THE ALGORITHM TO CALCULATE THE PERIOD MATRIX OF THE CURVE x m þ y n ¼ 1

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1 TSUKUA J MATH Vol 26 No (22), 5 37 THE ALGORITHM TO ALULATE THE PERIOD MATRIX OF THE URVE x m þ y n ¼ y Abstract We show how to take a canonical homology basis and a basis of the space of holomorphic -forms on the curve x m þ y n ¼, and we show how to calculate its explicit period matrix Introduction Let R be a compact Riemann surface and g be the genus of R Let fo j g, ð j ¼ ; 2; ; gþ be a basis of the space of holomorphic -forms of R and fa k ; b k g, ðk ¼ ; 2; ; gþ be a basis of the first homology group of R over Z, the ring of integers Suppose that the intersection numbers of the closed paths a k s and b k s satisfy Iða j ; b k Þ¼ Iðb k ; a j Þ¼d jk ; Iða j ; a k Þ¼Iðb j ; b k Þ¼ Such a basis fa k ; b k g, ðk ¼ ; 2; ; gþ is called a canonical homology basis In other words, a canonical homology basis is a set of closed paths whose intersection matrix is J ¼ O I ; I O where O and I are the g g zero and identity matrices We denote by W the period matrix (or Riemann matrix) of R with respect to fo j g and fa k ; b k g, that is, Ð Ð Ð Ða g o W ¼ Ð a o b o b g o Ð Ð Ða g o g a o g b o g b g o g A Received April 3, 2 Revised October 3, 2

2 6 We write this matrix as ða; Þ by two g g matrices Multiplying W by A, we obtain the normalized period matrix ði; TÞ ¼ðI; A Þ The matrix T, called the theta matrix of R with respect to fo j g and fa k ; b k g, is non-singular symmetric and has positive definite imaginary part Let L be the lattice of g generated by the column vectors of ði; TÞ Then the torus JðRÞ ¼ g =L is an algebraic variety called the Jacobian variety of R One of the advantages of computing T lies in the theorem of Torelli which states that given Riemann surfaces R and R 2 are isomorphic if and only if their Jacobian varieties JðR Þ and JðR 2 Þ are isomorphic In general it is not so easy to give explicitly a canonical homology basis on R Recently the combinatorial group theoretic method for general curves is presented by L Tretko and M D Tretko [2] Unfortunately the method needs to draw complicated figures and requires some knowledge of graph theory If R is a hyperelliptic curve, a general method to compute its normalized period matrix is well-known (cf [], pp ) If R is Fermat curve, only its period lattice can be known by David E Rohrlich ([5], p 79) ut the representation is not so simple, and the method can not present the intersection matrix of the homology basis on R In the usual definition of the theta function attached to R, the period matrix is normalized Our method can present not only the period lattice but also the intersection matrix and, hence, the normalized period matrix Our method is an extension of the method for the case of hyperelliptic curves Let R be the curve y n ¼ x m, ðn; m A NÞ, and g ðbþ be the genus of R If we regard R as a n-ply branched covering of ^ ¼ U fyg and apply the Riemann-Hurwitz formula, we have g ¼fðn Þðm Þþ dg=2; where d is the GD of m and n Let ^R be the projective imbedding of R into P 2, the complex projective plane That is, ^R ¼f½X; Y; ZŠ A P 2 j Y n ¼ Z n X m Z n m g The points at infinity ðx ¼ yþ of R correspond to the set E ¼ f½x; Y; Š A P 2 g V ^R We classify R into the following three cases (i) n ¼ m (E ¼f½; r j ; Š A P 2 j r ¼ expðpi=nþ; j ¼ ; 2; ; ng), (ii) n b m þ and d ¼ (E ¼f½; ; Š A P 2 g The point ½; ; Š is a branch point of order n), (iii) n > m þ and d > (E ¼f½; ; Š A P 2 g The point ½; ; Š is a branch point of order n=d) I would like to thank professor Kimio Watanabe for his instruction and helpful advice, and the referee for his careful reading and detailed suggestions

3 The algorithm to calculate the period matrix 7 2 Topological Model and Homology asis We regard R as a Riemann surface of n-ply branched covering of the Riemann sphere ^ Let X ; X 2 ; ; X n be these ^ s h j ð j ¼ ; 2; ; mþ are the branch points of R, where h is one of the primitive m-th roots of unity Since the ramification indices of these branch points are equal to n, there is no ambiguity to choose the branch points on X ; ; X n Therefore any round trip path on R between two of these branch points is a closed path and belongs to H ðr; ZÞ, the first homology group There exist dim H ðr; ZÞ ¼2g closed paths which are linearly independent over Z It is clear that any closed path on R is homologous to the union of the round trip paths between two branch points on R We cut open the lines between h j and h jþ, ð j ¼ ; 2; ; m Þ on X ; X 2 ; ; X n Now we consider the minimal loop that is the union of the branch cuts ½h j ; h jþ Š k from h j to h jþ on X k and ½h jþ ; h j Š kþ from h jþ to h j on X kþ We denote such loops by a j; k ¼½h j ; h jþ Š k þ½h jþ ; h j Š kþ for j ¼ ; 2; ; m and k ¼ ; 2; ; n, where X nþ ¼ X ase (i) In this case, all closed paths on R are generated by the a j; k s Since the cut lines ½h; h 2 Š ; ½h 2 ; h 3 Š ; ; ½h m ; Š belong to X, each of the other bank of these branch cuts belongs to another X k We denote the path of the opposite bank of the path ½h j ; h jþ Š k by ½h j ; h jþ Š k, X k A fx ; ; X n g Here X k is selected by the analytic continuation of y, n-ply valued function of x A ^ Since the union of the loops a ; ; a ; 2 ; ; a ; n is homologous to zero on R, any one of them is not required to obtain a homology basis So we discard the loop a ; n ecause of the same reason we discard the loops a 2; n ; a 3; n ; ; a m ; n Now consider the following path on R l ½h; h 2 Š þ½h 2 ; h 3 Š þþ½h m ; Š þ½; h m Š þþ½h 3 ; h 2 Š þ½h 2 ; hš The path l is obviously closed and homologous to zero on X Since l is a sum of the loops ½h j ; h jþ Š þ½h jþ ; h j Š ; j ¼ ; 2; ; m ; in order to get independent loops, at least one of them has to be discarded According to the same argument for X 2 ; X 3 ; ; X n, we discard the loops ½h m ; Š k þ½; h m Š k ; k ¼ ; 2; ; n

4 8 In other words, by the rearrangement of the branch cuts, we discard the minimal loops a m ; k ¼½h m ; Š k þ½; h m Š kþ ; k ¼ ; 2; ; n Now the remaining minimal loops are a j; k, ð j ¼ ; 2; ; m 2 and k ¼ ; 2; ; n Þ The number of them is ðn Þðm 2Þ Since ðn Þ ðm 2Þ ¼2g ¼ dim H ðr; ZÞ, these loops must be independent over Z Therefore they form a basis of H ðr; ZÞ ase (ii) If once we add the point at x ¼ y to the set of the branch points, we can obtain a homology basis in the same way as in the case (i) From the minimal loops a j; k ¼½h j ; h jþ Š k þ½h jþ ; h j Š kþ ; a m; k ¼½; yš k þ½y; Š kþ we discard the loops j ¼ ; 2; ; m and k ¼ ; 2; ; n; where X nþ ¼ X ; a ; n ; a 2; n ; ; a m; n ; a m; ; a m; 2 ; ; a m; n to eliminate the linear dependence over Z Then the number of the remaining minimal loops is mn fmþðn Þg ¼ ðn Þðm Þ ¼2g So the remaining minimal loops a j; k, ð j ¼ ; 2; ; m and k ¼ ; 2; ; n Þ form a basis of H ðr; ZÞ ase (iii) As the ramification index of x ¼ y is n=d, there are d points at infinity; hence, the minimal loops a m; k, ðk ¼ ; 2; ; nþ are not always closed Since the number of the loops corresponding to the remaining minimal loops in the case (ii) is ðn Þðm Þ > 2g ¼ðn Þðm Þþ d, they are linearly dependent over Z To remove the dependent loops, we have to construct a topological model of R by gluing X ; ; X n together To do so we investigate how the branch cuts join one another Since each of the branch cuts ½h j ; h jþ Š k and ½; yš k belongs to only one side of X k, the paths of the opposite bank to X k do not belong to X k These branch cuts are joined with the corresponding branch cut by the analytic continuation of the branch of y Let ~X k, ðk ¼ ; 2; ; nþ be X k fh; h 2 ; ; h m ; ; yg cut open all the branch cuts ½h j ; h jþ Š k and ½; yš k Let z be one of the primitive n-th roots of unity and f be a branch of y on ~X Since the interchange of the branches of y does not occur by the analytic continuation along any closed path on ~X k, we can assume that the branches z k f, ðk ¼ ; 2; ; nþ are assigned to ~X k by the analytic continuation In the first place, we consider the analytic continuation of f along a path

5 The algorithm to calculate the period matrix 9 which starts from any point on the branch cut ½h; h 2 Š, encircles the branch point h counterclockwise, ends at the point on ½h; h 2 Š opposite to the starting point Then the branch turns from f to z f So the end point of the path should be on ½h; h 2 Š 2 and the branch cut ½h; h 2 Š is identified with the branch cut ½h; h 2 Š 2 Similarly we see the branch cuts ½h; h 2 Š 2 ; ; ½h; h 2 Š n are identified with the branch cuts ½h; h 2 Š 3 ; ; ½h; h 2 Š, respectively In the next place, we take a path which starts from any point on ½h 2 ; h 3 Š, encircles the branch points h 2 and h counterclockwise, ends at the point on ½h 2 ; h 3 Š opposite to the starting point Since this path encircles two branch points, the branch turns from f to z 2 f by the analytic continuation along this path Therefore the branch cut ½h 2 ; h 3 Š is identified with the branch cut ½h 2 ; h 3 Š 3 Similarly we see the branch cuts ½h 2 ; h 3 Š 2 ; ; ½h 2 ; h 3 Š n are identified with the branch cuts ½h 2 ; h 3 Š 4 ; ; ½h 2 ; h 3 Š 2, respectively Successively using the same process, we see that the branch cuts ½h j ; h jþ Š k and ½; yš k on each X k are glued by ½h j ; h jþ Š k ¼½h j ; h jþ Š hkþji ; ½; yš k ¼½; yš hkþmi for j ¼ ; 2; ; m and k ¼ ; 2; ; n, where hai denotes the element of f; 2; ; ng congruent to a modulo n X k ; X hkþmi ; ; X hkþðn=d Þmi are glued together by the identification ½; yš hkþami ¼½; yš hkþðaþþmi, for k ¼ ; 2; ; n and a ¼ ; ; ; n=d Since any two of the numbers ; 2; ; d are not congruent each other modulo m, the set of all Y k ¼ X k U X hkþmi U U X hkþðn=d Þmi ; k ¼ ; 2; d glued by the above identification is a topological model of R Since the sum of the loops ½h j ; h jþ Š hþami þ½h jþ ; h j Š hþami ; j ¼ ; 2; ; m and a ¼ ; ; ; n=d is homologous to zero on Y, we discard one of these loops Here we discard the loop ½h; h 2 Š þ½h 2 ; hš ¼½h; h 2 Š þ½h 2 ; hš 2 ¼ a ; y the same reason for Y ; ; Y d the loops a ; 2 ; a ; 3 ; ; a ; d are discarded We can not discard a ; d since Y d contains X n and the loop a ; n is already discarded Now the number of the remaining minimal loops is ðn Þðm Þ ðd Þ ¼2g Therefore the remaining minimal loops a ; d ; a ; dþ ; ; a ; n ; a j; k, ð j ¼ 2; 3; ; m and k ¼ ; 2; ; n Þ form a basis of H ðr; ZÞ

6 2 3 Holomorphic -Forms We shall find the condition that the di erential -form o ¼ x a y b nþ dx, ða; b A ZÞ is holomorphic For this purpose we consider the representation of o by a local coordinate at each point on ^R (cf [6], pp 89 92) If x ¼ a A is not a branch point, then t ¼ x a is a local coordinate in a neighbourhood of a Ifa, o is obviously holomorphic in this neighbourhood If a ¼, o is holomorphic for a b p In a neighbourhood of the branch point x ¼ h j, t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi n x h j is a local coordinate Then o ¼ uðtþt b dt, where uðtþ is a non-vanishing holomorphic function on this neighbourhood Hence o is holomorphic for b b in this neighbourhood To describe o in a neighbourhood of x ¼ y, we consider o in the a ne part ^R V f½x; Y; ZŠ A P 2 j X g We rewrite the defining equation and o w n ¼ z n z n m ; o ¼ x a y b nþ dx ¼ z a bþn 3 w b nþ dz; where z ¼ Z=X ¼ =x; w ¼ Y=X ¼ y=x ase (i) The points at infinity x ¼ y on R correspond to the points ðz; wþ ¼ð; r j Þ, r ¼ expðpi=nþ, j ¼ ; 2; ; n We choose a local coordinate p t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi n w r j vanishing at ðz; wþ ¼ð; r j Þ Then o ¼ t a bþn 3 uðtþ dt, where uðtþ is a non-vanishing holomorphic function on a neighbourhood of t ¼ Hence o is holomorphic everywhere on ^R if and only if a; b b and n 3 b a þ b ecause the number of the pairs of integers ða; bþ which satisfy these inequalities is X n 3 j¼ ðn 2 jþ ¼ðn Þðn 2Þ=2 ¼ g; the di erentials x a y b nþ dx with a; b b, n 3 b a þ b form a basis of the vector space of holomorphic -forms ase (ii) The unique point at infinity x ¼ y on R corresponds to the point ðz; wþ ¼ð; Þ We choose a local coordinate t ¼ np ffiffi z vanishing at ðz; wþ ¼ð; Þ Then o ¼ t an bm m nþmn uðtþ dt, with uðtþ as above Hence o is holomorphic everywhere on ^R if and only if ðþ a; b b ; an bm m n þ mn b These inequalities are equivalent to being m ðm þ Þ=n b a þ bm=n b a b and n fða þ Þn þ g=m b b b Since < ðm þ Þ=n a, a a a m 2

7 The algorithm to calculate the period matrix 2 Let N be the number of the pairs of integers ða; bþ that satisfy () Then we have 2N ¼ X m 2 ½n fðj þ Þn þ g=mšþx m 2 ½n fðj þ Þn þ g=mš j¼ j¼ ¼ X m 2 ½n fðj þ Þn þ g=mšþx m 2 ½n fðm j Þn þ g=mš; j¼ j¼ where the symbol ½Šis the Gauss symbol Now suppose ð j þ Þn ¼ q j m þ r j, q j ; r j A Z, a r j a m Then ½n fðj þ Þn þ g=mš ¼½n q j ðr j þ Þ=mŠ ¼n q j ; Therefore ½n fðm jþn þ g=mš ¼½q j þðr j Þ=mŠ ¼q j n N ¼ X m 2 ðn q j¼ j Þþ X m 2 j¼ q j o =2 ¼ðn Þðm Þ=2 ¼ g Hence the di erentials x a y b nþ dx with a; b satisfying () form a basis of the vector space of holomorphic -forms ase (iii) The unique point at infinity x ¼ y on R corresponds to the point ðz; wþ ¼ð; Þ Then o is equal to t ð an bmþmn m n dþ=d uðtþ dt and holomorphic everywhere on ^R if and only if a; b b ; an bm þ mn m n d b We rewrite this condition as follows ð2þ a; b b ; an bm þ m n d m n b ; where n ¼ n=d; m ¼ m=d We denote by N the number of pairs ða; bþ that satisfy (2) If m ¼ ðd ¼ mþ, then (2) is equivalent to the condition a a a d 2 and a b a n d n 2 an So N ¼ X d 2 j¼ ðn d n jn Þ ¼fðn d n Þðd Þ n ðd 2Þðd Þ=2g ¼ðm Þðn 2Þ ¼ g

8 22 If m, then the conditions are a a a m 2 ¼ m d 2 and a b a n fða þ Þn þ dg=m ¼ n d fða þ Þn þ g=m So we have 2N ¼ 2 X dm 2 ½n d fðj þ Þn þ g=m Š j¼ ¼ 2 X d k¼ X ðkþþm 2 j¼km ½n d fðj þ Þn þ g=m Š þ 2 X d j¼ ½n d fðjm Þn þ g=m Š ¼ X d k¼ X ðkþþm 2 j¼km f½n d fðj þ Þn þ g=m Š þ½n d fð2km þ m j Þn þ g=m Šg þ 2 X d j¼ ðn d jn Þ Suppose ð j þ Þn ¼ q j m þ r j, q j ; r j A Z, a r j a m Then ½n d fðj þ Þn þ g=m Š¼½n d q j ðr j þ Þ=m Š¼n d q j ; ½n d fð2km þ m j Þn þ g=m Š¼½n d 2kn n þ q j þðr j Þ=m Š Therefore 2N ¼ X d k¼ X ðkþþm 2 j¼km ¼ n d 2kn n þ q j ð2n d 2kn n Þþ2 X d j¼ ðn d jn Þ ¼ðm Þfð2n d n Þd 2n ðd Þd=2g þ 2fðn d Þðd Þ n ðd Þd=2g ¼ nm m n þ 2 d ¼ðn Þðm Þþ d ¼ 2g ðso N ¼ gþ Hence the di erentials x a y b nþ dx with a; b satisfying (2) form a basis of the vector space of holomorphic -forms 4 Period Lattice In this section we compute the period lattice relative to the di erential -forms and the homology basis obtained in the previous Sections

9 The algorithm to calculate the period matrix 23 Let z; h be one of the primitive n-th, m-th roots of unity and f be a branch of y on ~X Then z k f, a branch of y, is assigned by the analytic continuation The integral along a j; k of the di erential -form o l ¼ x a l y bl nþ dx is ð ð h jþ ð h j o l ¼ a j; k h j x a l ðz k f Þ bl nþ dx þ x a l ðz k f Þ bl nþ dx h jþ ¼ h jða lþþ z ðk Þðb l nþþ ð h ¼ h jða lþþ z kðb l nþþ ðz ðb l nþþ Þ x a l f b l nþ dx þ h jða lþþ z kðb l nþþ ð h x a l f b l nþ dx From this equality, we find the relations ð ð ð o l ¼ z b l nþ a j; kþ o l ¼ z kðb l nþþ a j; k o l ; a j; ð o l ¼ h a lþ a jþ; k ð o l ¼ h jða lþþ a j; k ð o l a ; k ð h x a l f b l nþ dx We compute the period lattice for the case (i) and (ii) at the same time Let P be the matrix whose ðl; kþ-component is Ð a ; k o l ðk ¼ ; 2; ; n ; l ¼ ; 2; ; gþ Then we have Ð Ð P ¼ Ð s ¼ O a ; o a ; n o Ð a ; o g a ; n o g A O ðz b nþ Þ ðz b nþ Þ n 2 A A ; s g ðz bg nþ Þ ðz bg nþ Þ n 2 ð where s l ¼ o l a ; Similarly the matrix P 2 whose ðl; kþ-component is Ð a 2; k o l ðk ¼ ; 2; ; n ; l ¼ ; 2; ; gþ is given as follows s O h ða þþ O ðz b nþ Þ ðz b nþ Þ n 2 P 2 ¼ A A A O s g O h ða gþþ ðz bg nþ Þ ðz bg nþ Þ n 2 In the same way we can also obtain P 3 ; P 4 ; ; P m 2 P 3 ; P 4 ; ; P m for the case (ii) Letting for the case (i),

10 24 and s S ¼ O O A; A ¼ s g h ða þþ O O h ða gþþ ðz b nþ Þ ðz b nþ Þ n 2 ¼ A ; ðz bg nþ Þ ðz bg nþ Þ n 2 A we consider a g 2g matrix P ¼ðP ; P 2 ; ; P m 2 Þ¼Sð; A; ; A m 3 Þ P ¼ðP ; P 2 ; ; P m Þ¼Sð; A; ; A m 2 Þ for the case ðiþ; for the case ðiiþ The set of the 2g column vectors of P is obviously a basis of the period lattice relative to the holomorphic -forms selected in Section 3 For the case (iii) we define S as above Then the period lattice P relative to the holomorphic -forms selected in Section 3 is obtained by removing the first ðd Þ columns from the period lattice P of the case (ii) In any case (i), (ii), or (iii), we change the basis of the space of the holomorphic -forms by multiplying S f ^o ; ^o 2 ; ; ^o g g¼fo ; o 2 ; ; o g gs ¼fs o ; s 2 o 2; ; s g o gg Thus we can obtain the following theorem (cf [5], p 79) Theorem Let S; P, and f ^o ; ^o 2 ; ; ^o g g be as above Let P ¼ S P The period lattice of the curve defined by x m þ y n ¼ relative to the basis of the space of the holomorphic -forms f ^o ; ^o 2 ; ; ^o g g is spanned by the 2g column vectors of P Note that each of the components of P is just a monomial of the primitive root of unity 5 Intersection Matrix In this section, we show how to obtain the intersection matrix ase (i) In order to read the intersection numbers between a j; k s, we deform a j; k, ð j ¼ ; 2; ; m 2; k ¼ ; 2; ; n Þ around the branch points like the Appendix If the minimal loops are chosen such a way, we have Iða j; k ; a j; kþ Þ¼; Iða j; k ; a jþ; k Þ¼; Iða j; kþ ; a jþ; k Þ¼; Iða j; k ; a jþ; kþ Þ¼

11 The algorithm to calculate the period matrix 25 learly the other intersection numbers are Iða j; k ; a j ; k Þ¼ for j j j j b 2 or jk k j b 2 The intersection matrix L of the minimal loops a j; ; a j; 2 ; ; a j; n, namely, the ðn Þðn Þ matrix whose ðs; tþ-component is Iða j; s ; a j; t Þ, is given by L ¼ A The matrix L 2 of the intersection numbers of a j; ; a j; 2 ; ; a j; n and a jþ; ; a jþ; 2 ; ; a jþ; n, namely, the ðn Þðn Þ matrix whose ðs; tþ-component is Iða j; s ; a jþ; t Þ, is given by L 2 ¼ A Then the intersection matrix of the minimal loops a j; k, ð j ¼ ; 2; ; m 2; k ¼ ; 2; ; n Þ can be represented by L L 2 O O t L 2 O K ¼ ; O L 2 A O O t L 2 L where O is the ðn Þðn Þ zero matrix This is the ðm 2Þðm 2Þ matrix of the ðn Þðn Þ minor matrices ase (ii) The intersection matrix of the same representation is obtained in the same way, and the matrix is the ðm Þðm Þ matrix of the ðn Þðn Þ minor matrices

12 26 ase (iii) The intersection matrix is obtained by removing the first ðd Þ rows and columns from the intersection matrix of the case (ii) To get a canonical homology basis we transform the intersection matrix into J For the case (i), for example, this transformation is equivalent to finding a 2g 2g unimodular matrix M which satisfies ða ; ; a g ; b ; ; b g Þ¼ða ; ; ; a ; n ; a 2; ; ; a 2; n ; ; a m 2; n ÞM; where fa j ; b j g is a canonical homology basis It is also equivalent to finding a 2g 2g unimodular matrix M which satisfies J ¼ t MKM Finally, we obtain a normalized matrix from PM as we mentioned in Introduction It is guaranteed by the theorem of Frobenius that we can transform the intersection matrix K into J within finitely many operations (cf [3] p 65, [4] p 56) We explain the process of finding M in the next Section 6 Examples () The curve defined by y 4 ¼ x 4 This curve is of genus 3 The pairs of integers ða; bþ which satisfy a; b b, b a þ b b areð; Þ; ð; Þ; ð; Þ Therefore the set of o ¼ y 3 dx; o 2 ¼ y 2 dx; o 3 ¼ xy 3 dx is a basis of the space of holomorphic -forms on Let h be one of the primitive fourth roots of unity The matrices A, and P are h h 3 ðh 3 Þ 2 h h 2 A ¼ h A; ¼ h 2 ðh 2 Þ 2 A ¼ h 2 A; h 2 h 3 ðh 3 Þ 2 h h 2 h h 2 h h 2 h 3 P ¼ð; AÞ ¼ h 2 h h 3 h A h h 2 h 2 h 3 The intersection matrix of the minimal loops a i; j ði ¼ ; 2; j ¼ ; 2; 3Þ is L L 2 K ¼ t ¼ L 2 L A

13 The algorithm to calculate the period matrix 27 Now we explain the method to find a unimodular matrix M which transform K into J The i-th column of the intersection matrix is called c i ði ¼ ; ; 6Þ in each step Step hoose a column whose the first element is non-zero, for instance, c 2 Subtract c 2 from c 4 and add c 2 to c 5 so that the first element of each column except c 2 vanishes (ie c 4! c 4 c 2, c 5! c 5 þ c 2 ) This operation substitutes a 2; a ; 2 for a 2; ; a 2; 2 þ a ; 2 for a 2; 2 The matrix that represents the change of the minimal loops is M ¼ A Then the intersection matrix changes into K ¼ t M KM ¼ A Step 2 hoose a column whose the second element is non-zero and the first element is zero, for instance, c Add c to c 3 ; c 5 and subtract c from c 6 so that the second element of each column except c vanishes (ie c 3! c 3 þ c, c 5! c 5 þ c, c 6! c 6 c ) This operation substitutes a ; 3 þ a ; for a ; 3 ; a 2; 2 þ a ; for a 2; 2 and a 2; 3 a ; for a 2; 3 We see the matrix that represents the change of the minimal loops is M 2 ¼ A

14 28 Then the intersection matrix changes into K 2 ¼ t M 2 K M 2 ¼ A Step 3 hoose a column whose the third element is non-zero and the first and the second elements are zero, for instance, c 4 Add c 4 to c 5 and subtract c 4 from c 6 so that the third element of each column except c 4 vanishes (ie c 5! c 5 þ c 4, c 6! c 6 c 4 ) This operation substitutes a 2; 2 þ a 2; for a 2; 2 ; a 2; 3 a 2; for a 2; 3 We see the matrix that represents the change of the minimal loops is M 3 ¼ A The intersection matrix changes into K 3 ¼ t M 3 K 2 M 3 ¼ A Step 4 hoose a column whose the fourth element is non-zero and the elements from the first to the third are zero, for instance, c 3 Add c 3 to c 6 so that the fourth element of each column except c 3 vanishes (ie c 6! c 6 þ c 3 ) This operation substitutes a 2; 3 þ a ; 3 for a 2; 3 We see the matrix that represents the change of the minimal loops is M 4 ¼ A

15 The algorithm to calculate the period matrix 29 The intersection matrix changes into K 4 ¼ t M 4 K 3 M 4 ¼ A Here each column of K 4 has only one nonzero element Letting g ; ; g 6 be the loops corresponding to the columns c ; ; c 6, we have the intersection numbers of these loops Therefore we arrange Iðg ; g 2 Þ¼; Iðg 3 ; g 4 Þ¼; Iðg 5 ; g 6 Þ¼ ðg ; g 3 ; g 5 ; g 2 ; g 4 ; g 6 Þ¼ðg ; g 2 ; g 3 ; g 4 ; g 5 ; g 6 ÞM 5 ; where M 5 ¼ A Through out these steps, we have a canonical homology basis ða ; a 2 ; a 3 ; b ; b 2 ; b 3 Þ¼ða ; ; a ; 2 ; a ; 3 ; a 2; ; a 2; 2 ; a 2; 3 ÞM where M ¼ M M 2 M 3 M 4 M 5 ¼ A If we multiply P by M, then we have þ h 2 þ h þ h 2 h h 2 þ h 3 PM ¼ 2 þ h þ h 3 h 2 h h 2 h 2 þ A þ h 2 þ h 2 þ h 3 h h 2 h hþ

16 3 Finally using the relation h 2 ¼, we obtain a normalized period matrix h ðh þ Þ=2 W ¼ ðh þ Þ=2 ð2h þ Þ=2 ð hþ=2 A ð hþ=2 h The theta matrix T of W is certainly non-singular (Det T ¼ð 2h þ Þ=2 ), symmetric, and has positive definite imaginary part (because of h ¼ i) (2) The curve 2 defined by y 5 ¼ x 3 This curve is of genus 4 The pairs of integers ða; bþ which satisfy a; b b, 6 b 5a þ 3b b are ð; Þ; ð; Þ; ð; 2Þ; ð; Þ Therefore the set of o ¼ y 4 dx; o 2 ¼ y 3 dx; o 3 ¼ y 2 dx; o 4 ¼ xy 4 dx is a basis of the space of the holomorphic -forms on 2 Let h; z be one of the primitive cubic, fifth roots of unity The matrices A; and P are h z 4 ðz 4 Þ 2 ðz 4 Þ 3 z z 2 z 3 h A ¼ h A ; ¼ z 3 ðz 3 Þ 2 ðz 3 Þ 3 z 2 z 4 z ¼ z 2 ðz 2 Þ 2 ðz 2 Þ 3 A z 3 z z 4 A ; h 2 z 4 ðz 4 Þ 2 ðz 4 Þ 3 z z 2 z 3 z z 2 z 3 h hz hz 2 hz 3 z 2 z 4 z h hz 2 hz 4 hz P ¼ð; AÞ ¼ z 3 z z 4 h hz 3 hz hz 4 A z z 2 z 3 h 2 h 2 z h 2 z 2 h 2 z 3 Let s be one of the primitive fifteenth roots of unity Then the components of P can be expressed by the monomials of s s 3 s 6 s 9 s 5 s 8 s s 4 s 6 s 2 s 3 s 5 s s 2 s 8 P ¼ s 9 s 3 s 2 s 5 s 4 s 8 s 2 A s 3 s 6 s 9 s s 3 s s 4 The intersection matrix of the minimal loops a i; j ði ¼ ; 2; j ¼ ; 2; 3; 4Þ is

17 L L 2 K ¼ t ¼ L 2 L A If we execute the steps as in the first example, we can obtain two unimodular matrices M ¼ A and The algorithm to calculate the period matrix 3 M ¼ A which satisfy t M t M KM M ¼ t M M ¼ J A

18 32 Then the matrix M ¼ M M ¼ A transforms K into J and gives a canonical homology basis PM ða ; a 2 ; a 3 ; a 4 ; b ; b 2 ; b 3 ; b 4 Þ¼ða ; ; a ; 2 ; a ; 3 ; a ; 4 ; a 2; ; a 2; 2 ; a 2; 3 ; a 2; 4 ÞM We have þs 6 s 3 þs 5 s 9 þs 3 þs 8 þs 9 s 3 s 9 s 9 þs s 6 þs þs 4 þs 2 s 3 þs 5 s 6 þs 3 þs 6 þs s 6 s 3 s 3 þs 2 s 2 þs 8 þs 2 ¼ þs 3 s 5 s 9 s 2 þs 9 þs 2 þs 4 s 9 s 2 þs 8 s 2 s 2 þs 3 þs 8 A þs 6 s 3 s 9 þs þs 3 þs 9 þs 3 s 3 s 9 þs s 9 sþs 4 þs 6 Using the relation s 8 ¼ s 7 s 5 þ s 4 s 3 þ s, we can represent each component of the theta matrix T by the quotient of the polynomials of degree at most seven The result is T ¼ 2s 7 s 5 3s 4 s 3 s 2 2s ðt T 2 T 3 T 4 Þ; where 5s 7 s 6 þ 3s 5 2s 4 þ 4 4s 7 2s 5 þ 3s 4 þ s 2 T ¼ s 5 þ s þ A ; 2s 7 s 5 þ s 4 þ s 4s 7 2s 5 þ 3s 4 þ s 2 4s 7 þ s 6 þ 2s 5 3s 4 þ 2s 3 s þ 4 T 2 ¼ 2s 7 s 5 þ 2s 4 A ; s 7 2s 4 2s

19 The algorithm to calculate the period matrix 33 s 5 þ s þ 2s 7 s 5 þ 2s 4 T 3 ¼ 5s 7 þ s 6 þ 3s 5 3s 4 þ 2s 3 s 2 þ 4 A ; s 7 þ 2s 5 þ s 4 þ 2s þ 2 2s 7 s 5 þ s 4 þ s s 7 2s 4 2s T 4 ¼ s 7 þ 2s 5 þ s 4 þ 2s þ 2 A 2s 7 þ 2s 5 þ s 4 þ 2s 3 s 2 þ 2s þ 4 ertainly, T is non-singular (Det T ¼ðs 7 2s 4 þ s 3 s 2 2Þ=ð2s 7 þ s 5 s 4 þ 2s 3 Þ ), symmetric with positive definite imaginary part (because of p s ¼ cosð2p=5þ þi sinð2p=5þ ¼ð2 þ 2 ffiffiffi pffiffiffiffiffiffiffi pffiffi pffiffiffiffiffiffiffiffiffi p 5 6ð5 þ 5 Þþ 3ð5 þ ffiffi 5 ÞÞ=6 þ p ð2 ffiffiffi pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 þ 2 5 þ 2ð5 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 5 Þ ð5 þ 5 Þ Þi=6Þ (3) The curve 3 defined by y 6 ¼ x 3 This curve is of genus 4 The pairs of integers ða; bþ which satisfy a; b b, 2 b 2a þ b b are ð; Þ; ð; Þ; ð; 2Þ; ð; Þ Therefore we choose o ¼ y 4 dx; o 2 ¼ y 3 dx; o 3 ¼ y 2 dx; o 4 ¼ xy 4 dx for a basis of the space of holomorphic -forms on 3 Let h; z be one of the primitive cubic, sixth roots of unity The matrices A; and P are h z 5 ðz 5 Þ 2 ðz 5 Þ 3 ðz 5 Þ 4 h A ¼ h A ; ¼ z 4 ðz 4 Þ 2 ðz 4 Þ 3 ðz 4 Þ 4 z 3 ðz 3 Þ 2 ðz 3 Þ 3 ðz 3 Þ 4 A h 2 z 5 ðz 5 Þ 2 ðz 5 Þ 3 ðz 5 Þ 4 z z 2 z 3 z 4 z 2 z 4 z 2 ¼ z 3 z 3 A ; z z 2 z 3 z 4 z 2 z 3 z 4 h hz hz 2 hz 3 hz 4 z 4 z 2 h hz 2 hz 4 h hz 2 P ¼ z 3 h hz 3 h hz 3 h A z 2 z 3 z 4 h 2 h 2 z h 2 z 2 h 2 z 3 h 2 z 4 We simplify the representation of P by using the relations h ¼ z 2 z 3 ¼ and

20 34 z 2 z z 2 z z 2 z z 2 z 2 z z 2 z P ¼ z 2 z 2 z 2 z 2 z 2 A z 2 z z z 2 z z 2 The intersection matrix of the minimal loops a i; j ði ¼ ; 2; j ¼ ; 2; 3; 4; 5Þ is A Therefore the intersection matrix of the minimal loops a ; 3 ; a ; 4 ; a ; 5 ; a 2; ; a 2; 2 ; a 2; 3 ; a 2; 4 ; a 2; 5 is K ¼ A If we execute the steps as in the first example, we obtain two unimodular matrices M ¼ A

21 The algorithm to calculate the period matrix 35 and M ¼ A which satisfy t M t M KM M ¼ t M M ¼ J A Then the matrix M ¼ M M ¼ A transforms K into J and gives a canonical homology basis ða ; a 2 ; a 3 ; a 4 ; b ; b 2 ; b 3 ; b 4 Þ¼ða ; 3 ; a ; 4 ; a ; 5 ; a 2; ; a 2; 2 ; a 2; 3 ; a 2; 4 ; a 2; 5 ÞM Then we have

22 36 z 2 z 2 z z 2 z 2 z þ z 2 z z 2 z z 2 z z 2 þ 2z 2 z z 2 z z PM ¼ 2 z 2 þ z 2 þ 2z 2 z 2 z 2 A z 2 z 2 z z þ z 2 2 z z 2 2 þ z 2 z z z þ z z z z z z ¼ 2 z z 2z z þ z þ A z z z 2 z z þ z 3 z 6 4 2z z T ¼ð5z 2Þ 4 2z 4 2z 6z þ z 2z 4 3 2z A 6z þ 3 2z 3z 4 We used the relation z 2 z þ ¼ for simplification The theta matrix T is certainly non-singular (Det T ¼ 2ð48z 6Þ= ð85z 46Þ ), symmetric with positive definite imaginary part (because of p z ¼ðþ ffiffiffi 3 iþ=2) References [ ] H F aker, Multiply Periodic Functions, ambridge University Press (97) [ 2 ] L Tretko, M D Tretko, ombinatorial group theory, Riemann surfaces and Di erential equations, ontemporary Mathematics, Vol 33 (984), pp [ 3 ] L Siegel, Topics in omplex Function Theory (volume 3), Wiley-Interscience, 973 [ 4 ] A I Markushevich, Introduction to the lassical Theory of Abelian Functions, Translations of Mathematical Monographs Vol 96, American Mathematical Society (992) [ 5 ] S Lang, Introduction to Algebraic and Abelian Functions, Graduate Texts in Mathematics 89, Springer-Verlag (982) [ 6 ] T Higuchi, Y Tashiro, S Yamazaki and K Watanabe, Function theory on Riemann surface (in Japanese), Morikita Publications Inc (997) Institute of Mathematics, Tsukuba University Tsukuba, Ibaraki , Japan address kamatamathtsukubaacjp

23 Appendix The algorithm to calculate the period matrix 37