Analytical and number-theoretical properties of the two-dimensional sigma function

This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\bf u};\lambda)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})$. It is an analogue of the Weierstrass sigma function $\sigma(u;g_2,g_3)$ of a family of elliptic curves. Logarithmic derivatives of order 2 and higher of the function ${\sigma({\bf u};\lambda)}$ generate fields of hyperelliptic functions of ${\bf u} = (u_1,u_3)$ on the Jacobians of curves with a fixed parameter vector $\lambda$. We consider three Hurwitz series $\sigma({\bf u};\lambda)=\sum_{m,n\ge 0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\bf u};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ and $\sigma({\bf u};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. The survey is devoted to the number-theoretic properties of the functions $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ and $\mu_k(u_3;\lambda)$. It includes the latest results, which proofs use the fundamental fact that the function ${\sigma ({\bf u};\lambda)}$ is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.


Introduction
Deep results on the number-theoretic properties of fields of hyperelliptic functions were obtained in the papers of V.P. Platonov, where he gave answers to long-standing questions. The fields of meromorphic functions on the Jacobian of curves of genus 2 occupy one of the main places in these papers (see [40], [41] and [42]). Abelian functions, including meromorphic functions on the Jacobians of algebraic curves, were a central topic of the 19th century mathematics. In this review, we mainly discuss the results obtained due to a new direction in the study of fields of Abelian functions. This direction arose in the mid-seventies of the last century in response to the discovery that Abelian functions provide a solution to a number of challenging problems of modern theoretical and mathematical physics. The elliptic sigma function, which was defined and investigated by Weierstrass, is important in many fields in mathematics and physics. This function is closely related to the theory of the elliptic curves. In [28] and [29], F. Klein posed the problem of the construction of multi-dimensional sigma functions associated with the hyperelliptic curves. He obtained important results in this direction. Many years later, F. Klein wrote a paper and a survey in which he acknowledged that the theory of his sigma functions is still far from complete (see [30] and [31]). The theory of the hyperelliptic sigma functions was developed by H. F. Baker in [5], [6], [7], and [8]. Recently, by a series of work of V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, the theory of the hyperelliptic sigma functions was developed significantly and they were generalized to the large family of algebraic curves called (n, s) curves, which include the hyperelliptic curves as special cases (see [12], [13], [14], [15], [17]). After the publications of Buchstaber, Enolskii, and Leykin, many papers appeared on the theory and applications of multi-dimensional sigma functions. Our survey is devoted to the sigma functions of curves of genus 2. The focus of our attention is the number-theoretic aspects of the results on these functions. Throughout the present survey, we denote the sets of positive integers, integers, rational and complex numbers by N, Z, Q, and C, respectively.
Let V be a hyperelliptic curve of genus g defined by The sigma function σ(u; λ), where u = (u 1 , u 3 , . . . , u 2g−1 ) and λ = (λ 4 , . . . , λ 4g+2 ), associated with V , is an entire function in u ∈ C g . It is shown that the coefficients of the power series expansion of σ(u) around u = 0 are polynomials of the coefficients λ 4 , . . . , λ 4g+2 over the rationals ( [14], [15], [17], [33]). Let R be an integral domain with characteristic 0, u 1 , u 3 , . . . , u 2g−1 be indeterminates, and R u 1 , u 3 . . . , u 2g−1 =    i 1 ,i 3 ,...,i 2g−1 ≥0 a i 1 ,i 3 ,...,i 2g−1 u i 1 1 u i 3 3 · · · u i 2g−1 2g−1 If a power series belongs to R u 1 , u 3 . . . , u 2g−1 , then it is said to be Hurwitz integral over R. In [39], Y.Ônishi proved that the power series expansion of σ(u) around u = 0 is Hurwitz integral over the ring Z[λ 4 , . . . , λ 4g+2 ] by using the expression of the sigma function in terms of the tau function of KP-hierarchy given in [34]. In [37], in the case of g = 1, the Hurwitz integrality of the sigma function is proved in a different way from [39] and relationships with number theory are discussed. In [21], in the case of g = 1, it is conjectured that the power series expansion of the sigma function is Hurwitz integral over Z[2λ 4 , 24λ 6 ]. The focus of our survey is on the above fundamental fact, i.e., the power series expansion of the sigma function around the origin is Hurwitz integral over Z[λ 4 , . . . , λ 4g+2 ]. In this survey, we will discuss in detail expansions of the sigma functions of curves of genus 1 and 2, including theÔnishi's proof for Hurwitz integrality (see Sections 2.2 and 2.3).
Weierstrass [45] showed that the elliptic sigma function σ(u; λ 4 , λ 6 ) satisfies the fol-lowing system of equations 4λ 4 σ λ 4 + 6λ 6 σ λ 6 − uσ u + σ = 0, The second equation of this system is the heat equation or, equivalently, the Schrödinger equation of type ℓ 2 σ = 1 2 H 2 σ, where ℓ 2 = 6λ 6 ∂ ∂λ 4 − 4 3 λ 2 4 ∂ ∂λ 6 and H 2 = ∂ 2 ∂u 2 − 1 3 λ 4 u 2 . Weierstrass gave recurrence relations of the coefficients of series expansion of the elliptic sigma function. Buchstaber and Leykin succeeded in generalizing the theory of the heat equations to the sigma functions of higher genus curves ( [18], [19], and [20]). In [24], the detailed proof of their theory is given. In [19] and [24], the recurrence relations of the coefficients of series expansion of the two-dimensional sigma function are given based on the heat equations. In [25], the theory of the heat equations is constructed for the elliptic curves defined by the most general Weierstrass equation. In [23], for g = 1, 2, it is shown that the holomorphic solution of the heat equations around (u 0 , 0) ∈ C 3g for some u 0 ∈ C g is the sigma function up to a multiplicative constant. We consider the case of g = 2. For λ = (λ 4 , λ 6 , λ 8 , λ 10 ), we set In Section 3, we will derive the differential equations satisfied by ξ k and µ k from the heat equations. From these results, we will prove that two-dimensional sigma function is Hurwitz integral over Z[λ 4 , λ 6 , λ 8 , 2λ 10 ] (Corollary 3.2).
For (x, y) ∈ V , let which are a basis of the vector space of holomorphic one forms on V . We have two ultra-elliptic integrals P ∞ du 1 and P ∞ du 3 obtained with the help of two holomorphic differentials du 1 and du 3 . We take a point P * ∈ V and an open neighborhood U * of this point P * such that U * is homeomorphic to an open disk in C. Let us fix a path γ * on the curve V from ∞ to the point P * . We consider the holomorphic mappings defined by the ultra-elliptic integrals where as the path of integration we choose the composition of the fixed path γ * from ∞ to the point P * and some path in the neighborhood U * from P * to the point P . When we consider the map I 3 , we assume P * = ∞. When we consider the map I 1 , we assume P * = (0, ± √ λ 10 ). If U * is sufficiently small, then the maps I 1 and I 3 are biholomorphisms. In [4], the inversion problems of the maps I 1 and I 3 are considered. In Section 4, we will summarize the results of [4]. Proposition 4.8 in the present survey gives the recurrence formula of the coefficients of series expansion of the solution of the inversion problem with respect to I 1 in the case of P * = ∞. This result is not included in [4].
These relations play an important role in describing the coefficient ring of the universal formal group (see [32], [16]) and in the algebraic-topological applications of the formal group in the theory of complex cobordisms (see [36], [43]). The polynomials B n = B n (α 1 , . . . ,α n ) ∈Â which generating series is given by the Hurwitz exponential series over the ringÂ are called universal Bernoulli numbers. For example, . The classic Bernoulli numbers are obtained by substituting α n = (−1) n . Substitutinĝ Classical Bernoulli numbers entered into algebraic geometry and algebraic topology due to the fact that the generating series (2) defines the Hirzebruch genus, which associates to any smooth complex manifold an integer equal to the Todd genus of this manifold (see [26]). The generating series (5) of universal Bernoulli numbers defines the universal Todd genus, which associates to any smooth complex manifold an integer polynomial (see details in [9]). In [22], F. Clarke generalized the von Staudt-Clausen theorem for the classical Bernoulli numbers to the universal Bernoulli numbers. The Kummer's congruence for the classical Bernoulli numbers was generalized to the universal Bernoulli numbers ( [1], [2], [3], [38]).
For a hyperelliptic curve of genus g defined by equation (1), in a neighborhood of point (∞, ∞), we can choose a local coordinate u such that the functions x(u) and y(u) can be expanded around u = 0 as Then C n and D n are called generalized Bernoulli-Hurwitz numbers. In [38], the von Staudt-Clausen theorem and the Kummer's congruence for the classical Bernoulli numbers are extended to the generalized Bernoulli-Hurwitz numbers in the case of the curves y 2 = x 2g+1 − 1 and y 2 = x 2g+1 − x. We will extend the methods of [38] to the curve y 2 = x 5 + λ 4 x 3 + λ 6 x 2 + λ 8 x + λ 10 and show some number-theoretical properties for the generalized Bernoulli-Hurwitz numbers associated with this curve (Theorem 5.2). These results will give the precise information on the series expansion of the solution of the inversion problem of the ultra-elliptic integrals.

The sigma function
For a positive integer g, we set and B = C 2g \ ∆. We consider the non-singular hyperelliptic curve of genus g where (λ 4 , λ 6 , . . . , λ 4g+2 ) ∈ B. In this paragraph we recall the definition of the sigmafunction for the curve V (see [15]) and give facts about it which will be used later on.

Hurwitz integrality of the expansion of the elliptic sigma function
In [39], Hurwitz integrality of the expansion of the sigma functions is proved. In this subsection, we will explain the proof of [39] for g = 1.
In this subsection, we assume g = 1. For simplicity, we denote u 1 and du 1 by u and du, respectively. For an integral domain R with characteristic 0 and a variable u, let For n < 0 let p n (u) = 0 and for n ≥ 0 let p n (u) = u n n! .
For an arbitrary partition µ = (µ 1 , µ 2 , . . . , µ l ), we define the Schur polynomial s µ (u) by Then t is a local parameter of V around ∞. We have Denote by Z ≥r the set of integers that are not less than r.
Lemma 2.2. We have the following expansion of s in terms of t around t = 0 where β i is a homogeneous polynomial in Z[λ 4 , λ 6 ] of degree i. In particular, we have Proof. By substituting (10) into y 2 = x 3 + λ 4 x + λ 6 , we have The expansion of s with respect to t around t = 0 takes the following form where β i ∈ C. By substituting the above expansion into (11), we have By comparing the coefficients, we obtain β 0 = 1, β 4 = λ 4 , β 6 = λ 6 , and β n = 0 for n = 1, 2, 3, 5. For n ≥ 6, we have Therefore we obtain the statement of the lemma. (10), we have the expansions

From Lemma 2.2 and
where a i is a homogeneous polynomial in Z[λ 4 , λ 6 ] of degree i. We enumerate the monomials x m y n , where m is a non-negative integer and n = 0, 1, according as the order of a pole at ∞ and denote them by ϕ j , j ≥ 1. In particular we have ϕ 1 = 1. We set e i = t i+1 . We expand tϕ j around ∞ with respect to t. Let . For a partition µ = (µ 1 , µ 2 , . . . , ), we define where m i = µ i −i and the infinite determinant is well defined. Then we have ξ µ ∈ Z[λ 4 , λ 6 ]. We define the tau function τ (u) by where the sum is over all partitions. From Lemma 2.1, we obtain the following proposition.
Lemma 2.3. The expansion of du around ∞ with respect to t takes the form Proof. From (12), we have We take the algebraic bilinear form where where q ij ∈ C and t 1 , t 2 are copies of the local parameter t.
Proof. From (13) and (14), we have By substituting the expansions of x 1 , x 2 , y 1 , y 2 into the above equation and multiplying the both sides of this equation by t 7 1 t 7 By comparing the coefficient of t 6 2 in the above equation, we obtain q 11 = 0. We define c i by the following relation Lemma 2.5. We have c 1 = 0.
Proof. From Lemma 2.3, we have the following expansion On the other hand, we have Thus we have c 1 = 0.
From Proposition 2.2 and (15), we obtain the following theorem.

Hurwitz integrality of the expansion of the two-dimensional sigma function
In this subsection, we will explain the proof of [39] for Hurwitz integrality of the expansion of the sigma function for g = 2.
From Lemma 2.7, we have the expansions i are homogeneous polynomials in Z[λ 4 , λ 6 , λ 8 , λ 10 ] of degree i. We enumerate the monomials x m y n , where m is a non-negative integer and n = 0, 1, according as the order of a pole at ∞ and denote them by ϕ j , j ≥ 1. In particular we have ϕ 1 = 1.
We set e i = t i+1 . We expand t 2 ϕ j around ∞ with respect to t. Let For a partition µ = (µ 1 , µ 2 , . . . , ), we define where the sum is over all partitions. From Lemma 2.6, we obtain the following proposition.
The expansions of du i around ∞ with respect to t take the following form Proof. We have Therefore we obtain b 11 = 1 and b 13 = 0. We have Therefore we obtain b 31 = 0 and b 33 = 1.
We define c i by the following relation Proof. We have the following expansion Therefore we have the following expansion On the other hand, we have Thus we have c 1 = c 2 = c 3 = 0.
We take the algebraic bilinear form where q ij ∈ C and t 1 , t 2 are copies of the local parameter t.
Proof. We have ε 1 = 1 ∈ Z. Assume ε n ∈ Z. Then we have By mathematical induction, we obtain the statement.
From Lemma 2.11, for any k, ℓ ∈ Z ≥0 , we have Therefore we obtain the statement.

Universal Bernoulli numbers
In this subsection, we will describe the definition of the universal Bernoulli numbers and their properties according to [22,38].
Let f 1 , f 2 , . . . be infinitely many indeterminates. We consider the power series f n z n+1 n + 1 and its formal inverse series namely, the series such that u(z(u)) = u.
For a rational number α, we denote by ⌊α⌋ the largest integer which does not exceed α. If p is a prime and the p-part of given rational number r is p e , then we write e = ord p r. If τ is a polynomial (possibly in several variables) with rational coefficients, then we denote by ord p τ the least number of ord p r for all the coefficients r of τ . For a prime number p and an integer a, let a| p = a/p ordpa . For positive integers a, b and a prime number p, we have For a positive integer a and a prime number p, the following formula is well known If positive integers a and b are relatively prime, we denote it by a ⊥ b.
Proof. For the sake to be complete and self-contained, we give a proof of this lemma. By using (22) and (23), we have By the assumption of the lemma, there exists a positive integer i such that i ≥ 3 and 1), (2, 1), (1, 2)}. By the assumption of the lemma, we have U εp k −1 = 0 for any (ε, k) ∈ T p . For any integers ε ≥ 1 and k ≥ 1 such that (ε, k) / ∈ T p and ε ⊥ p, we can check Therefore we have Lemma 2.14. Let U = (U 1 , U 2 , . . . , ) be an element of S such that U i = 0 for any odd integer i, U 2 = 0, and d(U) = 0. Then we have Proof. We can prove this lemma in the same way as [38] Proposition 3.11. For the sake to be complete and self-contained, we give a proof of this lemma. By using (22), (23), and ord 2 (Λ U ) = 0, we have Since j ≥ 4, we have 2j − (j + 1) ≥ 3. Therefore we have In [22], F. Clarke showed the following resutls, which are a generalization of the von Staudt-Clausen theorem for the classical Bernoulli numbers to the universal Bernoulli numbers (cf. the paper of Onishi [38]). These results were used in the proof of our Theorem 5.2 in the present survey.

Differential equations for the coefficients of the expansions of the two-dimensional sigma function
In this section, we assume g = 2.
Proposition 3.2. The function ξ 0 satisfies the following differential equation Proof. From (25) with k = 0, we have (28) From (26) with k = 0, ξ ′ 1 can be expressed in terms of ξ 0 and its derivatives. We take the derivative with respect to u 1 of (28) and substitute into it the expression for ξ ′ 1 . As a result, we obtain We substitute the above equation into (28) and finally obtain the statement of the proposition.
We set Lemma 3.2. If ℓ is even, then p ℓ = 0. We have p 1 = 0 and p 3 ∈ C.
Proof. From Lemma 3.1 and (24) with k = 0, we obtain the statement of the lemma. Proposition 3.3. For ℓ ≥ 2, the following recurrence relation holds : where p i,λ 2j denotes the derivative of p i with respect to λ 2j .
Proof. By substituting (29) into the differential equation in Proposition 3.2 and comparing the coefficients of u ℓ 1 /ℓ!, we obtain the statement of the proposition.

Expansions of ξ k
In this subsection we will calculate the expansions of ξ k . From (9), we have p 3 = 2. The initial terms of ξ i , i = 0, 1, 2, 3, 4, are as follows.

The coefficients of u 1
We set Proposition 3.4. For k ≥ 0, the functions µ 0 , µ 1 , . . . satisfy the following hierarchy of systems where the prime denotes the derivation with respect to u 3 and µ k,λ 2j denotes the derivation of µ k with respect to λ 2j .
Proof. In the same way as Lemma 3.1, we obtain the statement of the lemma. Proof. By substituting (34) for k = 0 into (33) for k = 0, we obtain the statement of the proposition.
Proof. From Lemma 3.3 and (31), we obtain q ℓ = 0 for any non-negative even integer ℓ and q 1 ∈ C. Further, we find that the coefficient of u 0 3 in µ 1 is equal to 0. By comparing the coefficient of u 3 in the equation (34) for k = 0, we obtain q 3 = λ 6 q 1 . Proposition 3.6. For ℓ ≥ 2, the following recurrence relation holds : where q i,λ 2j denotes the derivative of q i with respect to λ 2j .
Proof. By substituting (35) into the differential equation in Proposition 3.5 and comparing the coefficients of u ℓ 3 /ℓ!, we obtain the statement of the proposition. Proof. From Lemma 3.4 and Proposition 3.6, we find that all the coefficients q ℓ are determined from q 1 . Note that Lemma 3.4 and Proposition 3.6 follow from (31), (33), and (34). By (34), all the functions µ k are determined from µ 0 . As mentioned in the proof of Proposition 3.4, (31), (33), (34) follow from Q 0 σ = 0, Q 4 σ = 0, Q 6 σ = 0. Therefore we obtain the statement of this corollary.
Remark 3.2. In [11], the following expression is proved : From this result, the statement of Corollary 3.3 can be also proved.

Expansions of µ k
In this subsection we will calculate the expansions of µ k . From (9), we have q 1 = −1.

The ultra-elliptic integrals
In [4], the inversion problem of the ultra-elliptic integrals is considered. In this section, we will summarize the main results in [4]. Proposition 4.8 is not described in [4].
In this section, we assume g = 2. Let us take a point P * ∈ V and an open neighborhood U * of this point that is homeomorphic to an open disk in C. We fix a path γ * on the curve V from ∞ to P * . Let us consider the holomorphic mappings where as the path of integration we choose the composition of the path γ * from ∞ to the point P * and any path in the neighborhood U * from P * to the point P . We consider the meromorphic function on We assume P * = ∞. If we take the open neighborhood U * sufficiently small, then I 3 is injective. Let ϕ(u) be the implicit function defined by σ(ϕ(u), u) = 0 around (I 1 (P * ), I 3 (P * )). We define the function F (u) = f (ϕ(u), u).  The function F (u) satisfies the following ordinary differential equations: From Proposition 4.1 and Theorem 4.1, one can obtain the series expansion of F (u). Since the function F (u) is holomorphic in a neighborhood of u * = I 3 (P * ), the expansion in the neighborhood of this point has the form We set degp 2 = 2 and degp 5 = 5.
By comparing the coefficient of u n for n ≥ 12, we obtain the statement of the proposition.
where g(u) is a holomorphic function that in a neighborhood of the point u = 0 is given by a series Here the coefficientτ n+2 is a homogeneous polynomial in Q[λ 4 , λ 6 , λ 8 , λ 10 ] of degree n + 2 ifτ n+2 = 0.
Denote by G d (u) the formal Laurent series obtained from G(u) by substitution λ 8 = λ 10 = 0 in the series expansion of this function in a neighborhood of the point u = 0.

Number-theoretical properties of the generalized
Bernoulli-Hurwitz numbers for the curve of genus 2 In this section, we assume g = 2. Let V be a hyperelliptic curve of genus 2 defined by We take an open neighborhood U * of ∞ such that U * is homeomorphic to an open disk in C. We consider the map where as the path of integration we take any path in U * from ∞ to P . For P = (x, y) ∈ U * , let u = I 1 (P ). If U * is sufficiently small, then the map I 1 is biholomorphism. Therefore, we can regard x and y as functions of u. From Proposition 4.4, we have x(u) = G(u) and y(u) = −G(u)G ′ (u)/2. From Proposition 4.7, the function x(u) can be expanded around u = 0 as where the coefficient C n is a homogeneous polynomial in Q[λ 4 , λ 6 , λ 8 , λ 10 ] of degree n if C n = 0.
where (x(u) 2 ) ′ denotes the derivative of x(u) 2 with respect to u.
Proof. From Proposition 4.4, we obtain the statement of the lemma.
From (44) and Lemma 5.1, we find that y(u) can be expanded around u = 0 as where the coefficient D n is a homogeneous polynomial in Q[λ 4 , λ 6 , λ 8 , λ 10 ] of degree n if D n = 0. Then C n and D n are called generalized Bernoulli-Hurwitz numbers. In particular, we find that C n = D n = 0 for any odd integer n. In [38], the hyperelliptic curve of genus 2 defined by y 2 = x 5 − 1 is considered and the following formulae are proved.
In this section, we will generalize the method of [38] to the curve V defined by (43) and derive some number-theoretical properties of the generalized Bernoulli-Hurwitz numbers for the curve V .
Proposition 5.1. It is possible to take a local parameter z of V around ∞ such that where a n is a homogeneous polynomial in Z[ 1 2 , λ 4 , λ 6 , λ 8 , λ 10 ] of degree n if a n = 0. Proof. It is possible to take a local parameter z 1 such that The expansion of y around ∞ with respect to z 1 takes the following form By substituting the above expansions into (43), multiplying the both sides by z 10 1 , and comparing the coefficient of z 0 1 , we obtain α 2 = 1.
From the above equation, we can find that a n is a homogeneous polynomial in Z[ 1 2 , λ 4 , λ 6 , λ 8 , λ 10 ] of degree n if a n = 0 recursively.
We can regard u(z) as a function defined around z = 0.
Proof. From Proposition 5.1, we have From Proposition 5.1, we obtain the statement of the proposition.
For positive integers n and k such that n ≥ k, we use the notation (n) k = n(n − 1) · · · (n − k + 1).
Lemma 5.2. For k = 1, 2, 3, we have Proof. For k = 1, 2, 3, we have By differentiating the both sides of (46) with respect to u, we obtain By dividing the both sides of the above equation by z k+1 , we have Therefore we have Thus we have Lemma 5.3. ( [27], [38]) Let R 1 and R 2 be two integral domains with characteristic 0 satisfying R 1 ⊂ R 2 . We consider a formal power series of z h(z) = ∞ n=0 α n z n n! , α n ∈ R 2 .
If α 0 , . . . , α n−1 belong to R 1 , and there is a polynomial F of n variables over R 1 such that where h (n) (z) is the n-th derivative of h(z) with respect to z, then we have h(z) ∈ R 1 z .
Then for any positive integer m, h(z) m m! also belongs to R z . Lemma 5.5. ( [27], [38], [44]) Let R be an integral domain with characteristic 0 and Then, the formal inverse series z(w) = w + O(w 2 ) belongs to R w .
Lemma 5.6. For n ≥ 4, we have the following relations.
Lemma 5.7. It is possible to take a local parameter s of V around ∞ such that where α n is a homogeneous polynomial in Z[ 1 5 , λ 4 , λ 6 , λ 8 , λ 10 ] of degree n if α n = 0. Proof. It is possible to take a local parameter s 1 of V around ∞ such that The expansion of x around ∞ with respect to s 1 takes the following form (1 + O(s 1 )), α ∈ C.
By substituting the above expansions into (43), multiplying the both sides by s 10 1 , and comparing the coefficient of s 0 1 , we obtain 1 = α 5 .
We can regard u(s) as a function defined around s = 0.
where g 1 = g 2 = g 3 = 0 and g n is a homogeneous polynomial in Z[ 1 5 , λ 4 , λ 6 , λ 8 , λ 10 ] of degree n if g n = 0. We find that the coefficient of s −1 in Proof. We can prove this lemma in the same way as Lemma 5.6.
By multiplying the above equation by x −2 and differentiating this equation with respect to u, we obtain the statement of the lemma.
Let p ≥ 5 be a prime.
(iii) If p − 1 ∤ n, then we have ord p C n n ≥ 0, ord p D n n ≥ 0.
(iv) If p − 1 | n, then we have ord p C n n ≥ −1 − ord p a, ord p D n n ≥ −1 − ord p a.
Remark 5.1. Theorem 5.2 gives the precise information on the series expansion of the solution of the inversion problem of the ultra-elliptic integrals given in Proposition 4.7.