ′ j ! = σ ( ˆ u o ( P ) − ˆ u ( P 1 , P 2 )) σ ( ˆ u o ( Q ) − ˆ u ( P ′ 1 , P ′ 2 )) σ (( ˆ u o ( Q ) − ˆ u ( P 1 , P 2 )) σ ( ˆ u o ( P ) − ˆ u ( P ′ 1 , P ′ 2 )) . Using the differen tial iden tity , 2 X i,j =1 φ H 1 i − 1 ( P ′ 1 ) φ H 1 j − 1 ( P ′ 2 ) ∂ 2 ∂ ˆ u i ( P ′ 1 ) ∂ ˆ u j ( P ′ 2 ) = 9 y 4 ,P ′ 1 y 5 ,P ′ 1 y 4 ,P ′ 2 y 5 ,P ′ 2 ∂ 2 ∂ x ′ 1 ∂ x ′ 2 , taking logarithm of b oth sides of the relation and differen tiat- ing them along P ′ 1 = P and P ′ 2 = P a , w e ha ve the differen tial expressions of the relation, as men tioned in [20, I: Prop. 4.5]: Prop osition 4.5. F or ( P , P 1 , P 2 ) ∈ X × S 2 ( X ) \ S 2 1 ( X ) and u := ˆ u ( P 1 , P 2 ) , the e quality 2 X i,j =1 ℘ i,j ( ˆ u o ( P ) − u ) φ H 1 i − 1 ( P ) φ H 1 j − 1 ( P a ) = F ( P , P a ) ( x − x a ) 2 holds for every a = 1 , 2 , wher e we set ℘ ij ( u ) := − σ i ( u ) σ j ( u ) − σ ( u ) σ ij ( u ) σ ( u ) 2 ≡ − ∂ 2 ∂ u i ∂ u j log σ ( u ) . 4.4. Jacobi in version form ulae. As in [20], w e in tro duce meromorphic functions on the curv e X : Definition 4.6. F or P , P 1 , . . . , P n ∈ ( X \∞ ) × S S n ( X \∞ ) , ( n = 1 , 2) , we define µ 1 ( P ; P 1 ) := y 5 − y 5 , 1 y 4 , 1 y 4 , µ 2 ( P ; P 1 , P 2 ) := xy 4 − y 4 , 1 x 2 y 4 , 2 − y 4 , 2 x 1 y 4 , 1 y 4 , 1 y 4 , 2 − y 4 , 2 y 4 , 1 y 5 + y 5 , 1 x 2 y 4 , 2 − y 5 , 2 x 1 y 4 , 1 y 4 , 1 y 4 , 2 − y 4 , 2 y 4 , 1 y 4 . W e note that µ n for X is characterize d b y the condition on a p olynomial µ n = P n i =0 a i φ H 1 i ( P ), a i ∈ C and a n = 1, whic h has a zero at eac h po int P i and has the smallest p o ssible order suc h that it m ultiplied b y d x/ 3 y 4 y 5 b elongs to H 1 ( X \ ∞ , O X ). F o r giv en P 1 , the solution of µ 1 ( P ; P 1 ) = 0 corresp o nds to a p oint Q 1 = [ − 1] P 1 with B a ( a = 0 , 1 , 2) , a nd for giv en P 1 and P 2 , t he solution of µ 2 ( P ; P 1 , P 2 ) = 0 giv es tw o p oin ts Q 1 , Q 2 with B a ( a = 0 , 1 , 2) suc h that Q 1 + Q 2 = [ − 1]( P 1 + P 2 ). Here w e use B 0 + B 1 + B 2 − 3 ∞ ∼ 2 B 0 − 2 ∞ . Using µ n , w e hav e our main t heorem in this article: Theorem 4.7. 1) F or ( P , P 1 , P 2 ) ∈ X × ( S 2 ( X ) \ S 2 1 ( X )) , we have 11 1-1) µ 2 ( P ; P 1 , P 2 ) = xy 4 − ℘ 22 ( ˆ u ( P 1 , P 2 )) y 4 + ℘ 21 ( ˆ u ( P 1 , P 2 )) y 5 . 1-2) ℘ 22 ( ˆ u ( P 1 , P 2 )) = y 4 , 1 x 2 y 4 , 2 − y 4 , 2 x 1 y 4 , 1 y 4 , 1 y 4 , 2 − y 4 , 2 y 4 , 1 ℘ 21 ( ˆ u ( P 1 , P 2 )) = y 5 , 1 x 2 y 4 , 2 − y 5 , 2 x 1 y 4 , 1 y 4 , 1 y 4 , 2 − y 4 , 2 y 4 , 1 · 2) F or ( P , P 1 ) ∈ X × ( X \ S 1 1 ( X )) and u = ˆ u ( P 1 ) ∈ κ − 1 ( W 1 ) , µ 1 ( P ; P 1 ) = y 5 − σ 1 ( u ) σ 2 ( u ) y 4 , and σ 1 ( u ) σ 2 ( u ) = y 5 y 4 · Pr o of. 1) is the same as [20, I: Prop. 4.6]. As in [20, I: Theorem 5.1], by conside ring lim P 2 →∞ ℘ 21 ( ˆ u ( P 1 , P 2 )) ℘ 22 ( ˆ u ( P 1 , P 2 )) , w e hav e the second result.  F o llowing the statemen t b y Buchs tab er, Leykin and Enolskii, Nak ay ashiki sho wed that the leading of the sigma function for ( r, s ) curv e is express ed b y Sc h ur function [21]. Noting ( 3 .2) and degrees of u , the ab o ve Jacobi in v ersion f orm ula e giv es an extension that σ ( u ) = 1 2 u 2 2 − u 1 + X | α | > 2 a α u α where a α ∈ Q [ b 1 , · · · , b 5 ], α = ( α 1 , α 2 ), | α | = α 1 + α 2 and u α = u α 1 1 u α 2 2 . The prefactor c is determined b y this relatio n. Since for a Y oung diagram Λ, S Λ and s Λ are the Sc hur functions define d b y (4.1) S Λ ( T 1 , T 2 ) = t 1 t 2 = 1 2 T 2 1 − T 2 where T 1 := t 1 + t 2 and T 2 := 1 2 ( t 2 1 + t 2 2 ), we ha ve σ ( u ) = S Λ ( u 1 , u 2 )+ X | α | > 2 a α u α . Remark 4.8. W e sho w ed that the EEL construction w orks w ell eve n for a space curv e, and the sigma function asso ciated with t he curve is naturally defined. Since this construc- tion is very natural, this study sheds a new ligh t on the w a y to construction of the sigma functions for space curv es. W e conjectured that the EEL construction could b e applied to ev ery space curv e if it is W eierstrass. As an in teresting example of a space c urv e, w e will g iv e a commen t on a problem as follo ws, fo r whic h w e started to study sigma functions for affine space curv es. McKa y considers a relation b etw een disp ersionless KP hierarc h y and the replicable functions in order to obtain a further pro found in terpretation of the moonshine phenomena of Monster group [22]. He conjectured that it migh t b e related to the quantis ed elastica [18, 1 9]. By studying a relation b etw een a replicable function and an algebraic curv e asso ciated with elastica, Matsutani fo und tha t a semigroup H 12 generated b y M 12 := h 6 , 13 , 14 , 15 , 1 6 i has gap sequenc e, L ( H 12 ) = { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 11 , 1 7 , 2 3 } , whic h is 12 SHIGEKI MA TSUT ANI A ND JIR YO KOMED A iden tical to the Norton n um b er, N 12 := { 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 17 , 19 , 23 } by exc hanging 10 and 19. The Norton num b er play s t he essen tial role in the mo onshine phenomena for the Monster group [22]. The replicable f unction is giv en as an elemen t of Q [ a 1 , a 2 , a 3 , a 4 , a 5 , a 7 , a 8 , a 9 , a 11 , a 17 , a 19 , a 23 ][[ t ]]. The replicable function is a generalization of the elliptic J -function, whic h causes the mo onshine phenomena of the Monster group. After then, Komeda prov ed that H 12 is the W eierstrass se migroup and ga ve t he fun- damen tal relations Prop o sitions A.2 as men tioned in App endix, whic h is rep orted more precisely in [13]. Then w e applied the EEL-construction to the curv e and obtain a sigma function for a Ja cobi v ariet y J 12 for C ( H 12 ) [13]. Since the Jacobi v ariety J 12 is giv en as 12-dimensional complex torus whose real dimension is 24, it might remind us of Witten conjecture asso ciated with Monster group problem [8]; Witten conjectured that a 24 di- mensional manifold exists suc h that the Monster group acts on it via W eierstrass sigma function. A. App endix: Weierstrass pr oper ties of (6 , 1 3 , 14 , 15 , 16) by Jir yo K omeda The pro ofs of these pr o p ositions are giv en in the article [13] in detail. W e show only the sk etc h of the first one b ecause the second one is not difficult. Prop osition A.1. The numeric al semigr oup h 6 , 13 , 1 4 , 15 , 16 i is Weierstr ass. Pr o of. Let ( C , P ) b e a p ointed curv e with H ( P ) = h 3 , 7 , 8 i . Then 2 = h 0 (4 P ) = 4 + 1 − 4 + h 0 ( K − 4 P ) = 1 + h 0 ( K − 4 P ) whic h implies that K − 4 P ∼ P 1 + P 2 for some p oints P 1 and P 2 ∈ C . Here K is a canonical divisor on C . Moreo ver, 2 = h 0 (5 P ) = 5 + 1 − 4 + h 0 ( K − 5 P ) = 2 + h 0 ( K − 5 P ) whic h implies that h 0 ( K − 5 P ) = 0 . Hence, w e get P i 6 = P for i = 1 , 2. Th us, K ∼ 4 P + P 1 + P 2 with P i 6 = P for i = 1 , 2. W e set D = 7 P − P 1 − P 2 . Then deg(2 D − P ) = 9 = 2 × 4 + 1, whic h implies that the complete linear system | 2 D − P | is v ery ample, hence base-p oin t free. There fore, 2 D ∼ P + Q 1 + . . . + Q 9 (= a reduced divisor). Let L b e the in vertible sheaf O C ( − D ) on C and φ a n isomorphism L ⊗ 2 ≈ O C ( − P − Q 1 −· · ·− Q 9 ) ⊂ O C . Then the v ector bundle O C ⊕ L has an O C -algebra structure through φ . The canonical morphism π : ˜ C = Sp ec ( O C ⊕ L ) → C , is a double co v ering. Its branch lo cus of π is { P , Q 1 , ..., Q 9 } . Let ˜ P b e the ramification p o in t of π ov er P . Then it can b e show ed that H ( ˜ P ) = h 6 , 13 , 14 , 1 5 , 16 i using the form ula, h 0 (2 n ˜ P ) = h 0 ( nP ) + h 0 ( nP − D ) for an y non-negativ e integer n . By considering h 0 (2 n ˜ P ) for n = 3 , 4 , 5 , 6 , 7 , 8 , 9, w e show H ( ˜ P ) = h 6 , 13 , 14 , 1 5 , 16 i .  Prop osition A.2. L et B 12 a mon o mial ring which is given by k [ t a ] a ∈ M 12 for the nume ric al semigr oup H 12 . F or a k -alg e b r a hom o morphism, ϕ 12 : k [ Z ] := k [ Z 6 , Z 13 , Z 14 , Z 15 , Z 16 ] → k [ t a ] a ∈ M 12 13 wher e Z a is the weight of a = 6 , 13 , 14 , 15 , 16 , the kern el of ϕ 12 is gener ate d by the fol l o wing r elations f ( Z ) 12 ,b ( b = 1 , · · · , 9) , f ( Z ) 12 , 1 = Z 2 13 − Z 2 6 Z 14 , f ( Z ) 12 , 2 = Z 13 Z 14 − Z 2 6 Z 15 , f ( Z ) 12 , 3 = Z 2 14 − Z 13 Z 15 , f ( Z ) 12 , 4 = Z 2 14 − Z 2 6 Z 16 , f ( Z ) 12 , 5 = Z 13 Z 16 − Z 14 Z 15 , f ( Z ) 12 , 6 = Z 2 15 − Z 5 6 , f ( Z ) 12 , 7 = Z 14 Z 16 − Z 5 6 , f ( Z ) 12 , 8 = Z 15 Z 16 − Z 3 6 Z 13 , f ( Z ) 12 , 9 = Z 2 16 − Z 3 6 Z 14 . A cknowledgements One of the authors (S.M.) t ha nks John McKa y for p osing my attention to the Norton problem and his encouragemen t. This w ork started in a seminar at Y okohama National univ ersit y 2008, and w as stim ulated b y the in ternational conferenc e at HWK 2011. 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Shigeki Mat sutani: 8-21-1 Higa shi-Link a n Minami-ku, Sagamihara 2 5 2-0311, JAP AN. e-mail: rxb01142@nifty .com Jiry o Komeda Departmen t of Mathematics Cen ter for Basic Education and In tegrat ed Learning Kanagaw a Institute of T ec hnology A tsugi, 243- 0 292 JAP AN. e-mail: k omeda@g en.k anagaw a-it.ac.jp