# Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018, Pages 86 - 105

# Rosenhain-Thomae formulae for higher genera hyperelliptic curves

Authors
Keno Eilers
Faculty of Mathematics, University of Oldenburg, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany,keno.eilers@uni-oldenburg.de
Received 2 August 2017, Accepted 7 September 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1440744How to use a DOI?
Keywords
Theta Functions; Rosenhain formula; Thomae Formula
Abstract

Rosenhain's famous formula expresses the periods of first kind integrals of genus two hyperelliptic curves in terms of θ-constants. In this paper we generalize the Rosenhain formula to higher genera hyperelliptic curves by means of the second Thomae formula for derivative non-singular θ-constants.

Open Access

## 1. Introduction

The developments of the theory of algebraic curves (and related theories) in the XIX-th century led to the idea of describing and classifying objects relevant to algebraic curves and their Jacobians in terms of their modular forms, the Riemann θ-functions, which depend on the Riemann period matrix τ. In this respect, a lot of work was accomplished for (hyper-)elliptic curves of genus 1 and 2. In this paper we want to generalize the existing results primarily due to Rosenhain and discuss here such representations of periods of higher genera hyperelliptic integrals.

The Riemann period matrix τ is defined as the quotient, τ = 𝒜−1ℬ of the 𝒜- and ℬ- period matrices of holomorphic integrals. Here, the leading question is the inverse problem: Given the Riemann period matrix τ, how can we express the period matrix 𝒜 in terms of θ-constants and, possibly, invariants of the curve?

The θ-constant representation of a complete elliptic integral,

K=π2θ32(0),
was known since Jacobi’s times. Rosenhain, Jacobi’s student, obtained a generalization of this formula to genus-2-curves in terms of θ-constants with characteristics [15]. To remind this result we introduce a genus two curve,
y2=x(x1)(xa1)(xa2)(xa3),ai. (1.1)

For a given Riemann matrix τ, we denote as 𝒜 and ℬ = 𝒜τ the period matrices. Also we define θ-constants with even characteristics [ε]=[ε1ε2ε1ε2],εεT=0mod2,εi,εj+ as the Riemann-θ-functions (with characteristics) evaluated at zero. That is, specifically:

θ[ε]=n2expiπ{(n1+ε12)2τ1,1+2(n1+ε12)(n2+ε22)τ1,2+(n2+ε22)2τ2,2+ε1(n1+ε12)+ε2(n2+ε22)}0. (1.2)

The derivated odd θ-constants for odd characteristics [δ]=[δ1δ2δ1δ2],δδT=1mod2,δi,δj+ , are defined as the derivation of the Riemann-θ-functions, evaluated at zero:

θi[δ]=2iπn2(ni+δi2)expiπ{(n1+δ12)2τ1,1+2(n1+δ12)(n2+δ22)τ1,2+(n2+δ22)2τ2,2+δ1(n1+δ12)+δ2(n2+δ22)}0,i=1,2 (1.3)

For genus-2-curves, there are 16 characteristics. 6 of them are odd and 10 even and we denote the sets of characteristics as 𝒮6 and 𝒮10, correspondingly. Odd characteristics are in 1 – 1 correspondence with the branching points, (0, 1, a1, a2, a3, ∞), in a way which will become clear in the subsequent sections.

Rosenhain’s central theorem is taken from his only known publicationa, where it was indeed unproven. We, too, give it here without proof:

### Theorem 1.1 (Rosenhain’s modular representation of the period matrix 𝒜).

In the homology basis given on Fig. 1, there are characteristics

[α1]=[1100],[α2]=[0010],[α3]=[1001][β1]=[0110],[β2]=[1000],[β3]=[0011] (1.4)
and
[δ1][α1]+[α2]+[α3]mod2=[0111],[δ2][β1]+[β2]+[β3]mod2=[1101], (1.5)
such that
𝒜1=12π2Q2(Pθ2[δ1]Qθ2[δ2]Pθ1[δ1]Qθ1[δ2]),a1a2a3=P4Q4 (1.6)
with the quantities P and Q as abbreviations for:
P=θ[α1]θ[α2]θ[α3],Q=θ[β1]θ[β2]θ[β3] (1.7)

This formula was proven by H. Weber [19] during his course of deriving special case solutions of the Clebsh problem on the motion of a rigid body in an ideal liquid (see also the discussion by O.Bolza in his dissertation devoted to the reduction of genus-2 holomorphic integrals to elliptic integrals, [3], a shorter version was published in [4]).

In more recent times, the problem of a θ-constant representation of 𝒜 was discussed within NovikovŠs program of Şeffectivization of finite-gap integration formulaeŤ, see e.g. Dubrovin [5]. E.Belokolos and V.Enolskii [2] implemented this representation in their approach to the reduction of θ-functional solutions of completely integrable equations to elliptic functions. Nart and Ritzenthaler ([14]) used a Thomae-type formula for non-hyperelliptic genus-3 curves, derived from Weber’s formula ([18] and more recently [11]), but did not apply the found θ-constants to the problem of representation of 𝒜. An attempt of a generalization of Rosenhain’s work can be found in [16] and below in the Corollary 2.1.

Some of the build-up of this work can also be found in [7], in especially the recovery of Rosenhain’s formula by Thomae’s second formula. Indeed, it was V. Enolskii, who brought the topic to our attention, and we believe to have generalized their previous contributions.

Below, one of our goals is to elucidate the role that these specific characteristics play in Rosenhain’s formula. For that purpose, the next section is dedicated to the first and second Thomae Formulae in higher genera. In the 3rd section we go on with the attempt to express the period matrix 𝒜 solely by θ-constants, which will then be completed exemplarily in the 4th and 5th section for genus 2 and 3 and in doing so we will broader the class of characteristics which fulfill eq. (1.6) and its higher-genus analogues.

We believe our results will be of general interest as both for theory and for numerical calculations of complete hyperelliptic integrals.

## 2. Thomae formulae for hyperelliptic curves

The seminal paper from Thomae [Tho870] is mostly known for the formula relating branching points to even θ-constants of a genus-g hyperelliptic curve C. But the paper also contains a formula for non-singular derivated odd θ-constants without a proof. In this section we give an elementary proof.

## 2.1. Curve and differentials

Let the curve C be of the form

y2=λ2g+2x2g+2+λ2g+1x2g+1++λ0=λ2g+2(xe1)(xe2g+2) (2.1)

Fix a basis of holombrphic differentials du(P) = (du1(P), …, dug(P))T,

dui(P)=xi1y dx,i=1,,g, (2.2)
and a canonical homology basis (a, b). Denote a- and b-periods
𝒜=(ak dui)i,k=1,,g,=(bk dui)i,k=1,,g. (2.3)

The normalised holomorphic differentials dv(P) = (dv1(P), …, dvg(P))T are defined as

dv=𝒜1 duaj dvi=δi,j,bj dvi=τi,j, (2.4)
where the g × g Riemann matrix τ = 𝒜−1ℬ belongs to the Siegel upper half-space 𝔖 = {τT = τ, Imτ > 0}. Denote Jac(C) = ℂg / Γ the Jacobi variety of the curve C, where Γ = 1gτ. Any point v on the Jacobi variety can be represented in the form
v=12ε+12τεε,εg (2.5)

The vectors ε and ε′ combine to a 2 × g matrix named the characteristic [ε] of the point v. If v is a half-period then all entries of the characteristic are equal 0 or 1 modulo 2.

## 2.2. Theta-functions

Next we introduce in greater detail the Riemann-θ-function θ[ε](z; τ), z ∈ ℂg, τ ∈ 𝔖g

θ[ε](z;τ)=e14iπεTτε+εT(z+12ε)θ(z+12τε+12ε)=mgexp{iπ(m+12ε)Tτ(m+12ε)+2iπ(z+12ε)T(m+12ε)}. (2.6)
with the binary characteristic
[ε]=[εTεT]=[ε1,,εgε1,,εg],εi,εj=1 or 0

It possesses the periodicity property

θ[ε](v+n+τn;τ)=e2iπnT(v+12τn)eiπ(nTεnTε)θ[ε](v;τ). (2.7)

The property (2.7) implies

θ[ε](v;τ)=eπiεεTθ[ε](v;τ). (2.8)

Therefore θ[ε](z; τ) is even if εεT is even and odd otherwise. The corresponding characteristic is called even or odd. Among 4g characteristics there are (4g + 2g) / 2 even and (4g−2g) / 2 odd.

Following the notion of Krazer [13], p.283, a triplet of characteristics [ε1], [ε2], [ε3] is called azygetic if

(1)ε1ε1T+ε2ε2T+ε3ε3T+(ε1+ε2+ε3)(ε1+ε2+ε3)T=1. (2.9)
and a sequence of 2g + 2 characteristics [ε1],…, [ε2g + 2] is called a special fundamental system if the first g characteristics are odd, the remaining are even and any triple of characteristics in it is azygetic. This last notion is taken from [12], and in our examples below we deal with a particular special fundamental system, where one of the even characteristics is the zero characteristic.b

The values θ[ε](0; τ) = θ[ε] are called θ-constants. An even characteristic [ε] is non-singular if θ[ε] ≠ 0, an odd characteristic [δ] is called non-singular if the derivative θ-constants, θk[δ]=θ(z;τ)/zk|z=0 , are not vanishing at least for one index k.

As it is implied in eq. (2.5) we can identify any branching point ei of the curve C with a half-period,

𝕬j=P0(ej,0)dv=12εj+12τεj (2.10)
where P0 is the base point of the Abel map which is supposed to be a branching point and the integer 2 × 2g-matrix [ε] is a characteristic of 𝔄j. We agree to denote with [𝔄j] the characteristic of the j-th half-period 𝔄j.

### Proposition 2.1.

[FK980] The homology basis (a, b) is completely defined by the characteristics [𝔄j], j = 1, …, 2g + 2. g of these characteristics are odd and the remaining g + 2 are even. The vector of Riemann constants KP0 with a base point P0 from the set of branching points is defined in the given homology basis as

KP0=all godd [𝕬j]𝕬j (2.11)

### Proposition 2.2.

Let the 2g + 2 characteristics [𝔄j] be ordered into a sequence, for which the first g characteristics are odd and the remaining are even. Then such a system of characteristics is a special fundamental system.

Proof. In the light of Proposition 2.1, it is clear that there are g odd and g + 2 even characteristics. Now, the first part of the exponent of eq. (2.9) is 0 mod 2 if none or two of the three characteristics are odd, and it equals 1 mod 2 if one or three characteristics are odd. The second part of the exponent asks for the parity of a sum of three characteristics. If none or two of them are odd, the sum is odd and hence the said part of the exponent is 1 mod 2. If one or three characteristics are odd, the sum is even and the second part of the exponent is 0 mod 2. In total, the exponent is always odd and hence any triple is azygetic.

Fay ([9], p. 13) describes a one-to-one correspondence between the characteristics [ε] and the partitions of indices of branching points {1,…, 2g + 2},

m𝒥m={i1,,ig+12m}{j1,,jg+1+2m}, (2.12)
where m is any integer between 0 and [g+12] . Characteristics with given m are defined by the vectors
kIm𝕬ikK=12εm+12τεm. (2.13)

Clearly, the following notation for characteristics is useful:

[ε(Im)]=[kIm𝕬ikK][εm,1,,εm,gεm,1,,εm,g]{Im},m=0,1, (2.14)

m is called the index of speciality of the branching point divisor and we will be interested in the cases m = 0, that deals with even non-singular θ-constants, and m = 1, the case of non-singular odd θ-constants. Here, we are considering hyperelliptic curves with a branching point at ∞ and we fix in what follows P0 = ∞. The defined sets can be written as

0={i1,,ig},𝒥0={j1,,jg+1}
along with the condition:
0𝒥0=,0𝒥0={1,2,,2g+1}.

From the set 0 2g sets 1 and 𝒥1 can be defined:

1(n)=0\{in},𝒥1(n)=𝒥0{in},1ng. (2.15)

It is convenient to denote the Vandermonde determinants,

Δ(m)=ik<ilm(eileik),Δ(𝒥m)=jk<jlJm(ejlejk) (2.16)
and we will write as a short form:
(0)=Δ(0)Δ(𝒥0),(1)=Δ(1)Δ(𝒥1)

## 2.3. Thomae theorems

We are now in the position of having set up our notation. The odd curve C will be realized as:

y2=f(x),f(x)=(xe1)(xe2g+1),ek

C has a branching point at infinity, e2g+2=, and we agree to take away from the products in eq. (2.16) all factors containing e2g + 2.

The next theorem is one of the key points of [17] and its proof is well-documented in the literature (e.g. [9]):

### Theorem 2.1 (First Thomae theorem).

Let ℐ0 ∪ 𝒥0 be a partition of the set of indices of the finite branching points and [ε(0)] the corresponding characteristic. Then

θ[ε(0)]=(det𝒜2gπg)1/21/4(0), (2.17)
where ϵ is the 8th root of unit, ϵ8 = 1.

To find ϵ, which does not depend on τ but rather on the ordering of the branching points in ∇(ℐ0), the classical way is to use a diagonal period matrix τ and use Jacobi’s θ-constants relation on the separated equations. However, we believe the quickest way to determine ϵ is to compute the θ-constants at a very low precision.

There are various corollaries of the Thomae formula (2.17). The following two are easy to prove. Their formulation is taken from [7], but the first one is also topic in [16].

### Corollary 2.1.

Let 𝒮 = {n1, …, ng−1} and 𝒯 = {m1, …, mg−1} be two disjoint sets of non-coinciding integers taken from the set 𝒢 of indices of the finite branching points. Then for any two kl from the set 𝒢 \ (𝒮 ∪ 𝒯) the following formula is valid

elemekem=θ2{k,𝒮}θ2{k,𝒯}θ2{l,𝒮}θ2{l,𝒯}, (2.18)
where m is the remaining index when 𝒮, 𝒯, k, l are taken away from 𝒢, and ϵ4 = 1.

### Corollary 2.2.

Let0 = {i1, …, ig} and 𝒥0 = {j1, … jg + 1} be a partition of 𝒢. Choose k, nI0 and i, j ∈ 𝒥0 and define the sets 𝒮k = ℐ0 \ {k}, 𝒮k, n = ℐ0 \ {k, n}, 𝒯i,j = 𝒥0 \ {i, j}. Then

jl𝒥0(ekejl)ilI0,ilk(ekeil)(eken)2=±θ4{i,𝒮k}θ4{j,𝒮k}θ4{n,𝒯i,j}θ4{i,j,𝒮k,n}θ4{i,𝒯i,j}θ4{j,𝒯i,j}. (2.19)

One can assure oneself of the correctness of these corollaries by a straightforward calculation and use of Thomaes first theorem.

Thomae’s paper contains also an important theorem describing non-singular derivated odd θ-constants. It is this, which is most significant for the course of the paper at hand:

### Theorem 2.2 (Second Thomae theorem).

Let 1(n)𝒥1(n) be a partition of the set of indices of the finite branching points. Then

θj[ε(1(n))]=(det𝒜2g+2πg)1/2(1(n))1/4i=1g𝒜j,isgi(1(n)),j=1g, (2.20)
where ϵ8 = 1 and sj(1(n)) is the elementary symmetric function of degree j built in the branching points ei, i1(n) , and alternated in the sign.

We give here an elementary proof of this theorem. For this we first examine a helpful lemma.

### Lemma 2.1.

Let 𝔄k be the Abelian image of the branching point ek and [𝔄k] its characteristic. Let v=j=1gPj dv=𝒜1u with Pj = (xj, yj). Then at k = 1, …, 2g + 1

(θ[𝔄k](vK)θ(vK))2=4f(ek)j=1g(ekxj), (2.21)
with 44=1 .

Proof. Consider the expression

F(P1,,Pg)=θ2((ek,0)dv+vK)θ2(vK) (2.22)

As the function of P¯1 (here P¯=(x,y) whilst P = (x, y)) it has, according to the Riemann vanishing theorem, zeros of second order in the points (ek, 0), P2,…, Pg and poles of second order in ∞, P2,…, Pg. Thus F(P1,…, Pg) ∼ c1(x1ek). Considering in the same way other the variables P2,…, Pg we conclude

(θ[𝔄k](vK)θ(vK))2=c(ekx1)(ekxg) (2.23)

To find the constant c we use (2.17). Therefore we fix v at some branching points Pi1,,Pig , Pij=(eij,y(eij)) , and rewrite eq. (2.23) using the ε-notation for the characteristics:

(θ[𝔄k](vK)θ(vK))2=θ2[εki1ig]θ2[εi1ig]=4({k,i1,,ig})({i1,,ig})=4f(ek)(ekei1)(ekeig).

### Proof of Theorem 2.2.

Coming back to the proof of Theorem 2.2 we introduce the functions

F(x)=k=1g(xxk),Fi(x)=F(x)/(xxi)=xg1+s1(i)xg2++sg1(i),i=1,,g, (2.24)
where sj(i) are the elementary symmetric functions of order j built in the elements {x1, …, xg} /{xi} and alternated in sign, namely,
s0(i)=1s1(i)=xp,p{1,,g}/{i}s2(i)=xpxq+,p,q{1,,g}/{i}sg1(i)=±xr,r{1,,g}/{i}

First, we use these functions to compute the Jacobian xv . Differentiating the Abel map,

x1(u1,,ug)dxy++xg(u1,,ug)dxy=u1x1(u1,,ug)xg1 dxy++xg(u1,,ug)xg1 dxy=ug,
with respect to u1 we get
1y1x1u1++1ygxgu1=1x1g1y1x1u1++xgg1ygxgu1=0 (2.25)
and similar for the other variables. Solving these equations with respect to xiuj , we arrive at
xv=(x1,,xg)(v1,,vg)=𝒜T(x1,,xg)(u1,,ug)=𝒜T(yisgj(i)F(xi))i,j=1,,g (2.26)

Aside from that, we compute the derivative of eq. (2.21):

viθ[𝔄k](vK)θ(vK)=f(ek)1/412j=1g(ekxj)vij=1g(ekxj),i=1,,g (2.27)
which can be processed for our purposes to:
vij=1g(ekxj)=j=1gxjviFj(ek)=(x1vi,,xgvi)(F1(ek)Fg(ek)) (2.28)

To write this relation for θ-constants, we proceed like in the previous proof and fix v at certain branching points: xj = el, j = 1,…, g, l = 1,…, 2g + 1. Again we can adopt the ε-notation and write:

(v1vg)θ[εk;l1lg]θ[εl1lg]=f(ek)1/412j=1g(ekelj)𝒜T(yisgj(i)F(eli))T(F1(ek)Fg(ek)). (2.29)
where the minus sign of eq. (2.28) was absorbed in ϵ. Of course, the different yi=j=12g+1(xiej) will become zero if xi=eli and hence the whole expression cancels unless ek coincides with that specific eli so that these factors in the numerator and denominator can cancel. Without loss of generality we choose k = lg and hence we have [εk;i1ig]=[εi1ig1] . These g − 1 elements shall now constitute the set ℐ1 for they form all the non-singular and odd characteristics. The characteristics from the left hand side’s denominator, [εi1ig] , we merge into the set 0.θk[εI1] does not vanish, but θk[εI0] does. The derivative hence becomes:
(v1vg)θ[εl1lg1]θ[εl1lg]=1θ[εI0](θ1[εI1]θg[εI1])

However, on the right hand side of eq. (2.29) all remaining yi cancel and the residual zeros of j=1g(ekelj) will be canceled by the factors of Fi(ek) .

Plugging all this together, we get:

(θ1[ε1]θg[ε1])=θ[ε0]χk4𝒜T(sg1(1)s1(1)1), (2.30)
where it is denoted
χk=j𝒥1,jk(ekej)iT1(ekei),k=1,,2g+1, (2.31)
with 𝒥1 the opposite partition of ℐ1 as usual. Note also, that sj(i) was renamed here to sj(1), 1 = 0 \ {xi} to keep track of which partition is in use.

On θ[ε0] we can use Thomae’s first theorem (2.17). Recognizing, that ∇(ℐ0) ⋅ χk = ∇(1), we arrive at the statement of the theorem.

Example: The genus-1 case

Let C be the Weierstrass cubic,

y2=4(xe1)(xe2)(xe3),e1+e2+e3=0

In this case we have: 1(1)=,𝒥1(1)={1,2,3},

Δ(1(1))=1,Δ(𝒥1(1))=(e1e2)(e1e3)(e2e3),
s0(I1(1))=1 and ε(1(1))=K=[11]

Using further [1], vol 3, Sect 13.20

(e1e2)1/2=π2ωϑ42(0),(e1e3)1/2=π2ωϑ32(0),(e2e3)1/2=π2ωϑ22(0),
then (2.20) takes the form of the Jacobi derivative relation
θ1(0)=πϑ2(0)ϑ3(0)ϑ4(0). (2.32)

Example: The genus-g case

With much the same method, one can obtain a generalization of eq. (2.32) to arbitrary genus, which is known as the Riemann-Jacobi-formula (for hyperelliptic curves). For that to formulate we introduce the g + 1 sets

𝒯n=𝒥0\{jn},1ng+1, (2.33)
and also write 𝒯0 = 𝒥0. The characteristics [ε(1(n))] are non-singular and odd, the characteristics [ε(𝒯n)] non-singular and even.

Further, we need the Jacobi matrix J:

J=(θ[ε(1(1))](v),,θ[ε(1(n))](v))(v1,,vg)|v=0. (2.34)

Then, the following relation is valid:

detJ=±πgn=0g+1θ[ε(𝒯n)] (2.35)

This is a long-known result and part of a more general Riemann-Jacobi formula, see e.g. [12] for a modern outline, or [10]. The formulation used here is at least valid for θ-functions with a given hyperelliptic curve in a fixed homology basis, see [7] for a check within the methods described here. There, we find also a useful matrix notation for the second Thomae formula, which we will adopt in the next section:

## 2.4. Matrix form of the Second Thomae formula

The formula (2.20) can be written in matrix form. With all the definitions above, one immediately recognises that this comes to:

JT=(Det𝒜2g+2πg)1/2𝒜T.𝒮.𝒟, (2.36)
where 𝒮 is the invertible matrix
𝒮=(si(j))i,j=1,,g, (2.37)
and 𝒟 is a diagonal matrix:
𝒟=Diag[(1(1))1/4,,(1(g))1/4].

This formula can also be rewritten as

𝒜=2g+2πgdet𝒜𝒮1T𝒟1J (2.38)

This can be treated further. To do that we write our given curve of genus g (with a branching point at infinity) in the form:

y2=ϕ(x)ψ(x) (2.39)
with
ϕ(x)=ik0(xeik),ψ(x)=jl𝒥0(xejl), (2.40)

After inverting eq. (2.38), we get:

Next, we want to use the Riemann-Jacobi formula on det J. For that we need the g + 1 sets 𝒯k = 𝒥0 \ {jk}, so that we can write:

with
Θ0=l=1g+1θ[ε(𝒯l)]. (2.43)

Note, that 𝒯0 is excluded from the product, which will be useful later, as well as the label of Θ. We can process θ[ε(𝒯0)] further with the help of the first Thomae theorem, so that most of the prefactors cancel and only a ∇1 / 4(0) remains in the denominator. As det𝒜 cancels, we get also rid of one possible source of a prefactor. Now using the previous found relation (2.31)

Δ(1(n))Δ(0)Δ(𝒥1(n))Δ(𝒥0)=ψ(ein)ϕ(ein)=χin,n=1,,g (2.44)
and defining
𝒟1=Diag[χi14,,χig4] (2.45)
we get the final form:

Of course, one can consider also the not-inverted, original period matrix 𝒜, and using the same steps on 𝒟 as before, we can rewrite eq. (2.38) as:

𝒜=2θ[ε(𝒥0)]𝒮1T𝒟11J. (2.47)

But we decided to work primarily on the inverted matrix, because this is, what Rosenhain’s formula gives us. Another advantage is that we can quickly recover and generalize Bolza’s formula. For that purpose we write our result in the following way:

### Proposition 2.3.

Let C be a hyperelliptic curve of genus g with one branching point at infinity. Let0 ∪ 𝒥0 = {i1,…,ig} ∪ {j1, …, jg + 1} be a partition of 2g + 1 indices of branching points. Then the columns Um of the matrix 𝒜−1 are of the form

## 2.5. Bolza formulae

Let Uk be the directional derivative along the vector Uk at zero argument:

Ukf(v)=j=1gUk,jvjf(v)|v=0,k=1,,g.

For a genus-2 curve with branching points e1,…, e2g + 1, Bolza ([4]) found that

ei=U1θ[𝕬i+KP0]U2θ[𝕬i+KP0]. (2.49)

Enolskii et al. reproved and generalized this formula in [6] by using Kleinian σ-functions. With the help of eq. (2.48) we can find a quicker proof and write it in a concise way:

As we have done before, we choose P0 = ∞ and keep the notation of ε instead of 𝔄. For a general hyperelliptic curve of genus g we consider the expression

Umθ[ε1(j)]Unθ[ε1(j)],m,n=1,,g

There are g different sets 1(j)=0\{ij} , which also constitute the matrices J and 𝒮. Inserting eq. (2.48) into this expression, we find that all θ-constants cancel out, as well as the prefactors of U and the 4th. roots. We arrive at the

### Corollary 2.3.

Let Uk be directional derivatives, 1(j)g sets of g − 1 branching point indices and sk(1(j)) alternating elementary symmetric functions of order k over the elements ei, i1(j) . Then the following generalized Bolza formulae are valid:

sgm(1(j))sgn(1(j))=Umθ[ε1(j)]Unθ[ε1(j)],m,n=1,,g (2.50)

Example: genus 3

Take ℐ0 = {1,2,3} and hence 1(1)={2,3},1(2)={1,3} and 1(3)={1,2} . We find:

U1θ[ε12]U3θ[ε12]=s2(e1,e2)s0(e1,e2)=e1e2U2θ[ε12]U3θ[ε12]=s1(e1,e2)s0(e1,e2)=e1e2, (2.51)
and all other combinations of m and n can be derived from these both.

## 3. A general θ-constant form of 𝒜−1

Though eq. (2.48) gives us a good tool, our final goal is to completely express 𝒜−1 with θ-constants for those cases where only τ is known. Thus, we want to work more on χk. We can achieve that by the use of eq. (2.21).

### Theorem 3.1.

Let 0 = {n, i1, …, ig−1} and 𝒥0 = {j1, …, jg + 1} be a partition of branching points, such that y2 = ϕ(x) ψ(x) with

ϕ(x)=i0(xei),ψ(x)=j𝒥0(xej).

Let further 1 = ℐ0 \ {n}. For χn=ψ(en)ϕ(en),n0 , we find:

χn4=Θ1Θ0g12g2i0,in(enei), (3.1)
where
Θ0=j𝒥0θ[ε(𝒥0\{j})],Θ1=j𝒥0θ[ε(1{j})]

Proof. Take eq. (2.21) and evaluate v at the branching points ej1,,ejg :

θ2[εnj1jg]θ2[εj1jg]=4f(en)l=1g(enejl)=4(enej1)(enejg)(enei1)(eneig1)(enejg+1).

Squaring this and iterating the procedure for every left-over jg + 1 we get:

j𝒥0θ4[ε{n}𝒥0\{j}]θ4[ε𝒥0\{j}]Θ14Θ𝒥04=±j𝒥0(enej)g1i0,in(enei)g+1=±χng1i0,in(enei)2. (3.2)

This equality comes from the fact that there are g times g + 1 terms in the numerator and every linear factor occurs g times, but is canceled once by the denominator. The residual parts fit the definition of 𝛘n.

Finally, we recognize ε({n} ∪ 𝒥0 \ {j} = ε({n} ∪ 0 ∪ {j}) = ε(ℐ1 ∪ {j}) to arrive at the definition of Θ1 .

Note: In eq. (3.2) are as much factors in the numerator as in the denominator. Therefore, we can interchange the ordering of the en and ej without changing the global prefactor ϵ, if one simultaneously changes the ordering of the en and ei in the denominator.

## 4.1. The Rosenhain derivatives

Consider the case g = 2 and the curve given as

y2=(xe1)(xe2)(xe3)(xe4)(xe5)f(x) (4.1)

In the homology basis drawn on Fig. 1 we have

[𝕬1]=[1000],[𝕬2]=[1010],[𝕬3]=[0110],[𝕬4]=[0111],[𝕬5]=[0011],[𝕬6]=[0000] (4.2)

The characteristic of the vector of Riemann constants reads

[K]=[𝕬2]+[𝕬4][𝕬1]+[𝕬3]+[𝕬5]=[1101] (4.3)

The characteristics in question here are:

[εi]=[𝕬i][K],i=1,,6[εij]=[𝕬i]+[𝕬j][K],i,j=1,,6,ij, (4.4)
and analogously for three indices, if necessary. The first line represents the 6 odd characteristics, the second line the 10 even characteristics. Due to the addition mod 2 one easily sees that [ε2] = [𝔄4], [ε4] = [𝔄2] and [ε24]=2[𝕬2]+2[𝕬4]=[0000]=[𝕬6]

One also has to hold in mind, that the sum of all characteristics Ai is zero, so that 2-indexed ε and 3-indexed ε can be interchanged (as shown for instance in eq. (4.3)).

We are now in the position to exemplary investigate the sets 𝒯l of eq. (2.33) and henceforward Θ0 of eq. (2.43). We therefore split f(x) = ϕ(x)ψ(x) like before and specify ϕ and ψ by fixing ℐ0 = {1,2} and 𝒥0 = {3,4,5}, so that:

𝒯1={3,4},𝒯2={3,5},𝒯3={4,5}.

The already defined quantity Θ0 becomes:

Θ{1,2}=θ[ε34]θ[ε35]θ[ε45]. (4.5)

This choice of the sets leads us directly to the following Rosenhain derivative formula as a consequence of the Riemann-Jacobi-formula:

θ1[ε2]θ2[ε1]θ1[ε1]θ2[ε2]=π2θ[ε34]θ[ε35]θ[ε45]θ[ε345]π2Θ{1,2}θ[ε345] (4.6)

In general, for the different choices of εi, εj as odd characteristics Riemann-Jacobi gives us (2g+1g)=(52)=10 different Rosenhain derivative formulae (up to a minus sign due to the antisymmetry of the determinant), and 5 more, if one includes ε6 K. These last 5 equations belong to the 5 possible sets 0 = {i, 6}, which are not covered by our notation, though they are valid anyway. All these 15 relations are shown in the Appendix A with their correct ordering to fix the sign.

For any triple {i, j, k} ⊂{1,…, 6} we can regard the three Rosenhain derivative formulae belonging to the sets 01={i,j},02={i,k} and 03={j,k} . Among the even characteristics on the righthand-side of them there will be precisely one characteristic εlmp,{l, m, p} = {1,…, 6} \ {i, j, k}, which appears in all three formulae. We therefore write:

θ1[εi]θ2[εj]θ1[εj]θ2[εi]=π2θ[εlmp]Θ{i,j}θ1[εj]θ2[εk]θ1[εk]θ2[εj]=π2θ[εlmp]Θ{j,k}θ1[εi]θ2[εk]θ1[εk]θ2[εi]=π2θ[εlmp]Θ{i,k} (4.7)

Here, Θ{i, j} = θ[εklp]θ[εkmp]θ[εklm], as it is apparent from the construction. 3-indexed ε can be changed to 2-indexed ε if convenient. Each two of eq. (4.7) can be used to solve for θn[εi], θn[εj] or θn[εk], n = 1, 2, and the third one provides a useful substitution. In the course, εlmp cancels and we arrive at the following lemma:

### Lemma 4.1.

For any odd genus-2 curve C and one from 20 triples {i, j, k} ⊂{1, …, 6} (with.ε6K) the following relation holds:

θn[εi]Θ{j,k}±θn[εj]Θ{i,k}=θn[εk]Θ{i,j}, (4.8)
with n = 1, 2 and Θ{i, j} = θ[εklp]θ[εkmp]θ[εklm] and analogously. The characteristics in Θ{i, j} sum up to εk.

For the choice above, ℐ0 = {1,2}, we deliberately pick as the third index 6. Eq. (4.8) gives us:

θn[ε1]θ[ε135]θ[ε145]θ[ε134]θn[ε2]θ[ε235]θ[ε245]θ[ε234]=θn[K]θ[ε346]θ[ε356]θ[ε456]θn[ε1]θ[ε24]θ[ε23]θ[ε25]θn[ε2]θ[ε14]θ[ε13]θ[ε15]=θn[K]θ[ε34]θ[ε35]θ[ε45]. (4.9)

For other partitions one has to keep in mind the sign in eq. (4.8) and switch the order of the characteristics if required.

## 4.2. General Rosenhain Theorem

With our chosen partitions eq. (2.46) reads

𝒜1=2π2Θ{1,2}(θ2[ε1]θ2[ε2]θ1[ε1]θ1ε2])Diag(χ14,χ24)(e21e11). (4.10)

Theorem 3.1 gives us:

χ14=e2e1Θ2Θ1,2χ24=e2e1Θ1Θ1,2, (4.11)
where the previous mentioned reordering of the branching points was applied. Note that Θ1 = Θ{1,6} and Θ2 = Θ{2,6}.

To compare this result with the Rosenhain-memoir [15] we apply a Moebius transformation to the curve, which sets e1 = 0 and e2 = 1. Now using Θ{1,2}, Θ{1,6} and Θ{2,6} as well as eq. (4.9) we find:

𝒜1,11=Θ{2,6}2π2Θ{1,2}2θ2[ε1],𝒜2,11=Θ{2,6}2π2Θ{1,2}2θ1[ε1]𝒜1,21=12π2Θ{1,2}2(Θ{2,6}θ2[ε1]Θ{1,6}θ2[ε2])=12π2Θ{1,2}θ2[K]𝒜2,21=12π2Θ{1,2}2(Θ{2,6}θ1[ε1]Θ{1,6}θ1[ε2])=12π2Θ{1,2}θ1[K] (4.12)

We now can identify δ1 = ε1, δ2 = K, P = Θ{2,6} and Q = Θ{1,2} and hence we have recovered Rosenhain’s theorem, eq. (1.6), along with the extra identity (1a1)(1a2)(1a3)=Θ{1,6}4Q4 .

We used here the partition {1, 2} ∪ {3, 4, 5} in order to compare it to Rosenhain’s original theorem. But the techniques of Theorem 3.1 and Lemma 4.1 allow for a more general statement:

We take the sets 0 = {i, j} and 𝒥0 = {k, l, m}, all indices mutually disjoint. Again, we normalize the curve to ei = 0 and ej = 1 by means of a Moebius transformation.

One can see, that for a set 0 = {i, j} it is always necessary to pick 6 as the third index for Lemma 4.1 to be applicable in this context. We arrive at the following theorem:

### Theorem 4.1 (General genus-2 Rosenhain Theorem).

For an odd genus-2 curve C with normalized branching points ei = 0, ej = 1 and arbitrary branching points ek, el, em the inverse period matrix 𝒜−1 is given as:

𝒜1=2π2Θ{i,j}2[Θ{j,6}θ2[εi]Θ{i,j}θ2[K]Θ{j,6}θ1[εi]Θ{i,j}θ1[K]]. (4.13)

In the same fashion one can indicate 𝒜 if desired. We therefore invert eq. (4.13) using eq. (4.7) one time. We conclude:

𝒜=2Θ{i,j}Θ{i,6}Θ{j,6}θ[εij][Θ{i,j}θ1[K]Θ{i,j}θ2[K]Θ{j,6}θ1[εi]Θ{j,6}θ2[εi]]. (4.14)

Note that this formula incorporates all 10 even characteristics. Also, the three characteristics in Θ{i, j} sum up to (the odd) K and the three characteristics in Θ{j, 6} sum up to (the odd) εi.

## 5. A genus-3 Rosenhain formula

We take a hyperellipticc curve in the form,

y2=k=17(xek)f(x)=ϕ(x)ψ(x),ek (5.1)
where
ϕ(x)=(xe1)(xe2)(xe3),ψ(x)=(xe4)(xe5)(xe6)(xe7), (5.2)
in especially we fixed ℐ0 = {1, 2, 3}. The homology basis is the apparent generalization of Fig. 1.

The characteristics of the Abelian images of branching points are

[𝕬1]=[100000],[𝕬2]=[100100],[𝕬3]=[010100],[𝕬4]=[010110],[𝕬5]=[001110],[𝕬6]=[001111],[𝕬7]=[000111],[𝕬8]=[000000]

The vector of Riemann constants K with base point at P8 = ∞ is given in this homology basis as

K=[𝕬2]+[𝕬4]+[𝕬6]=[𝕬1]+[𝕬3]+[𝕬5]+[𝕬7]=[111101] (5.3)

The important characteristics are here:

[εi]=[𝕬i][K],i=1,,8[εij]=[𝕬i]+[𝕬j][K],i,j=1,,8,ij,[εijk]=[𝕬i]+[𝕬j]+[𝕬k][K],i,j,k=1,,8,kijk. (5.4)

The Riemann-Jacobi formula for this choice of ℐ0 (and hence 𝒥0 = 𝒯0 = {4, 5, 6, 7}) reads

Det(θ[ε23],θ[ε13],θ[ε12])(v1,v2,v3)|v=0=π3θ[ε567]θ[ε467]θ[ε457]θ[ε456]θ[ε4567]=π3Θ{1,2,3}θ[ε4567]. (5.5)

Following the necessary steps, eq. (2.48) gives us for 𝒜−1 = (U1, U2, U3):

where we normalized again to e1 = 0, e2 = 1, but e3 can’t be expressed within our technique in the resulting formulae. We now can insert χk from Lemma 3.1 into eq. (5.6):

If required, we could use eq. (2.47) to arrive at 𝒜. But currently we see no further simplifications and therefore didn’t depict it here.

## 6. Concluding remarks

Without any major changes, one can adopt the method shown for genus 3 to higher genera. Unfortunately we were not able to find a generalization to Lemma 4.1, which could bring eq. (5.7) down to a structure like in eq. (1.6). It seems unlikely that there exists one as simple as in genus 2.

Our next steps in this work could be to unfix the base point, which was infinity throughout this work. And we see a chance to develop Thomae type formulae expressing higher derivative θ-constants. We hope to come back to this topic in the near future.

## Acknowledgments

The author wants to thank Victor Enolskii and Chris Eilbeck for providing the idea of the work and many suggestions for useful techniques as well as the constant interest in the work. Also the author gratefully acknowledges the Deutsche Forschungsgemeinschaft (DFG) for financial support within the framework of the DFG Research Training group 1620 Models of gravity.

## Appendix A. Rosenhain derivative formulae

For any two odd characteristics [δ1], [δ2] denote

D[[δ1];[δ2]]=θ1[δ1]θ2[δ2]θ2[δ1]θ1[δ2]

Then the following 15 Rosenhain derivative formulae are valid

D([0101],[1101])=π2θ[0010]θ[0011]θ[1111]θ[0110],([1000]);
D([1110],[1010])=π2θ[1111]θ[0001]θ[0011]θ[1001],([0100]);
D([1011],[0111])=π2θ[0110]θ[1001]θ[0010]θ[0001],([1100]);
D([0111],[0101])=π2θ[1000]θ[1001]θ[1100]θ[1111],([0010]);
D([0111],[1101])=π2θ[0000]θ[0001]θ[1111]θ[0100],([1010]);
D([1011],[1101])=π2θ[1100]θ[0011]θ[0001]θ[1000],([0110]);
D([1011],[0101])=π2θ[0100]θ[1001]θ[0000]θ[0011],([1110]);
D([1010],[1011])=π2θ[0100]θ[0110]θ[1111]θ[1100],([0001]);
D([1110],[0111])=π2θ[0011]θ[0010]θ[1100]θ[0100],([1001]);
D([1110],[1011])=π2θ[1111]θ[0000]θ[0010]θ[1000],([0101]);
D([1010],[0111])=π2θ[0110]θ[1000]θ[0011]θ[0000],([1101]);
D([1110],[1101])=π2θ[1001]θ[1000]θ[0110]θ[0100],([0011]);
D([1110],[0101])=π2θ[0001]θ[0000]θ[1100]θ[0110],([1011]);
D([1010],[1101])=π2θ[1100]θ[0010]θ[0000]θ[1001],([0111]);
D([1010],[0101])=π2θ[0100]θ[1000]θ[0001]θ[0010],([1111]).

We pointed at the right margin the characteristic, which is the sum of characteristics of each entry to the corresponding equality.

## Footnotes

a

Wikipedia tells us: “Rosenhain galt auch als begabt in Sprachen und Musik; allerdings bemerkten einige Beobachter, dass er die hohen Erwartungen seiner jungen Jahre nicht erfüllte und nach seiner preisgekrönten Arbeit keine nennenswerten Beiträge mehr veröffentlichte.”

b

This case generically occurs in our method if one chooses the partition I (see below) strictly even or odd with respect to the characteristics.

c

If not stated otherwise we always mean hyperelliptic curves.

## References

[1]H Bateman and A Erdélyi, Higher Transcendental Functions, Mc Graw-Hill Book Company, New York, Vol. 3, 1955.
[3]O Bolza, Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische insbesondere über die Reduction durch eine Transformation vierten Grades, Georg-August Universität zu Göttingen, 1885. Inaugural-Dissertation;
[10]J Fay, On the Riemann-Jacobi formula, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II, No. 5, 1979, pp. 61-73.
[13]A Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903.
[15]G Rosenhain, Abhandlung über die Functionen zweier Variablen mit vier Perioden welche die Inversion sind der ultra-elliptische Integrale erster Klasse, 1851. Translation to German from Latin manuscript published in: Ostwald; Klassiker der Exacten Wissenschaften; Nr. 65, 1-96; Verlag von Wilhelm Engelmann; Leipzig, 1895.
[18]H Weber, Theorie der Abelschen Functionen vom Geschlecht, Vol. 3, 1876.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 1
Pages
86 - 105
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1440744How to use a DOI?
Open Access

TY  - JOUR
AU  - Keno Eilers
PY  - 2021
DA  - 2021/01/06
TI  - Rosenhain-Thomae formulae for higher genera hyperelliptic curves
JO  - Journal of Nonlinear Mathematical Physics
SP  - 86
EP  - 105
VL  - 25
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1440744
DO  - 10.1080/14029251.2018.1440744
ID  - Eilers2021
ER  -