aa r X i v : . [ m a t h . C V ] N ov Thomae Formulae for General Fully Ramified Z n Curves
Shaul Zemel ∗ October 17, 2018
Introduction
The original Thomae formula is an assertion relating the theta constants ona hyper-elliptic Riemann surface X , presented as a double cover of P ( C ), tocertain polynomials in the P ( C )-values of the fixed points of the hyper-ellipticinvolution on X . They were initially derived by Thomae in the 19th century (see[T1] and [T2]). After laying dormant for more than a 100 years, these formulaereturned to active research, mainly due to the interest of the mathematicalphysics community. The first generalization of these formulae appears in [BR],who considered non-singular Z n curves, i.e., those compact Riemann surfaceswhich are associated with an equation of the type w n = Q nri =1 ( z − λ i ) for some r ≥
1, with λ i = λ j for i = j . The proof for this case was simplified by [N],using the Szeg˝o kernel function. A family of singular Z n curves has been treatedin [EG]. A very elementary proof for the original formulae was found in [EiF],where the case of non-singular Z curves was also seen to be covered by theseelementary techniques. The idea is that certain quotients of powers of thetafunctions can be identified as simple meromorphic functions on the Riemannsurface under consideration. These techniques were extended to arbitrary non-singular Z n curves in [EbF]. The book [FZ] presents in detail the proof of theThomae formulae for several families of Z n curves, namely the non-singularones, the families treated in [EG], as well as two other smaller families. Theproofs in this book follow the elementary methods of [EiF] and [EbF]. On theother hand, formulae for the general case have been obtained in [K], again usingthe Szeg˝o kernel function. Additional results on the Thomae formulae for Z curves are presented in [M] and [MT].An important step in the proof of the Thomae formulae in all the casesconsidered in [FZ] is the construction of non-special divisors of degree g on Z n curves, which are supported on the branch points on the Z n curve. Acharacterization of these divisors in the case of prime n is presented in [GD],using certain sums of residues modulo n . The first result of the present paper ∗ This work was supported by the Minerva Fellowship (Max-Planck-Gesellschaft). Z n curves for arbitrary n in terms of cardinalities of certain sets which are based onthe divisor in question. In our method, it is easier to work with the cardinalitiesthan with the residues, and our result is equivalent to that of [GD] in the caseof prime n . Next, certain operators defined in [EbF] and [FZ] are useful inthe derivations. We show how to define these operators for general Riemannsurfaces, and provide the formula to evaluate them in the case of a fully ramified Z n curve and a divisor which is supported on the branch points (see Theorem 2.3and Proposition 2.5). We remark that a result of [GDT] yields Z n curves withno non-special divisors of the sort required for us. However, it turns out thatthe Thomae formulae which we obtain, at least in the form presented here, areindependent of the cardinality conditions required for non-specialty. Thereforethey can be trivially extended to the cases considered in [GDT].We now indicate how the construction and proof of the Thomae formulaeare established in this paper. The process follows [FZ], as well as [EiF] and[EbF]. We begin by identifying quotients of powers of theta functions withcharacteristics as meromorphic functions on the Z n curve in order to deriverelations between pairs of theta constants whose characteristics are related byoperators of the form T Q,R , where Q is the branch point we choose as thebase point. Next we obtain, for every type of points (i.e., the power to whichthe point appears in the Z n equation defining the Z n curve), a quotient whichis invariant under all the operator T Q,R with our base point Q and R of thechosen type. A simple correction of the resulting denominator yields, for everybase point Q , a quotient which is invariant under the operators T Q,R for alladmissible points R . This quotient is called, following [FZ], the Poor Man’sThomae , or PMT for short. The next step, which is technically more difficult,is to obtain a denominator for which the quotient is invariant also under thenegation operator N Q (or N β ). A “base point change operator” M is alsointroduced, and the quotient is invariant also under M . If these operators acttransitively on the set of divisors under consideration, a fact which we provein several cases and should hold in general, then the quotient θ en [Ξ](0 , Π) h Ξ thusobtained is independent of the divisor Ξ. This is the Thomae formulae for the Z n curve.The resulting Thomae formulae are based on certain integral-valued func-tions, denoted here f β,α and later f ( n ) d . We investigate the propeties of thesefunctions, which have an interesting recursive definition (see Theorem 6.4). Weremark again that [K] obtained expressions for the Thomae formulae which in-clude sums of certain fractional parts. Comparing with our results gives a toolfor evaluating these sums, and it seems that our recursive relation yields a moreefficient way to obtain the actual values of these powers in every particular case.In Section 1 we describe the fully ramified Z n curves, and characterize thenon-special divisors of degree g supported on their branch points using the car-dinality conditions. Section 2 defines the operators whose action is necessary forestablishing the Thomae formulae. The first relations are obtained in Section 3,and the manipulations required in order to achieve the PMT are also performed.2he quotient which is invariant under the negation operator is given in Section4, where certain properties of the functions f β,α appearing in this quotient arealso proved. Section 5 introduces the base point change operator M , provessome partial results about transitivity, and states the final Thomae formulae.The recursive relation which is required in order to evaluate the functions f β,α is proved in Section 6, where explicit expressions for these functions are givenin a few cases. Section 7 presents examples of Thomae formulae for two familiesof Z n curves, including all the Z n curves which are treated in [FZ]. Finally, inSection 8 we discuss some remaining open questions. Z n Curves
Let X be a Z n curve, namely a cyclic cover of order n of P ( C ), with projectionmap z . We assume n > Z curve is just P ( C ) with achosen isomorphism z . Any such curve can be presented as the Riemann surfaceassociated with an equation of the sort w n = f ( z ) (called a Z n -equation ) forsome meromorphic function f ∈ C ( z ) which is not a d th power in C ( z ) forany d dividing n (so that the equation is irreducible). We call such a defininingequation for a Z n curve normalized if f is a monic polynomial which has no rootsof order n or more. Every Z n -equation can be made normalized by multiplying w by an appropriate function of z , and this function is unique as long as we keep w in a fixed component under the action of the cyclic Galois group (see the endof this paragraph). The field C ( X ) of meromorphic functions on X decomposesas L n − r =0 C ( z ) w r , or equivalently L n − k =0 C ( z ) · w k . The space of meromorphicdifferentials on X is 1-dimensional over C ( X ), and is spanned by any non-zero differential on X . By choosing the differential to by dz we obtain thatthis space decomposes as L n − k =0 C ( z ) dzw k . The cyclic Galois group of the map z : X → P ( C ) acts on C ( X ) and on the space of meromorphic differentials on X ,and the decompositions given here are precisely the decompositions accordingto the action of this Galois group. In particular, w generates a 1-dimensionalcomplex vector space which is invariant under the action of this group.Let ϕ : X → Y be a non-constant holomorphic map between compact Rie-mann surfaces. Recall that for every point x ∈ X there is a unique number b x ∈ N , called the ramification index or branching number of x , such that inlocal charts around x and ϕ ( x ) the map ϕ looks like z z b x +1 . The number b x is 0 for all x ∈ X except for a finite number of points, called the branch points of ϕ . The sum P x ∈ ϕ − ( y ) b x gives the same value for every y ∈ Y , and thisvalue equals the rank of the map ϕ . Thus, generically, the inverse image of apoint in Y consists of d points in X . The only points of Y whose inverse imagehas smaller cardinality are images of branch points of ϕ on X . The degree andbranching numbers appear in the Riemann–Hurwitz formula , relating the genus g X of X with the genus g Y of Y according to the equality2 g X − d (2 g Y −
2) + X x ∈ X b x .
3e will consider only the case where X is a Z n curve, Y = P ( C ), and ϕ is themeromorphic function z (which has degree n since the equation is irreducible).Since the genus of Y is 0, we write simply g for g X , with no confusion arising.Now, assume that X is given through a Z n -equation w n = f ( z ). For everypoint λ ∈ P ( C ), we let d be the greatest common divisor of n and ord λ ( f ).For each such λ there are d points of X lying over λ ∈ P ( C ), each of whichhas ramification index nd −
1. In particular, no point outside the divisor of f isa branch point. Let u be another meromorphic function on X , which generates C ( X ) over C ( z ) and spans a complex vector space which is invariant under theGalois group of z . Replacing w by u leaves the set of branch points, as wellas their ramification indices, invariant: Indeed, multiplying w by an element of C ( z ) changes the order of f at λ ∈ P ( C ) by a multiple of n , and replacing w by w k for k ∈ Z which is prime to n multiplies these orders by k and leaves thegreatest common divisors invariant. Thus we can assume that the Z n -equationis normalized. Then the branch points on which z is finite lie only over the rootsof f (and over all of them). Points lying over ∞ are branch points if and onlyif the degree of f is not divisible by n .Following [FZ], we call a Z n curve X fully ramified if any branch point on X has maximal ramification index (namely n − λ ∈ P ( C ), z − ( λ ) consists either of n points or of a unique branchpoint. Given a normalized Z n -equation defining the Z n curve X , this propertyis equivalent to all the roots of f appearing with orders which are prime to n ,and the degree of f is either divisible by n or also prime to n . In this case wehave the following Lemma 1.1.
Fix ≤ k ≤ n − , and fix a branch point P ∈ X . Everymeromorphic function of the form p ( z ) w k with p ∈ C ( z ) has the same order at P modulo n , and for l = k the classes modulo n of the orders of p ( z ) w k and q ( z ) w l at λ are distinct. The same assertion holds for differentials in C ( z ) dzw k .Proof. The first assertion (for both meromorphic functions and meromorpicdifferentials) follows from the fact the order of an element in C ( z ) at P isdivisible by n . The second assertion is established by taking the quotient ofthese functions or differentials, which is of the form ψ ( z ) w k − l with ψ ∈ C ( z )and k − l not divisible by n . Since the order of ψ ( z ) at λ is divisible by n , theorder of w at λ is prime to n , and n does not divide k − l , this completes theproof of the lemma.Let ∆ be a divisor on X (not necessarily integral). We recall from [FK](or [FZ]) that the space of meromorphic functions whose divisor is at least∆, denoted L (∆), is finite-dimensional, of dimension denoted r (∆). Similarly,the space of meromorphic differentials with this property (denoted Ω(∆)) isalso finite-dimensional (with i (∆) denoting this dimension). These numbers arerelated by the Riemann–Roch Theorem , stating that r (cid:0) (cid:1) = deg ∆+1 − g + i (∆).Using the decomposition of C ( X ) and C ( X ) dz from above, we denote r k (∆)the dimension of the space of meromorphic functions of the form p ( z ) w k (with4 ∈ C ( z )) which lie in L (∆), and i k (∆) denotes the dimension of the space ofmeromorphic differentials of the form ϕ ( z ) dzw k lying in Ω(∆). The inequalities P n − k =0 r k (∆) ≤ r (∆) and P n − k =0 i k (∆) ≤ i (∆) are clear. For divisors supportedon the branch points, Lemma 1.1 yields the following generalization of Lemma2.7 of [FZ]: Proposition 1.2.
Let ∆ be a divisor on X (not necessarily integral) which isbased on the branch points of z , and let h and ω be a meromorphic function anda meromorphic differential on X respectively. Decompose h and ω as P n − k =0 h k with h k ∈ C ( z ) · w k and P n − k =0 ω k with ω k ∈ C ( z ) dzw k respectively. Then h ∈ L (∆) (resp. ω ∈ Ω(∆) ) if and only if the same assertion holds for h k (resp. ω k ) forall ≤ k ≤ n − . In particular r (∆) = P n − k =0 r k (∆) and i (∆) = P n − k =0 i k (∆) . Before we prove Proposition 1.2, we introduce, following [FZ] and others,the useful notation e ( t ) = e πit for t ∈ C . Proof.
We prove the assertion only for functions, as the claim for differentialsis established by replacing every h by ω etc. Lemma 1.1 shows that the ordersof the different functions h k at every branch point (hence at every point atthe support of ∆) are distinct. Vanishing components h k , which have order+ ∞ at every point, can be ignored in all considerations. It follows that theorder of h (or ω ) at every branch point P equals min k ord P ( h k ), so that if ord P ( h ) ≥ ord P (∆) then the same inequality holds with h replaced by h k forany k . It remains to prove that if h has no pole at any point Q on X whichis not a branch point then neither do the functions h k , 0 ≤ k ≤ n −
1. Let V be an open subset of P ( C ) containing no z -image of a branch point. Hence z − ( V ) is a disjoint union of n open sets U i ⊆ X , 1 ≤ i ≤ n , with each U i homeomorphic to V via z . First assume ∞ 6∈ V , and observe that w doesnot vanish on S ni =1 U i . Moreover, we can choose the indices i such that theaction of the homeomorphism ( z | U j ) − ◦ z | U i multiplies each h k by e (cid:0) ( j − i ) kn (cid:1) .Let µ ∈ V , and let Q i ∈ U i be the point with z ( Q i ) = µ . If h k has a pole insome point Q j then it has a pole of the same order at all the points Q i . Let now a = − min k ord Q i ( h k ) (this number is independent of i ), so that a > h k has a pole at the points Q i . In coordinates we have h k ( P ) = ψ ki ( z ( P ))( z ( P ) − µ ) a for P ∈ U i , with ψ ki a function on V which has no pole at µ .Since h = P k h k has no pole at none of the points Q i we have P k ψ ki ( µ ) = 0 forall i . But the fact that h k lies in C ( z ) · w k yields the relation ψ kj = e (cid:0) ( j − i ) kn (cid:1) ψ ik ,so that the equality P k e (cid:0) ( j − i ) kn (cid:1) ψ ki ( µ ) = 0 holds for every i and j . As theVandermonde matrix whose jk -entry is e (cid:0) ( j − i ) kn (cid:1) , is non-singular, this implies ψ ik ( µ ) = 0 for all i and k , in contradiction to the choice of a . This implies a ≤
0, i.e., all the functions h k are holomorphic outside the branch points. Thesame argument but with h k ( P ) = z ( P ) a ψ ki (cid:0) z ( P ) (cid:1) proves the assertion for thecase in which ∞ ∈ V as well. The equalities involving r (∆) and i (∆) followdirectly from the previous assertions. 5e remark that the same argument shows that Proposition 1.2 holds also ifwe let the divisor ∆ contain points which are not branch points in its support,but insist that all the points with the same z -value appear to the same power in∆. In fact, in this case we can replace ∆ by a linearly equivalent divisor whichis supported only on the branch points, using a function from C ( z ) ∗ . Usingthis method one can show that Proposition 1.2 holds for any such divisor on a Z n curve, provided that there exists at least one fully ramified branch point.However, we consider only divisors supported on branch points on fully ramified Z n curves in this paper.The set of integers between 0 and n − N n . The set N n is also a good set ofrepresentatives of Z /n Z in Z .We now turn to bases for the holomorphic differentials on a fully ramified Z n curve. For simplicity and symmetry, we shall assume throughout that thereis no branching over ∞ (this can always be obtained by composing z with anautomorphism of P ( C )). We thus write the normalized Z n -equation defining X as w n = Y α r α Y i =1 ( z − λ α,i ) α , (1)where α runs over the set of numbers in N n which are prime to n . The as-sumption that no branch point lies over ∞ is equivalent to the assertion that n divides P α αr α . The genus g of X equals ( n − (cid:0) P α r α − (cid:1) / n is odd, and if n is even then so is P α αr α , and since we take only odd α , thesame assertion holds for P α r α . Proposition 1.2 implies that we can decomposethe space Ω(1) of holomorphic differentials on X as L n − k =0 Ω k (1). Now, Ω (1) isthe space of holomorphic differentials in C ( z ) dz , i.e., holomorphic differentialswhich are pullbacks of holomorphic differentials on P ( C ). As there are no suchdifferentials, the decomposition is in fact L n − k =1 Ω k (1). It follows that for an integral divisor ∆, Proposition 1.2 yields i (∆) = P n − k =1 i k (∆) (since i (∆) = 0).We shall denote, here and throughout, the poles of z on X by ∞ h , 1 ≤ h ≤ n .The (unique) branch point on X lying over λ α,i will be denoted P α,i . Then div ( z − λ α,i ) = P nα,i Q nh =1 ∞ h , div ( w ) = Q α Q r α i =1 P αα,i Q nh =1 ∞ t h , div ( dz ) = Q α Q r α i =1 P n − α,i Q nh =1 ∞ h where div denotes the divisor of a meromorphic function or differential and nt = P α αr α . We now introduce a convenient basis for the space C ( X ) dz over C ( z ). For any k ∈ Z and any α we define s α,k = (cid:4) αkn (cid:5) , where for a real number x the symbol ⌊ x ⌋ stands for the integral value of x , namely the maximal integer m satisfying m ≤ x . Then s α,k satisfies αk +1 − nn ≤ s α,k ≤ αkn . Moreover, thenumber αk − ns α,k , which lies in N n , depends only on the class of k modulo n . Let ω k = Q α Q r α i =1 ( z − λ α,i ) s α,k dzw k . This differential is well-defined for k ∈ Z /n Z , though we usually assume k ∈ N n . We remark that for k prime to n , the denominator under dz in ω k corresponds to the normalized Z n -equation6escribing X with w k . Moreover, if the greatest common divisor of k and n is d then this denominator corresponds to the normalized Z n/d -equation describingthe quotient of X by the subgroup of order d of the Galois group, which is a Z n/d curve. We evaluate div ( ω k ) = Y α r α Y i =1 P n − ns α,k − αkα,i n Y h =1 ∞ t k − h , t k = X α r α (cid:18) αkn − s α,k (cid:19) . (2)The numbers ns α,k − αk − n lie also in N n , and observe that t k ∈ Z (since n | P α r α ). Moreover, t k vanishes if n | k and is positive for every k not divisibleby n (or 1 ≤ k ≤ n − α which is prime to n ). In particular ω = dz with the divisorwritten above, and ω = dzw since s α, = 0 for all α . Hence the two formulaefor t coincide. We define, for every d ∈ Z with d ≥ −
1, the space P ≤ d ( z )of polynomials in z of degree not exceeding d (so that P ≤ ( z ) is the space ofconstant polynomials and P ≤− ( z ) = { } ). The dimension of P ≤ d ( z ) is d + 1(also for d = − Proposition 1.3.
The space
Ω(1) of holomorphic differentials on X decomposesas L n − k =1 P ≤ t k − ( z ) ω k .Proof. Proposition 1.2 and the paragraph below Equation (1) show that Ω(1)decomposes as L n − k =1 Ω k (1). It therefore suffices to show that a differentialin C ( z ) dzw k , or equivalently C ( z ) ω k , is holomorphic if and only if it lies in P ≤ t k − ( z ) ω k . Let ϕ ∈ C ( z ) and assume that ϕ ( z ) ω k is holomorphic. Thedivisor of ω k is supported only on the branch points and poles of z , and theformer points appear to non-negative powers which are smaller than n in thisdivisor. It follows that ϕ cannot have any pole in C , hence it must be a poly-nomial of some degree d . But then the order of ϕ ( z ) ω k at any point ∞ h is t k − − d , so that ϕ ( z ) ω k is holomorphic precisely when d ≤ t k − ϕ = 0, i.e., there exists no such holomorphic differential, if t k = 1). Thisproves the proposition.When we evaluate P n − k =1 t k = P α,k r α (cid:0) αkn − (cid:4) αkn (cid:5)(cid:1) , we observe that forany α which is prime to n the set of numbers (cid:8) αkn − (cid:4) αkn (cid:5)(cid:9) n − k =1 (or equivalently (cid:8) αk − ns α,k n (cid:9) n − k =1 ) is precisely the set (cid:8) ln (cid:9) n − l =1 . Hence the sum P n − k =1 ( t k −
1) of thedimensions of these spaces is indeed P α,k r α n ( n − n − ( n −
1) = g , as required.Lemma 1.1 and Proposition 1.3 imply that the set S n − k =1 { ( z − λ α,i ) l ω k } t k − l =0 is a basis for Ω(1) which is adapted to the point P α,i for any α and i . Thegap sequence at P α,i can be read from this basis. However, it is not needed forfinding non-special divisors or for the proof of the Thomae formulae. Moreover,in some of the examples in [FZ] some of the points P i have the usual gap sequenceand are not Weierstrass points. For these reasons we do not pursue this subjectfurther in this work.Using the notation | Y | for the cardinality of the finite set Y , we prove7 orollary 1.4. Let ∆ be an integral divisor on X which is supported on thebranch points, and assume that no branch point appears in ∆ to a power n or higher. For any α and any ≤ k ≤ n − denote A α,k the set of indices ≤ i ≤ r α such that P α,i appears in ∆ to a power larger than ns α,k − αk − n .Then i k (∆) = max { t k − − P α | A α,k | , } and i (∆) is the sum of these numbers.Proof. By Proposition 1.3, Ω k (1) is a space of differentials of the form p ( z ) ω k ,where p is a polynomial of degree not exceeding t k −
2. We claim that Ω k (∆)consists of those differentials in which p vanishes at all the values λ α,i with i ∈ A α,k . Indeed, since no branch point appears in ∆ to a power n or higher,if p ( λ α,i ) = 0 then ord P α,i ( p ( z ) ω k ) ≥ n > ord P α,i (∆). Hence simple zeroes of p suffice. For an index i not lying in A α,k we have ord P α,i ( ω k ) ≥ ord P α,i (∆), andmultiplying by any polynomial in z can only increase the order of the differentialat P α,i . On the other hand, if i ∈ A α,k then p must vanish at λ α,i in orderfor ord P α,i ( ϕ ( z ) ω k ) to reach ord P α,i (∆). This shows that Ω k (∆) is indeed theasserted space. Since the conditions p ( λ α,i ) = 0 are linearly independent (unlesswe reach the 0 space), the assertion about i k (∆) follows. The assertion about i (∆) is now a consequence of Proposition 1.2. This proves the corollary.The definition of A α,k extends to arbitrary k ∈ Z by considering the imageof k in Z /n Z . For k divisible by n all the sets A α,k are empty, and i (∆) = 0since t = 0.The following argument has been used in several special cases in [FZ]. Weinclude it here since it is simple, short, and general. Recall that an integraldivisor of degree g on a Riemann surface of genus g is called special if i (∆) > non-special otherwise. Lemma 1.5.
Any integral divisor ∆ of degree g on a fully ramified Z n curve X containing a branch point to power n or higher is special.Proof. Let Q be a branch point on X . Apart from the constant functions, thespace L (1 /Q n ) contains the meromorphic function z − z ( P ) if z ( Q ) ∈ C or thefunction z if z ( Q ) = ∞ (by full ramification). It follows that r (1 /Q n ) ≥ r (1 / ∆) ≥ Q n divides ∆. But then the Riemann–Roch Theoremimplies i (∆) ≥ g ), hence ∆ is special.We will be interested in non-special divisors supported on the branch pointson a fully ramified Z n curve. In the quest for such divisors, Lemma 1.5 allowsus to restrict attention to divisors in which the branch points appear only topowers at most n −
1, without losing possibilities. Every such divisor ∆ can bewritten as ∆ = Y α n − Y l =0 C n − lα,l , (3)where for every α the sets C α,l , l ∈ N n , form a partition of the set of points { P α,i } r α i =1 . 8e can now characterize the non-special divisors of degree g supported onthe branch points on X by appropriate cardinality conditions. Theorem 1.6.
Let ∆ be an integral divisor of degree g which is supported onthe branch points on X . Then ∆ is non-special if and only if it can be writtenas in Equation (3) and the cardinalities of the sets C α,l satisfy the equality X α αk − ns α,k − X l =0 | C α,l | = t k − for every ≤ k ≤ n − .Proof. Lemma 1.5 allows us to restrict attention to divisors ∆ in which thebranch points appear to powers not exceeding n −
1. We can thus define, forevery l ∈ N n , the set C α,l to contain those branch points P α,i appearing to thepower n − − l in ∆. Every point P α,i must lie in some set C α,l . Thus ∆ takesthe form given in Equation (3), and the sets C α,l form the required partitions.As l ≤ αk − ns α,k − n − − l > n − ns α,k − αk , it follows fromCorollary 1.4 that ∆ is non-special if and only if P α P αk − ns α,k − l =0 | C α,l | ≥ t k − ≤ k ≤ n −
1. But taking the sum over k yields g on the right handside, and we claim that the sum of the left hand sides equals the degree of ∆.Indeed, the set C α,l appears on the left hand side of the k th equality preciselyfor those 1 ≤ k ≤ n − αk − ns α,k is larger than l . Sincethe set { αk − ns α,k } n − k =1 consists precisely of the numbers between 1 and n − α is prime to n ), precisely n − − l of those numbers are larger than l .Since the degree of ∆ is g , all these inequalities must hold as equalities, whichcompletes the proof of the theorem.One can verify that Theorems 2.6, 2.9, 2.13, and 2.15 of [FZ], as well as theclaim in Section A.7 of that reference, are special cases of Theorem 1.6. Thisverification requires some care: The sets C j and D j of [FZ] correspond to oursets C ,j +1 and C n − ,n − − j respectively, and the j th cardinality condition inthese special cases is obtained by taking the difference of consecutive equalitiesin Theorem 1.6. It is also possible to verify that Theorems 6.3 and 6.13 of [FZ]follow from Theorem 1.6. As a point in C α,l appears to the power n − − l in ∆, we find that adding αk to it and then taking the number in N n which iscongruent to the result yields n − − l + αk − ns α,k − nχ ( l < αk − ns α,k ) , where χ of a given condition gives 1 if the condition is satisfied and 0 otherwise.It follows that the sum appearing in Theorem 1 of [GD] and Theorem 2 of[GDT] equals g + nt k − n P α P αk − ns α,k − l =0 | C α,l | , and for prime n Theorem 1.6is equivalent to the results given in these references. Moreover, this argumentshows that the results of [GD] and [GDT] extend to arbitrary n , provided thatthe Z n curve is fully ramified (which is always the case when n is prime). Notethat it can happen that no divisors satisfying the conditions of Theorem 1.6exist (see [GDT]). 9 Operators on Divisors
Let X be a compact Riemann surface of genus g >
0. By taking a canonical basisfor the homology of X , one obtains a symmetric matrix Π ∈ M g ( C ), the periodmatrix of X with respect to this basis, whose imaginary part is positive definite.We identify the Jacobian variety J ( X ) with the complex torus C g / Z g ⊕ Π Z g . Let Div ( X ) denote the group of divisors on X , and let Div ( X ) be the subgroupof Div ( X ) consisting of those divisors whose degree is 0. For a point Q on X , we denote ϕ Q the Abel–Jacobi map from Div ( X ) to J ( X ) with base point Q (see Chapter 3 of [FK] or Chapter 1 of [FZ] for some properties of thismap). It is related to the algebraic Abel–Jacobi map ϕ : Div ( X ) → J ( X ) by ϕ Q (∆) = ϕ (cid:0) ∆ Q deg ∆ (cid:1) . Hence on divisors of degree 0 the value of ϕ Q is independentof the choice of the base point Q (see also Equation (1.1) of [FZ]).Given two vectors ε and ε ′ in R g , one defines the theta function with char-acteristics (cid:20) εε ′ (cid:21) and period matrix Π as θ (cid:20) εε ′ (cid:21) ( ζ, Π) = X N ∈ Z g e (cid:20) (cid:18) N + ε (cid:19) t Π (cid:18) N + ε (cid:19) + (cid:18) N + ε (cid:19) t (cid:18) ζ + ε ′ (cid:19)(cid:21) . For the properties of this function see Chapter 6 of [FK] or Section 1.3 of [FZ]. Inparticular, up to a non-zero factor, the characteristics correspond to translationsof the variable ζ (see Equation (1.3) of [FZ]) in the classical theta function with ε = ε ′ = 0. We are interested in theta constants , i.e., the values of thetafunctions with rational characteristics at ζ = 0. The original Thomae’s formulais a relation between these theta constants on a hyper-elliptic Riemann surface(or, in our language, a Z curve). Here we extend this formula to arbitrary fullyramified Z n curves.Take a point e in J ( X ) (or in C g ), and consider the multi-valued function f ( P ) = θ (cid:0) ϕ Q ( P ) − e, Π (cid:1) on X . The Riemann Vanishing Theorem (see, e.g.,Theorem 1.8 of [FZ]) states that f either vanishes identically on X or hasprecisely g (well-defined) zeroes (counted with multiplicity). In the latter casethe divisor ∆ of zeroes of f is non-special and satisfies e = ϕ Q (∆) + K Q ,where K Q is the vector of Riemann constants associated with Q . Moreover,any element of e ∈ J ( X ) can be written as ϕ Q (∆) + K Q for some integraldivisor ∆ of degree g on X by the Jacobi Inversion Theorem. Proposition 1.10of [FZ] shows that f vanishes identically if and only if ∆ is special. Observe thatotherwise the presentation of e as ϕ Q (∆) + K Q is unique: Indeed, applying theRiemann–Roch Theorem for a non-special integral divisor ∆ of degree g yields r (cid:0) (cid:1) = 1. Hence L (cid:0) (cid:1) = C (the constant functions), and there is no otherintegral divisor Ξ of degree g such that ϕ Q (Ξ) = ϕ Q (∆) and e = ϕ Q (Ξ) + K Q .The following proposition about the vector of Riemann constants is veryuseful in the theory of Thomae formulae: Proposition 2.1. ϕ Q takes any canonical divisor on X to − K Q .Proof. See the theorem on page 298 of [FK], or page 21 of [FZ].10he dependence of K Q on the base point Q is given through the fact that ϕ Q (∆) + K Q is independent of Q if ∆ is a divisor of degree g − X (seeTheorem 1.12 of [FZ]).The following property of the vector of Riemann constants, in case the basepoint is a branch point on a fully ramified Z n curve, has been obtained in afew special cases in [FZ] (see Lemma 2.4, Lemma 2.12, Lemma 6.2, and Lemma6.12 of that reference). However, it turns out to hold in general: Lemma 2.2.
Let Q be a branch point on a fully ramified Z n curve of genus g ≥ . Then the vector K Q of Riemann constants associated with the base point Q has order dividing n in J ( X ) .Proof. Let µ = z ( Q ) ∈ C . Since g ≥
1, there exists some 1 ≤ k ≤ n − t k ≥
2, and then the divisor of ω = ( z − µ ) t k − ω k is supported only onthe branch points. But the fact that R n Q n is principal for any branch point R (as the divisor of z − z ( R ) z − µ ) implies that nϕ Q (cid:0) div ( ω ) (cid:1) = 0. In case z ( Q ) = ∞ thedivisor of every differential ω k is supported on the branch points, and R n Q n is thedivisor of z − z ( R ). The conclusion nϕ Q (cid:0) div ( ω ) (cid:1) = 0 follows also in this case.As ϕ Q (cid:0) div ( ω ) (cid:1) = − K Q by Proposition 2.1, the assertion follows.In all the cases considered in [FZ], the Thomae formulae have been provedusing two types of operators, denoted N and T R (with base point P ), actingon the set of non-special divisors of degree g which are supported on the branchpoints distinct from P . We now show that these operators exist in general (notonly on Z n curves!). Let X be an arbitrary Riemann surface of genus g ≥ v Q (∆) the power to which the point Q on X appears in the divisor∆ on X . Theorem 2.3. ( i ) Let ∆ be a non-special integral divisor of degree g ≥ on X ,and let Q be a point on X such that v Q (∆) = 0 . There exists a unique integraldivisor N Q (∆) of degree g on X satisfying ϕ Q (cid:0) N Q (∆) (cid:1) + K Q = − (cid:0) ϕ Q (∆) + K Q (cid:1) . (4) The divisor N Q (∆) is non-special, and satisfies v Q (cid:0) N Q (∆) (cid:1) = 0 . The operator N Q is an involution on the set of non-special integral divisors of degree g notcontaining Q in their support. ( ii ) Given any point R such that v R (cid:0) N Q (∆) (cid:1) = 0 ,there exists a unique integral divisor T Q,R (∆) of degree g on X such that theequality ϕ Q (cid:0) T Q,R (∆) (cid:1) + K Q = − (cid:0) ϕ Q (∆) + ϕ Q ( R ) + K Q (cid:1) (5) holds. The divisor T Q,R (∆) is also non-special, and we have the equalities v Q (cid:0) T Q,R (∆) (cid:1) = 0 and v R (cid:0) N Q (cid:0) T Q,R (∆) (cid:1)(cid:1) = 0 . The operator T Q,R , which isdefined on the set of non-special divisors on X not containing Q in their sup-port and such that R does not appear in N Q (∆) , is an involution on this set ofdivisors. roof. Denote by e ∈ J ( X ) the expression on the right hand side of Equation(4), and consider the (multi-valued) function f ( P ) = θ (cid:0) ϕ Q ( P )+ e, Π (cid:1) . Since − e equals ϕ Q (∆)+ K Q and i (∆) = 0, we find that f does not vanish identically, butrather vanishes only on points in the support of ∆. The condition v Q (∆) = 0thus implies θ ( e, Π) = 0, and since θ is an even function, we deduce θ ( − e, Π) = 0.But this implies that ψ ( P ) = θ (cid:0) ϕ Q ( P ) − e, Π (cid:1) does not vanish at P = Q , hencedoes not vanish identically. Thus e = ϕ Q (Ξ) + K Q for some non-special divisorΞ representing the zeroes of ψ , so that in particular v Q (Ξ) = 0. Since Ξ and N Q (∆) are both integral of degree g and have the same ϕ Q -images, the fact thatΞ is non-special implies Ξ = N Q (∆). The fact that N Q (∆) is non-special and v Q (cid:0) N Q (∆) (cid:1) = 0 yields the existence of a unique divisor N Q (cid:0) N Q (∆) (cid:1) satisfyingEquation (4) with ∆ replaced by N Q (∆). As ∆ satisfies this equation, theequality N Q (cid:0) N Q (∆) (cid:1) = ∆ follows, and N Q is an involution. This proves ( i ). Inorder to establish ( ii ) we denote the value on the right hand side of Equation(5) by d , and consider the multi-valued function ̺ ( P ) = θ (cid:0) ϕ Q ( P ) + d, Π (cid:1) . As ̺ ( R ) = f ( Q ) and the latter expression is non-vanishing, we find that − d canbe written as ϕ Q (Υ) + K Q where Υ is a non-special integral divisor of degree g representing the zeroes of ̺ (hence v R (Υ) = 0). Moreover, ̺ ( Q ) equals f ( R ) andis also non-vanishing by our assumption on R . This shows that v Q (Υ) = 0 aswell, and we define T Q,R (∆) = N Q (Υ). The equality Υ = N Q (cid:0) T Q,R (∆) (cid:1) (as N Q is an involution) and part ( i ) imply that T Q,R (∆) has the asserted properties. Inparticular, T Q,R (cid:0) T Q,R (∆) (cid:1) is defined, and since it is characterized by satisfyingEquation (5) with ∆ replaced by T Q,R (∆), we deduce that T Q,R is an involutionas in part ( i ). This completes the proof of the theorem.Note that part ( ii ) of Theorem 2.3 does not require that R = Q . However,if R = Q then the right hand side of Equation (5) reduces to that of Equation(4), implying that T Q,Q (∆) is simply N Q (∆). We shall therefore always assume R = Q in T Q,R .We are interested in the form of the operators N Q and T Q,R in the casewhere X is a fully ramified Z n curve and Q and R are branch points on X .Assume that X is associated with Equation (1), ∆ is given by Equation (3),and Q = P β,i for some β ∈ N n which is prime to n and some index i . Hence µ = z ( Q ) equals λ β,i , but we keep the notation µ . Let k β be an integer suchthat n | βk β − βk β − ns β,k β = 1). This characterizes the class of k β in Z /n Z (we rather not impose the assumption k β ∈ N n ). The point Q does notlie in the support of ∆ if and only if Q ∈ C β,n − in the notation of Equation(3). For any α and l we denote by a β,α ( l ) and b β,α ( l ) the elements of N n whichare congruent modulo n to αk β − − l and 2 αk β − − l respectively. Thesenumbers are of course independent of the choice of k β ∈ Z . We consider a β,α and b β,α as functions on N n , and these functions are involutions. Two usefulequalities concerning these involutions are given in the following Lemma 2.4.
The equality a β,α (cid:0) b β,α ( l ) (cid:1) = n − − a β,α ( l ) holds for every α , β , and l ∈ N n . It is equivalent to the equality a β,α (cid:2) b β,α (cid:0) a β,α ( l ) (cid:1)(cid:3) = n − − l holding for all such α , β , and l . roof. The first equality follows from the fact that both expressions are ele-ments of N n which are congruent to l − αk β modulo n . The second equality isobtained from the first by replacing l by a β,α ( l ) and using the fact that a β,α isan involution. This proves the lemma.It will turn our convenient to let the index l of C α,l to be any integer, whileidentifying C α,l with C α,l + n for every l ∈ Z . In this way we can consider theset C α,αk β , for example, without having to write C α,αk β − ns α,kβ .We now assume that the Z n curve has genus g at least 1, for the theory oftheta functions to be non-trivial. This means P α r α ≥ g . The following proposition generalizes Definitions 2.16, 2.18, 6.4, and 6.14 of[FZ], as well as Propositions 2.17, 2.19, 6.5, and 6.15 there: Proposition 2.5. If ∆ is given by Equation (3) and v Q (∆) = 0 then the divisor N Q (∆) is defined by the formula N Q (∆) = Y α n − Y l =0 C n − − a β,α ( l ) α,l /Q n − . Moreover, assume that the branch point R = Q does not appear in the support of N Q (∆) (this means R ∈ C γ,γk β if R = P γ,m for some index m ). Then T Q,R (∆) is given by T Q,R (∆) = Y α n − Y l =0 C n − − b β,α ( l ) α,l /RQ n − , and v R (cid:0) T Q,R (∆) (cid:1) = v R (∆) .Proof. Denote the asserted values of N Q (∆) and T Q,R (∆) by Ξ and Ψ respec-tively. By Theorem 2.3 it suffices to prove that Ξ and Ψ are of degree g andsatisfy Equation (4) and (5) respectively. The latter equations are equivalent to ϕ Q (cid:0) ∆ · Ξ (cid:1) = ϕ Q (cid:0) ∆ · R · Ψ (cid:1) = − K Q , so that by Proposition 2.1 it suffices to find differentials on X such that theirdivisors have the same ϕ Q -images as ∆Ξ or ∆ R Ψ. Consider the differentials( z − µ ) t kβ − ω k β and ( z − µ ) t kβ − ω k β . Equation (2) shows that their divisorsare Y α r α Y i =1 P n − ns α,kβ − αk β α,i Q n ( t kβ − and Y α r α Y i =1 P n − ns α, kβ − αk β α,i Q n ( t k β − respectively (in fact, our choice of k β shows that the total power of Q in thesedivisors are n ( t k β − − n ( t k β − − α and l the equalities( n − − l ) + ( n − − a ) = n − ns α,k β − αk β + nχ ( l < αk β − ns α,k β )13nd( n − − l ) + ( n − − b ) = n − ns α, k β − αk β + nχ ( l < αk β − ns α, k β )hold, where a and b stand for a β,α ( l ) and b β,α ( l ) respectively. Indeed, both sidesare congruent to − − ηαk β modulo n (with η being 1 for the first equation and2 for the second one), and the two numbers on the left hand side and the numberon the right hand side not involving the conditional expression are all elementsof N n . As an index i satisfies i ∈ A α,ηk β if and only if P i,α lies in a set C α,l with l < ηαk β − ns α,ηk β , it follows that∆ · Ξ = Y α " r α Y i =1 P n − ns α,kβ − αk β α,i Y i ∈ A α,kβ P nα,i /Q n − and ∆ · R · Ψ = Y α " r α Y i =1 P n − ns α, kβ − αk β α,i Y i ∈ A α, kβ P nα,i /Q n − . The number of points P α,i appearing to the power n is P α P αk β − ns α,kβ − l =0 | C α,l | or P α P αk β − ns α, kβ − l =0 | C α,l | . These numbers equal t k β − t k β − Q α Q r α i =1 P n − ns α,kβ − αk β α,i Q n ( t kβ − − ( n − , while ∆ R Ψ is linearly equivalentto Q α Q r α i =1 P n − ns α, kβ − αk β α,i Q n ( t kβ − − ( n − . These divisors are Q timesthe divisor of ( z − µ ) t kβ − ω k β and Q times the divisor of ( z − µ ) t kβ − ω k β given above. Since the degree of a canonical divisor is 2 g − ϕ Q ( Q ) = 0,this proves that Ξ and Ψ have the required properties. Hence Ξ = N Q (∆) andΨ = N Q (∆) as desired. Observe that for l = γk β − ns γ,k β we have a β,γ ( l ) = n − v R (cid:0) N Q (∆) (cid:1) = 0) and b β,γ ( l ) is congruent to γk β − n .It follows that v R (cid:0) T Q,R (∆) (cid:1) coincides with v R (∆) since the division by R coversfor this difference of 1 between l and b β,γ ( l ). This proves the proposition.The part of Proposition 2.5 concerning N Q relates to Proposition 6.2 of [K],with a simpler proof. Let Q be a point on a Riemann surface X with period matrix Π with respectto a canonical basis, and let ∆ be a divisor of degree g on X . If ϕ Q (∆) + K Q isthe J ( X )-image of Π ε + I ε ′ ∈ C g then we denote, following Section 2.6 of [FZ],the theta function with characteristics (cid:20) εε ′ (cid:21) by θ [ Q, ∆]( z, Π) . This functiondepends on the choice of the lift (i.e., on ε and ε ′ not up to 2 Z g ). However, if X is a fully ramified Z n curve, Q is a branch point, and ∆ is supported on the14ranch points, then the vectors ε and ε ′ lie in n Z g (see Lemma 2.2). In thiscase Equation (1.4) of [FZ] shows that changing the lift can only multiply thefunction by a constant which is a root of unity of order dividing 2 n . It followsthat θ n [ Q, ∆]( z, Π) is independent of the lift. The same assertion thus holdsfor θ en [ Q, ∆]( z, Π) where e is 1 for even n and 2 for odd n . The argumentsof Section 2.6 of [FZ] show that given two non-special divisors ∆ and Ξ ofdegree g which are supported on the branch points distinct from Q , the quotient θ en [ Q, ∆]( ϕ Q ( P ) , Π) θ en [ Q, Ξ]( ϕ Q ( P ) , Π) is a well-defined function on X , which is a constant multipleof Q α,i ( z − λ α,i ) en [ v Pα,i ( N Q (∆)) − v Pα,i ( N Q (Ξ))] (see Propositions 2.21, 6.6, and 6.16of that reference for special cases). Moreover, if R is some branch point suchthat v R (cid:0) N Q (∆) (cid:1) = v R (cid:0) N Q (Ξ) (cid:1) = 0 then the value of this function at R equals θ en [ Q,T
Q,R (∆)](0 , Π) θ en [ Q,T
Q,R (Ξ)](0 , Π) . This generalizes Equations (2.1) and (2.2) of [FZ] to thegeneral setting considered here, and choosing Ξ = T Q,R (∆) (as we shall soondo) yields the corresponding generalization of Equations (2.3) and (2.4) of thatreference.We now obtain relations between theta constants on X , following the methodused in all the special cases presented in [FZ]. By substituting P = Q in thequotient given in the previous paragraph we obtain the value of the constant,so that this quotient equals θ en [ Q, ∆](0 , Π) θ en [ Q, Ξ](0 , Π) · Q α,i ( µ − λ α,i ) env Pα,i ( N Q (Ξ)) Q α,i ( µ − λ α,i ) env Pα,i ( N Q (∆)) · Q α,i ( z − λ α,i ) env Pα,i ( N Q (∆)) Q α,i ( z − λ α,i ) env Pα,i ( N Q (Ξ)) . By choosing Ξ = T Q,R (∆) and substituting P = R we obtain the equality θ en [ Q, ∆](0 , Π) Q α,i ( µ − λ α,i ) env Pα,i ( N Q (∆)) Q α,i ( σ − λ α,i ) env Pα,i ( N Q ( T Q,R (∆))) == θ en [ Q, T
Q,R (∆)](0 , Π) Q α,i ( µ − λ α,i ) env Pα,i ( N Q ( T Q,R (∆))) Q α,i ( σ − λ α,i ) env Pα,i ( N Q (∆)) , (6)where σ = z ( R ).Write ∆ as in Equation (3) in order to express the latter equality using thesets appearing that Equation. The divisor N Q (∆) is given in Proposition 2.5,and using the fact that b β,α is an involution on N n we write the formula for T Q,R (∆) in Proposition 2.5 as Q α Q n − l =0 C n − − lα,b β,α ( l ) /RQ n − . Thus N Q (cid:0) T Q,R (∆) (cid:1) is Q α Q n − l =0 C n − − a β,α ( l ) α,b β,α ( l ) /QR n − , or equivalently Q α Q n − l =0 C a β,α ( l ) α,l /QR n − bythe involutive property of b β,α and Lemma 2.4. The powers of R and Q aredetermined by the condition that both points must not appear in the support of N Q (cid:0) T Q,R (∆) (cid:1) . Let S be a point in X and let Y and Z be (finite) disjoint subsetsof points on X . Following Definition 4.1 of [FZ], we introduce the notation[ S, Y ] = Y T ∈ Y,T = S (cid:0) z ( S ) − z ( T ) (cid:1) , [ Y, Z ] = Y S ∈ Y,T ∈ Z (cid:0) z ( S ) − z ( T ) (cid:1) , Y, Y ] = Y S Y, Z ] coincides with[ Z, Y ]. In order to ease notation in some expressions below we shorthand thesets C α,l \ { Q } and C α,l \ { Q, R } to simply C Qα,l and C Q,Rα,l . The denominatorsunder θ en [ Q, ∆](0 , Π) and θ en [ Q, T Q,R (∆)](0 , Π) in Equation (6) become Y α,l [ Q, C Q,Rα,l ] en [ n − − a β,α ( l )] Y α,l [ R, C Q,Rα,l ] ena β,α ( l ) and Y α,l [ Q, C Q,Rα,l ] ena β,α ( l ) Y α,l [ R, C Q,Rα,l ] en [ n − − a β,α ( l )] respectively. We prefer to use the sets C Qα,l and C Q,Rα,l rather than evaluating thepowers of ( σ − µ ) which have to be canceled since the symmetrization is easierin this way.In order to free the denominator under θ en [ Q, ∆](0 , Π) in Equation (6)from its dependence on R ∈ C Qγ,γk β we would like to divide that Equation bythe expression Y ( α,l ) =( γ,γk β ) [ C Q,Rγ,γk β , C Q,Rα,l ] ena β,α ( l ) · [ C Q,Rγ,γk β , C Q,Rγ,γk β ] en ( n − . The latter multiplier is obtained by setting α = γ and j = γk β , but as thebehavior of [ Y, Z ] for Y ∩ Z = ∅ is different from that of [ Y, Y ], we prefer toseparate this terms from the product. Now, Equation (6) is symmetric underinterchanging ∆ and T Q,R (∆), and we wish to preserve this symmetry. Bywriting the formula for T Q,R (∆) from Proposition 2.5 as Q α,l e C n − − lα,l we obtain,after omitting the problematic points Q and R , the equality e C Q,Rα,l = C Q,Rα,b β,α ( l ) .As b β,γ subtracts 1 from γk β − ns γ,k β , the fact that b β,α is an involution for every α and Lemma 2.4 allow us to write the expression by which we have dividedEquation (6) as Y ( α,l ) =( γ,γk β − [ e C Q,Rγ,γk β − , e C Q,Rα,l ] en [ n − − a β,α ( l )] · [ e C Q,Rγ,γk β − , e C Q,Rγ,γk β − ] en ( n − . In order to keep the symmetry, we must divide Equation (6) also by Y ( α,l ) =( γ,γk β − [ C Q,Rγ,γk β − , C Q,Rα,l ] en [ n − − a β,α ( l )] · [ C Q,Rγ,γk β − , C Q,Rγ,γk β − ] en ( n − . The following observations help to simplify the result. First, as R ∈ C γ,γk β and γk β − = γk β in Z /n Z , we can omit the superscript R from C Q,Rγ,γk β − . The16ame assertion holds for any set C α,l with ( α, l ) = ( γ, γk β ). Second, the sets C γ,γk β − and C γ,γk β do not contain Q (since Q ∈ C β,n − and if γ = β thenneither γk β ∈ n Z nor γk β − ∈ n Z are congruent to n − n ),so that we can omit Q from its notation as well. Third, the set C Q,Rγ,γk β (whichis the only set C Q,Rα,l which really differs from C Qα,l ) appears, in the expressioninvolving Q or C γ,γk β − , to the power 0 (as a β,γ ( γk β − ns γ,k β ) = n − R can be omitted from the notation there as well. Corollary 4.3of [FZ] now allows us, when considering the total product, to add R and Q tothe appropriate sets. Let C + Qγ,γk β − denote the set C γ,γk β − ∪ { Q } , and then theproduct of the denominator appearing under θ en [ Q, ∆](0 , Π) in Equation (6)and the correction terms considered above equals q Q,γ ∆ = Y ( α,l ) =( γ,γk β ) [ C γ,γk β , C Qα,l ] ena β,α ( l ) · [ C γ,γk β , C γ,γk β ] en ( n − ×× Y ( α,l ) =( γ,γk β − [ C + Qγ,γk β − , C Qα,l ] en [ n − − a β,α ( l )] · [ C + Qγ,γk β − , C + Qγ,γk β − ] en ( n − . This is the required form of the denominator under θ en [ Q, ∆](0 , Π) whichdepends on γ but no longer on R ∈ C γ,γk β . As we preserved the symmetry ofyielding the same equation from ∆ and from T Q,R (∆), we have established Proposition 3.1. The quotient θ en [ Q, ∆](0 , Π) q Q,γ ∆ is invariant under the operators T Q,R for all admissible branch points R of the form P γ,m . As a β,γ takes γk β − − ns γ,k β to 0 and γk β − ns γ,k β to n − 1, the power towhich the exceptional sets C γ,γk β − and C γ,γk β appear in in q Q,γ ∆ is determinedby the same rule as the other sets. Observe that Proposition 3.1 generalizesPropositions 4.4, 5.1 and 5.2 of [FZ], where the sets with superscript + Q aredenoted C − for γ = β = 1 and H for γ = − 1. We will ultimately expressour formulae in terms of the set C + Qβ, , which corresponds to the divisor Q n − ∆used for changing the base point below. We also remark that the fact thatonly products of the form αk β show up in our operators and denominators isnot coincidental. Indeed, by replacing w by w k (divided by the appropriatepolynomial in z ) for some k which is prime to n , all the indices α , β , etc. aredivided by k modulo n , so that only such products are independent of the choiceof the generator w of C ( X ) over C ( z ).We can now prove the Poor Man’s Thomae (PMT) for X . Recall that thePMT is a formula which attaches, given a branch point Q as base point, anexpression g Q ∆ to every non-special divisor ∆ supported on the branch pointsdistinct from Q , such that the quotient θ en [ Q, ∆](0 , Π) g Q ∆ remains invariant underall the operators T Q,R for admissible R . Our aim is to multiply q Q,γ ∆ (hencedivide Equation (6) further) by an expression which is invariant under all the17perators T Q,R with R ∈ C γ,γk β , and obtain an expression which is independentof γ as well. Consider the expression Y δ = γ " Y α,l = αk β [ C δ,δk β , C Qα,l ] ena β,α ( l ) · [ C δ,δk β , C δ,δk β ] en ( n − ×× Y α,l = αk β − [ C δ,δk β − , C Qα,l ] en [ n − − a β,α ( l )] · [ C δ,δk β − , C δ,δk β − ] en ( n − ×× Y { ( α,δ ) | α<δ,α = γ,δ = γ } [ C α,αk β − , C δ,δk β − ] en ( n − · [ C α,αk β , C δ,δk β ] en ( n − . We claim that this expression is invariant under T Q,R for all R ∈ C γ,k β . Thisfollows from the considerations regarding the sets e C Q,Rα,l above, together withthe fact that the only set in which C Q,Rα,j = C Qα,l is with α = γ and l = γk β (modulo n ). Since this set appears in our expression only once, with the powerinvolving a β,γ ( γk β − ns γ,k β ) = n − 1, and this power vanishes, we can replaceevery C Q,Rα,l by the R -independent notation C Qα,l . Therefore multiplying q Q,γ ∆ bythis expression gives a denominator g Q ∆ such that θ en [∆](0 , Π) g Q ∆ is invariant under T Q,R for all admissible points R = P γ,m (with our γ ). In order to analyze g Q ∆ we use the following generalization of Lemma 4.2 of [FZ]: Lemma 3.2. Assume the set Y is the union of the finite sets Z j , ≤ j ≤ d ,and let W be a finite set which is disjoint from Y . Then [ Y, W ] is the product Q dj =1 [ Z j , W ] up to sign, and [ Y, Y ] equals Q dj =1 [ Z j , Z j ] · Q ≤ i 1. Considering the expressions with Z d and Z d − we claim that we can replacethe product Q dj =1 [ Z j , Z j ] · Q ≤ i 1) to the one over l = αk β − α, αk β − α = γ . Using these considerations we find that g Q ∆ = Y α,l = αk β [ F β , C Qα,l ] ena β,α ( l ) · [ F β , F β ] en ( n − × Y α,l = αk β − [ E Q , C Qα,l ] en [ n − − a β,α ( l )] · [ E Q , E Q ] en ( n − . Since g Q ∆ does not depend on γ , this argument proves Proposition 3.3. The quotient θ en [ Q, ∆](0 , Π) g Q ∆ is invariant under all the admis-sible operators T Q,R , and it is the PMT of the Z n curve X . One can check that the PMT appearing in Propositions 4.4, 5.3, and 6.7 of[FZ] are special cases of Proposition 3.3, except that the isolated divisor P n − of Section 6.1 of [FZ] (on which no T P ,R can act) is now given the denominator( λ − λ ) n ( n − ( λ − λ ) n ( λ − λ ) n rather than 1. As for Propositions 6.17and 6.19 of that reference, our formula for g Q ∆ multiplies the expression giventhere for the divisor P si P sj for t = 1 (resp. P s +1 i P sj for t = 2) by the ens thpower of the T P ,P i -invariant (resp. T P ,P j -invariant) expressions ( λ − λ j )( λ − λ k )and ( λ i − λ j )( λ i − λ k ) (resp. ( λ − λ i )( λ − λ k ) and ( λ j − λ i )( λ j − λ k )). Henceour results are compatible also in these cases.As already remarked in Section 2.6 of [FZ], we can allow (full) ramificationat ∞ by assuming that P α r α is prime to n in Equation (1). Then the integers t k from Equation (2) (which are no longer integers) have to be replaced by theirupper integral values. All our further results hold also in this setting, whenwe omit any meaningless expression involving ∞ . This holds also when wesubstitute ∞ in a rational function, since every such substitution always yieldsthe value 1. The same assertion applies for what follows as well.We also observe that the formula for g Q ∆ (as well as the preceding expressions)is independent of the cardinality conditions on the set C α,l . Therefore the formof the Thomae formulae is unrelated to the actual set of divisors needed inorder to define the characteristics etc., but is only based on the general shapeof a divisor supported on the branch points distinct from Q containing no n thpowers or higher. In particular, the formulae are not connected to the questionwhether such divisors exist or not, and one might say that they hold in a trivialmanner in the latter case.We now turn to changing the base point Q (but leave the index β fixed).Although this change is not required at this stage, it helps to simplify thenotation. In the proof of Proposition 3.1 we have encountered the sets C + Qγ,γk β − ,namely C γ,γk β − ∪ { Q } , for various γ . Since Q = P β,i , it is natural to considerthis set for γ = β , namely C β, ∪ { Q } . Omitting Q from its original set C β,n − (which stands for the fact that v Q (∆) = 0) and including it in C β, (the set ofpoints P β,m appearing to the power n − Q n − ∆ of degree g + n − 1. This is the divisor appearing in thesymmetric notation of the Thomae formulae in Chapters 3, 4, and 5 of [FZ],since the second statement in Corollary 1.13 there implies that for such divisorsthe element ϕ P (Ξ) + K P of J ( X ) is independent of the choice of the branchpoint P . Its value coincides with ϕ Q (∆)+ K Q , as is easily seen by taking Q = P .We denote the appropriate theta constant θ [Ξ](0 , Π) (with no need to add the19ase point), and it coincides with θ [ Q, ∆](0 , Π). Taking D α,l to be C β, ∪ { Q } if α = β and l = 0 and C Qα,l otherwise, we obtain from Equation (3) thatΞ = Q n − ∆ = Y α n − Y l =0 D n − − lα,l . (7)Moreover, for every 1 ≤ k ≤ n − l = n − ≤ l ≤ βk − ns β,k − 1, while the value l = 0 does participatesin this summation. It follows that the point Q does not contribute to any ofthe cardinalities appearing in Theorem 1.6, but after replacing every set C α,l by D α,l it contributes to all of them. Therefore the divisors of degree g + n − D α,l satisfying X α αk − ns α,k − X l =0 | D α,l | = t k for every 1 ≤ k ≤ n − Q n − as well and using the fact that a β,α and b β,α are involutions, we can write theseoperators in terms of the divisors Ξ as N β (Ξ) = Y α,l D n − − a β,α ( l ) α,l = Y α,l D n − − lα,a β,α ( l ) (8)(the notation N β , rather than N Q , can be used here since the effect of thisoperator depends only on β and not on the choice of Q ∈ D β, ) and T Q,R (Ξ) = Q Y α,l D n − − b β,α ( l ) α,l /R = Q Y α,l D n − − lα,b β,α ( l ) /R (9)for Q ∈ D β, and R ∈ D γ,γk β . Moreover, the set C Qα,l is just D α,l unless α = β and l = 0. It follows that the set E Q appearing in g Q ∆ is simply S δ D δ,δk β − (since for δ = β we already have Q ∈ D β, ). This set depends on β , but nolonger on Q ∈ D β, , just like F β = S δ D δ,δk β . In addition, the set C Qβ, , whichis the only choice of indices α and l for which C Qα,l = D α,l , does not appear inthe expression for g Q ∆ : With E Q the index l = 0 is αk β − α = β (modulo n ), and with F β the power a β,β (0) vanishes (as n divides βk β − g Q ∆ does not depend on Q ∈ D β, in this setting, so we denote it g β Ξ . Furthermore,as a β,α ( αk β − ns α,k β ) = n − a β,α ( αk β − − ns α,k β ) = 0, the power en ( n − 1) to which the expressions [ D δ,δk β , D α,αk β ] and [ D δ,δk β − , D α,αk β − ](coming from [ F β , F β ] or [ E Q , E Q ] respectively) appear in g β Ξ obeys the samerule as with the other expressions [ D δ,δk β , D α,l ] or [ D δ,δk β − , D α,l ]. Expandingthe products using Lemma 3.2 we can write g β Ξ = Y { ( δ,α,l ) | δ ≤ α if l = αk β − ns α,kβ } [ D δ,δk β , D α,l ] ena β,α ( l ) × Y { ( δ,α,l ) | δ ≤ α if l = αk β − − ns α,kβ } [ D δ,δk β − , D α,l ] en [ n − − a β,α ( l )] (the condition δ ≤ α for the appropriate value of l is imposed to avoid undesiredrepetitions), and Proposition 3.3 takes the form Proposition 3.4. The quotient θ en [Ξ](0 , Π) g β Ξ is invariant under all the operators T Q,R with Q ∈ D β, and R ∈ D γ,γk β (with arbitrary γ ), and it is the base-point-invariant form of the PMT of X . We remark that a divisor Ξ takes the form Q n − ∆ for some integral divisor∆ of degree g and some base point Q only if some set D β, is not empty. Ingeneral, however, this condition might not be satisfied, and there exist divisors Ξsatisfying the cardinality conditions such that D β, = ∅ for all β . These divisorscannot be presented as Q n − ∆ for any branch point Q . The operators N β acton these divisors, but no T Q,R does so since Q ∈ D β, (for the appropriate β ) is required to define the action of these operators. Hence the assertion ofProposition 3.4 holds trivially for these divisors, at least at this point. Moredetails will be given in Section 5. N β Consider a Z n curve X , a branch point Q on X , and a non-special divisor ∆ ofdegree g on X which is supported on the branch points distinct from Q . Thecombination of Equation (1.5) of [FZ] and Equation (4) yields the equality θ N [ Q, ∆](0 , Π) = θ N [ Q, N Q (∆)](0 , Π)for any N divisible by 2 n . The condition 2 n | N is necessary to ensure indepen-dence of the lifts. Expressed in terms of the degree g + n − N = 2 en becomes θ en [ N β (Ξ)](0 , Π) = θ en [Ξ](0 , Π) , (10)holding for every β ∈ N n which is prime to n and for every divisor Ξ of the formpresented above. Hence our goal is to divide the quotient from Proposition 3.4(or equivalently, multiply g β Ξ ) by an expression which is invariant under all theadmissible operators T Q,R considered in that Proposition, such that the product h Ξ of g β Ξ with this expression will satisfy h N β (Ξ) = h Ξ . In case the expression h Ξ is independent also of β , the quotient θ en [Ξ](0 , Π) h Ξ will be invariant under allthe operators T Q,R as well as N β for all β .To achieve this goal, we need to compare g β Ξ with g βN β (Ξ) . According toEquation (8), moving from Ξ to N β (Ξ) is equivalent to replacing every set D α,l by D α,a β,α ( l ) . Now, a β,δ ( δk β − − ns δ,k β ) = 0 and a β,δ ( δk β − ns δ,k β ) = n − a β,δ is an involution. These considerations imply that g βN β (Ξ) equals Y { ( δ,α,l ) | δ ≤ α if l = n − } [ D δ,n − , D α,l ] enl · Y { ( δ,α,l ) | δ ≤ α if l =0 } [ D δ, , D α,l ] en ( n − − l ) . Observe that this expression does not depend on β , which suggests that we mighttake it as the denominator under θ en [Ξ](0 , Π) in the PMT using Equation (10).Nevertheless, we prefer to follow [FZ] and maintain the denominator g β Ξ .In order to motivate the following definition, we consider only those parts of g β Ξ and g βN β (Ξ) which involve the set D β, (which remains invariant under N β ).An expression of the form [ D β, , D α,l ] appears to the power en [ n − − a β,α ( l )]in g β Ξ . Assume that there exists an expression h Ξ with the properties stated inthe previous paragraph. Write the power to which [ D β, , D α,l ] appears in h Ξ as en [ c ( β, α ) − f β,α ( l )], where c ( β, α ) ∈ Z and f β,α : N n → Z is some function.By altering the constant c ( β, α ) if necessary, we can always assume f β,α (0) = 0.Then the N β -invariance of h Ξ yields the equality f β,α (cid:0) a β,α ( l ) (cid:1) = f β,α ( l ) (11)for every l ∈ N n . Moreover, the T Q,R -invariance of h Ξ g β Ξ implies that the equality f β,α (cid:0) b β,α ( l ) (cid:1) − a β,α (cid:0) b β,α ( l ) (cid:1) = f β,α ( l ) − a β,α ( l )holds for every l ∈ N n . The latter property becomes easier to work with whenwe replace l by a β,α ( l ). Indeed, Lemma 2.4, the fact that a β,α is an involution,and Equation (11) combine to show that the latter equality is equivalent to f β,α (cid:2) b β,α (cid:0) a β,α ( l ) (cid:1)(cid:3) + l = f β,α ( l ) + n − − l (12)holding for every l ∈ N n . Moreover, the common value of the two sides in Equa-tion (12) is left invariant under replacing l by n − − l , as follows from Equation(11) and the second assertion of Lemma 2.4. In particular, the normalization f β,α (0) = 0 implies f β,α ( n − 1) = n − α and β . Theorem 4.1. For any n and any α and β in N n which are prime to n thereexists a unique function f β,α : N n → Z which satisfies Equations (11) and (12) for every l ∈ N n and attains 0 on l ∈ N n .Proof. We first prove that there is a unique function f β,α : N n → Z satisfying f β,α (0) = 0 and Equation (12) for every l ∈ N n . Observe that b β,α ◦ a β,α adds αk β to l up to multiples of n , and that αk β is prime to n . Hence multipleapplications of b β,α ◦ a β,α takes any element of N n to any other. Since Equation(12) presents f β,α [ b β,α (cid:0) a β,α ( l ) (cid:1)(cid:3) as f β,α ( l ) plus another term, knowing the valueof f β,α on one element of N n determines the values of f β,α on all the elements of N n . Hence the normalization f β,α (0) = 0 determines f β,α uniquely. Note that n applications of b β,α ◦ a β,α takes every l ∈ N n to itself. Applying Equation(12) n times shows that while doing so we add to f β,α ( l ) the values n − − j j ∈ N n . As this sum is n ( n − − n ( n − = 0, theseequalities are consistent with one another, and the function f β,α indeed exists(and is unique).It remains to show that the function f β,α thus obtained satisfies also Equa-tion (11) for all l ∈ N n . First, Equation (11) holds if l is a fixed point of a β,α ,and we claim that a β,α must have at least one fixed point. Indeed, we are look-ing for l ∈ N n such that 2 l ≡ αk β − n ). For odd n such l exists and isunique. On the other hand, if n is even then so is αk β − 1, implying that thereare two such values of l . Assume that Equation (11) holds for some value of l .We claim that Equation (11) holds also for b β,α (cid:0) a β,α ( l ) (cid:1) . To see this, first sub-stitute l = a β,α (cid:0) b β,α ( j ) (cid:1) in Equation (12). Since a β,α and b β,α are involutions,Lemma 2.4 shows that this substitution yields the equality f β,α ( j ) + n − − a β,α ( j ) = f β,α (cid:2) a β,α (cid:0) b β,α ( j ) (cid:1)(cid:3) + a β,α ( j ) . Put now j = a β,α ( l ) and use the involutive property of a β,α again to obtain f β,α (cid:0) a β,α ( l ) (cid:1) + n − − l = f β,α (cid:8) a β,α (cid:2) b β,α (cid:0) a β,α ( l ) (cid:1)(cid:3)(cid:9) + l. Equation (12) and the assumption that Equation (11) holds for l now yieldEquation (11) for b β,α (cid:0) a β,α ( l ) (cid:1) . Since we have shown that Equation (11) holdsfor some l ∈ N n and that multiple applications of b β,α ◦ a β,α connect any twoelements of N n , this completes the proof of the theorem.Observe that altering the constants c ( β, α ) does not affect the invariance ofthe quotient θ en [Ξ](0 , Π) h Ξ under any operator, so one may choose these constantsarbitrarily. It is natural to normalize the constants such that h Ξ is a polynomial(i.e., excluding negative powers) and reduced (i.e., some [ D β,j , D α,l ] appearswith vanishing power). However, determining these constants depends muchmore delicately on the relations between n , α , and β : For example, such anormalizing constant c ( β, β ) depends on the parity of n while f β,β does not (seethe differences between the formulae for odd and even n in Chapters 4 and 5of [FZ]). As another example, if n is odd and αk β is 2 modulo n then the formof these constants depends on whether n is equivalent to 1 or to 3 modulo 4,while the form of the function f β,α does not depend on this congruence (see theexample in Section 6.1 of [FZ]).We now present several lemmas, which are needed to define the denominator h Ξ and to establish its properties. Lemma 4.2. Given three elements α , β , and δ of N n which are all prime to n and two elements l and r of N n , let j ∈ N n be the element which is congruentto l + rαk δ modulo n . Then the congruences a β,α ( j ) ≡ a δ,α ( l ) + a β,δ ( r ) αk δ (mod n ) , b β,α ( j ) ≡ a δ,α ( l ) + b β,δ ( r ) αk δ (mod n ) hold. roof. As in the proof of Proposition 2.5, we take η to be 1 when we work with a and 2 when we work with b . By definition, the left hand side of our expressionsis congruent to ηαk β − − l − rαk δ modulo n , while the right hand side iscongruent to αk δ − − l + αk δ ( ηδk β − − r ) modulo n . The latter expressioncontains − − l − rαk δ , the two terms with αk δ cancel, and the terms including η also coincide since δk δ ≡ n ). This proves the lemma. Lemma 4.3. For every l ∈ N n (given α and δ ) let y α,δ,l denote the number − lδk α − ns δ,lk α ∈ N n . Then the equality f α,δ ( y α,δ,l + nδ l − 1) + l = f α,δ ( y α,δ,l ) + n − − l holds. In this Lemma, δ l denotes Kronecker’s symbol (namely 1 if l = 0 and 0otherwise). It is included here to account for the fact that for y α,δ, = 0 thenumber y α,δ, − − N n but adding n to it yields n − ∈ N n . Inthis case the assertion of Lemma 4.3 reduces to the equality f β,α ( n − 1) = n − Proof. We prove the asserted equality by decreasing induction on l . We beginby observing that y α,δ,n − = a α,δ ( n − 1) while y α,δ,n − − b α,δ (cid:0) a α,δ ( n − (cid:1) (or alternatively, y α,δ,n − = b α,δ (cid:0) a α,δ (0) (cid:1) and y α,δ,n − − a α,δ (0)). Hencethe assertion for l = n − < l ≤ n − 1, and we wish to prove it for l − y α,δ,l − is b α,δ (cid:0) a α,δ ( y α,δ,l ) (cid:1) and y α,δ,l − + nδ l − b α,δ (cid:0) a α,δ ( y α,δ,l − (cid:1) ,Equation (12) shows that the left hand side and right hand side of the equationcorresponding to l − f α,δ ( y α,δ,l − 1) + n − y α,δ,l + l and f α,δ ( y α,δ,l ) + 2 n − − y α,δ,l − l respectively. But these expressions are obtained by adding n − y α,δ,l to bothsides of the equality corresponding to l . Hence if the equality holds for l it alsoholds for l − 1. This completes the proof of the lemma. Lemma 4.4. The equality f δ,α ( l ) = f α,δ ( − lδk α − ns δ, − lk α ) (namely f α,δ ( y α,δ,l ) in the notation of Lemma 4.3) holds for every α and δ and every l ∈ N n .Proof. As both sides attain 0 on l = 0, Theorem 4.1 reduces the assertion toverifying that the function of l given on the right hand side satisfies Equations(11) and (12) with the parameters δ and α . Substituting a δ,α ( l ) in place of l yields an argument of f α,δ which lies between 0 and n − δk α − lδk α modulo n (recall that αk α ≡ δk δ ≡ n )). Since thisnumber is (by definition) the a α,δ -image of − lδk α − ns δ, − lk α , Equation (11) forthe latter function confirms that Equation (11) is satisfied also with the requiredargument. For Equation (12) we consider the right hand side as f α,δ ( y α,δ,l ).Applying b δ,α ◦ a δ,α to l is the same as adding αk δ to it (modulo n ), and aftermultiplying by − δk α the argument of f α,δ becomes y α,δ,l + nδ l − δk δ and αk α are 1 modulo n ). The desired Equation (12) now follows fromLemma 4.3. This proves the lemma. 24he following lemma is not a part of the proof of the Thomae formulae(Theorem 4.6 below), but it will turn out to be useful for deriving explicitexpressions for the functions f β,α in Section 6. Lemma 4.5. The function f n − β,α is related to the function f β,α through theequality f n − β,α ( l ) = 2 l − f β,α ( l ) (holding for all l ∈ N n ).Proof. First we observe that the equalities n − − a n − β,α ( l ) = a β,α ( n − − l )and n − − b n − β,α ( l ) = b β,α ( n − − l ) hold for every α , β , and l ∈ N n . Indeed,all four numbers are in N n , the former two are congruent to αk β + l modulo n ,and the latter two are 2 αk β + l up to multiples of n . Consider now the function ψ n − β,α ( l ) = n − − f β,α ( n − − l ). Equation (11) for f β,α implies f β,α (cid:0) n − − a n − β,α ( l ) (cid:1) = f β,α (cid:0) a β,α ( n − − l ) (cid:1) = f β,α ( n − − l ) , which yields Equation (11) for ψ n − β,α with the parameters n − β and α . Usingthe equalities above and Equation (12) for f β,α we also obtain f β,α (cid:2) n − − b n − β,α (cid:0) a n − β,α ( l ) (cid:1)(cid:3) + n − − l = f β,α (cid:2) b β,α (cid:0) a β,α ( n − − l ) (cid:1)(cid:3) + n − − l == f β,α ( n − − l ) + n − − ( n − − l ) = f β,α ( n − − l ) + l. Subtracting both sides from 2 n − ψ n − β,α with the sameparameters. As f β,α ( n − 1) = n − α and β , we deduce that ψ n − β,α (0) = 0.Hence ψ n − β,α = f n − β,α by Theorem 4.1. As replacing l by n − − l leavesthe expression appearing in Equation (12) invariant, the expression defining ψ n − β,α ( l ) can be written as ψ n − β,α ( l ) = 2 l − f β,α ( l ) for every l ∈ N n . Thisproves the lemma.Fix an order on the set of pairs ( α, l ) with α ∈ N n prime to n and l ∈ Z /n Z .Choose, for every δ and α , an integral constant c ( δ, α ) such that c ( δ, α ) = c ( α, δ )for every α and δ . Define, for any divisor Ξ as in Equation (7), the expression h Ξ = Y ( δ,r ) ≤ ( α,l + rαk δ ) [ D δ,r , D α,l + rαk δ ] en [ c ( δ,α ) − f δ,α ( l )] . The inequality in the product is with respect to the chosen order. We now prove Theorem 4.6. The expression h Ξ is independent of the order chosen. Thequotient θ en [Ξ](0 , Π) h Ξ is invariant under all the operators N β as well as underall the admissible operators T Q,R .Proof. Changing the order means that for some pairs, we write [ D δ,r , D α,l + rαk δ ]as [ D α,s , D δ,j + sδk α ] for appropriate s and j . We need to see that the power towhich this expression appears in h Ξ is the same. But s ≡ l + rαk δ (mod n )and j ≡ r − sδk α (mod n ), so that j ≡ − lδk α (mod n ) since αk α and δk α arecongruent to 1 modulo n . Therefore the powers to which the two forms of thisexpression appear in h Ξ , namely c ( δ, α ) − f δ,α ( l ) and c ( α, δ ) − f α,δ ( j ), coincide25y Lemma 4.4 and the choice of the constants. This proves the independenceof h Ξ of the order chosen on the set of pairs ( α, l ).Proposition 3.4 and Equation (10) reduce the invariance assertions to thestatements that h Ξ is invariant under any operator N β , and for any β thequotient h Ξ g β Ξ is invariant under every admissible operator T Q,R with Q ∈ D β, (for this β ). Decompose h Ξ g β Ξ into the product of expressions involving some set D α,αk β or D α,αk β − and those which do not. The division by g β Ξ affects onlythe powers appearing in the first part in this decomposition. We start with theinvariance under N β , as well as the T Q,R -invariance of the second part of h Ξ g β Ξ (orsimply of h Ξ ). By Equations (8) and (9) this invariance reduces to verifying thattogether with any expression [ D δ,r , D α,j ], the expressions [ D δ,a β,δ ( r ) , D α,a β,α ( j ) ]and [ D δ,b β,δ ( r ) , D α,b β,α ( j ) ] appear to the same power in h Ξ . But if j ≡ l + rαk δ forsome l ∈ N n then the first expression appears to the power en [ c ( δ, α ) − f δ,α ( l )],and Lemma 4.2 implies that the other two expression must then appear to thepower en (cid:2) c ( δ, α ) − f δ,α (cid:0) a δ,α ( l ) (cid:1)(cid:3) . The two invariance assertions now follow fromEquation (11) for f δ,α .It remains to prove the invariance of the first part of h Ξ g β Ξ under the operators T Q,R (with Q ∈ D β, ). We may choose the order such that the expressions weconsider include only powers of [ D δ,δk β , D α,j ] and [ D δ,δk β − , D α,j ] (either with j taking the values αk β or αk β − j taking other values). Since theoperators T Q,R may mix D δ,δk β with D δ,δk β − , Equation (9) shows that for T Q,R -invariance of h Ξ g β Ξ , this quotient must contain all the expressions [ D δ,δk β , D α,j ],[ D δ,δk β − , D α,j ], [ D δ,δk β , D α,b β,α ( j ) ], and [ D δ,δk β − , D α,b β,α ( j ) ] raised to the samepower. Observe that this assertion holds regardless of whether j is congruentto one of αk β and αk β − n or not, since in the former case, whereadditional mixing may appear, b β,α interchanges the elements of N n which arecongruent to αk β and αk β − n with one another. We remark thatin the former case with α = δ the assertion refers to the three expressions[ D δ,δk β , D δ,δk β ], [ D δ,δk β , D δ,δk β − ], and [ D δ,δk β − , D δ,δk β − ].Now, h Ξ is given in terms of [ D δ,r , D α,l + rαk δ ] while g β Ξ is given in termsof [ D δ,r , D α,l ] for r being either δk β or δk β − 1. In the case r = δk β theindex l + rαk δ coincides modulo n with l + αk β (as δ and k δ cancel modulo n ) hence with b β,α (cid:0) a β,α ( l ) (cid:1) . We shall thus express g β Ξ also in terms of this set.In addition, D δ,δk β − is associated in h Ξ with D α,l + αk β − αk δ , which we write as D α,a δ,α ( b δ,α ( l ))+ αk β . By replacing l by b δ,α (cid:0) a δ,α ( l ) (cid:1) in the expressions involving D δ,δk β − in h Ξ we find that the part of h Ξ containing D δ,δk β or D δ,δk β − is Y { ( δ,α,l ) | l = n − ,δ ≤ α if l =0 } [ D δ,δk β , D α,l + αk β ] en [ c ( δ,α ) − f δ,α ( l )] ×× Y { ( δ,α,l ) | δ ≤ α if l = n − } [ D δ,δk β − , D α,l + αk β ] en { c ( δ,α ) − f δ,α [ b δ,α ( a δ,α ( l ))] } (this form is based on an order in which δ ≤ α implies ( δ, δk β ) ≤ ( α, αk β ) and( δ, δk β − ≤ ( α, αk β − 1) and in which ( δ, δk β − < ( α, αk β ) for all α and26 ). On the other hand, Lemma 2.4, the fact that a β,α is an involution, and thecongruence l + αk β ≡ b β,α (cid:0) a β,α ( l ) (cid:1) (mod n ) allow us to write g β Ξ as Y { ( δ,α,l ) | δ ≤ α if l =0 } [ D δ,δk β , D α,l + αk β ] en ( n − − l ) · Y { ( δ,α,l ) | δ ≤ α if l = n − } [ D δ,δk β − , D α,l + αk β ] enl (we can add the condition l = n − h Ξ ). The powers to which [ D δ,δk β , D α,l + αk β ]and [ D δ,δk β − , D α,l + αk β ] appear in the quotient h Ξ g β Ξ are now seen to be en times c ( δ, α ) − f δ,α ( l ) − n + 1 + l and c ( δ, α ) − f δ,α (cid:2) b δ,α (cid:0) a δ,α ( l ) (cid:1)(cid:3) − l respectively. Thesenumbers are equal by Equation (12). Applying b β,α to an element of N n whichis congruent to l + αk β modulo n yields the element of N n which is congruentto n − − l + αk β modulo n . Hence the action of b β,α in this setting takes l to n − − l . The invariance of the number appearing in Equation (12) under thisoperation now completes the proof of the theorem.We remark that the assertion of Theorem 4.6 holds also for divisors Ξ forwhich all the sets D β, are empty. In this case it refers only to the action ofthe operators N β . Moreover, the fact that the power to which an expression[ D δ,j , D α,l ] appears in h Ξ depends only on the index difference between l and j in some sense allows, in any particular case, for a pictorial description of thesepowers, in similarity to Chapters 4 and 5 of [FZ]. Another operation on the divisors Ξ from Equation (7) is related to changing thebase point. For any k ∈ N n (and even k ∈ Z ), we let w k = dzω k = w k Q α,i ( z − λ α,i ) sα,k be the “normalized k th power of w ”. w k is defined for k ∈ Z /n Z and its divisoris Q α Q r α i =1 P αk − ns α ,kα,i / Q nh =1 ∞ t k h (which is trivial if n divides k ). If k is primeto n then this function is the function whose n th power gives a normalized Z n -equation for X with the appropriate generator of C ( X ) over C ( z ). MultiplyingΞ by div ( w k ) yields a divisor with the same ϕ Q -image for any base point Q (byAbel’s Theorem), and the same claim holds for multiplication by div (cid:0) w k p ( z ) (cid:1) forany polynomial in z . For every k ∈ Z we define M k to be the operator whichtakes a divisor Ξ from Equation (7) and multiplies it by div (cid:0) w k p k, Ξ ( z ) (cid:1) , where p k, Ξ ( z ) = Q α Q i ∈ A α,k ( z − λ α,i ). Recall from Corollary 1.4 that A α,k denotesthe set of indices 1 ≤ i ≤ r α such that v P α,i (Ξ) > n − − αk + ns α,k , orequivalently P α,i ∈ D α,l for 0 ≤ l < αk − ns α,k (so that for k divisible by n allthe sets A α,k are empty). Thus, dividing by the divisor of p k, Ξ ( z ) ensures thatall the branch points appear in M k (Ξ) raised to powers from N n . Proposition 5.1. M k defines an operator on the set of divisors Ξ from Equa-tion (7) satisfying the cardinality conditions. Moreover, this operator leaves thequotient θ en [Ξ](0 , Π) h Ξ invariant. roof. By definition, the divisor M k (Ξ) contains no branch point to the power n or higher. Moreover, the k th cardinality condition on Ξ implies that the powersof the points ∞ h arising from w k and from p k, Ξ ( z ) cancel, so that M k (Ξ) alsotakes the form given in Equation (7). Its degree is g + n − 1, since we multipliedΞ by the divisor of a rational function on X , which thus has degree 0. The set D α,l now appears in M k (Ξ) to a power which lies in N n and is congruent to n − − l + αk modulo n . Hence we can write M k (Ξ) in the form of Equation(7) as Q α Q n − l =0 D n − − lα,l + αk . As for the cardinality conditions, we may assumethat n does not divide k (since otherwise M k (Ξ) = Ξ for which we know thatthe assertion holds). To see that M k (Ξ) satisfies the j th cardinality condition,we observe that for α such that αj − ns α,j + αk − ns α,k < n we take thecardinalities of the sets D α,l with αk − ns α,k ≤ l ≤ α ( k + j ) − ns α,k + j − 1, whileif αj − ns α,j + αk − ns α,k ≥ n then we consider the sets with αk − ns α,k ≤ l ≤ n − ≤ l ≤ α ( k + j ) − ns α,k + j − 1. Adding and subtracting P α P αk − ns α,k l =0 | D α,l | (which equals t k ) shows that the sum in question (whichwe need to be t j ) equals t k + j − t k + P { α | αj − ns α,j + αk − ns α,k ≥ n } r α . The fact that s α,k + s α,j is s α,k + j − αj − ns α,j + αk − ns α,k ≥ n and equals s α,k + j otherwiseand the definition of the numbers t k , t j , and t k + j completes the proof of thecardinality conditions for M k (Ξ). Note that the proof works also if n | k + j ,where t k + j = 0 and the sum of r α is taken over all α . This proves the firstassertion.We now turn to the second assertion. Since M k multiplies Ξ by a principaldivisor, M k (Ξ) and Ξ represent the same characteristic for all k and Ξ. Hencewe have to show that h M k (Ξ) = h Ξ . According to the formula for the actionof M k , this assertion is equivalent to the statement that [ D δ,r , D α,j ] appears in h Ξ to the same power as [ D δ,r + δk , D α,j + αk ] for every α , δ , r , and j . Write j as l + rαk δ modulo n , so that the first power is c ( δ, α ) − f δ,α ( l ). The congruence l + ( r + δk ) αk δ ≡ l + rαk δ + αk (mod n ) (since n divides δk δ − 1) shows that[ D δ,r + δk , D α,j + αk ] appears to the same power in h Ξ . This completes the proofof the proposition.We remark that the proof of the cardinality conditions in Proposition 5.1 canbe adapted to provide a direct proof of the appropriate assertions for N β (Ξ) (or N Q (∆)) above, as well as for the images of T Q,R . However, using Theorem 2.3,Proposition 2.5, and Theorem 1.6 we could establish these assertions withoutthe need of direct evaluations.From the formula for M k (Ξ) in the proof of Proposition 5.1, it is clear that M k is the k th power of the operator M = M , and that this operator is oforder n (recall that the index l of D α,l is considered in Z /n Z ). This operator M reduces to the operator denoted M and given explicitly in Propositions 1.14and 1.15 (as well as after Theorem A.2) and implicitly in Propositions 6.10 and6.12 of [FZ] in the appropriate special cases.The proof of Proposition 5.1 shows that rather than divisors of the form ofEquation (7) defining characteristics, M -orbits of such divisors (which are sets of n such divisors) provide better definitions for these characteristics. Moreover,28t follows from the formula for M k (Ξ) that for any branch point Q and any M -orbit, any two divisors in that orbit contain Q raised to different powersin N n . In particular, there is precisely one divisor Ξ in this orbit such that v Q (Ξ) = n − 1, and by Theorem 1.6 this divisor Ξ is of the form Q n − ∆with ∆ non-special. As indicated in Section 2, different non-special divisorsrepresent different characteristics (with a given base point). Hence every M -orbit represents another characteristic. In addition, this establishes the basepoint change formula: Let a non-special divisor ∆ supported on the branchpoints distinct from Q and another base point S be given. The non-specialdegree g divisor Γ not containing S in its support and satisfying the equality ϕ S (Γ) + K S = ϕ Q (∆) + K Q is Υ S n − , where Υ is the unique divisor in the M -orbit of Ξ = Q n − ∆ with v S (Υ) = n − 1. This generalizes Propositions 1.14,1.15, 6.10, and 6.21 of [FZ].We remark at this point that given a base point Q , we considered only thosedivisors from Theorem 1.6 whose support does not contain Q for characteristics.This is required for the theta constant θ [∆ , Q ](0 , Π) not to vanish. It follows thatif a non-special divisor of degree g contains all the branch points in its support,then it cannot represent a non-vanishing theta constant with any branch pointas base point. One such divisor shows up in Theorem 6.3 of [FZ].We present an assertion about the operators which are defined on all thedivisors Ξ. Lemma 5.2. The operator M and the operators N β for the various β all lie inthe same dihedral group G of order n .Proof. Given any β and k , both compositions M k (cid:0) N β (Ξ) (cid:1) and N β (cid:0) M − k (Ξ) (cid:1) yield the divisor Q α,l D n − − lα,α ( k β − k ) − − l . Hence the order 2 operator N β and theelements M k of the cyclic group M Z /n Z satisfy the relation M k ◦ N β = N β ◦ M − k defining the dihedral group of order 2 n . Moreover, replacing β by anotherelement δ ∈ N n which is prime to n and replacing k by j = k + k δ − k β yieldsthe same operator as above (namely M k ◦ N β = N β ◦ M − k coincides with M j ◦ N δ = N δ ◦ M − j ). Hence the dihedral group generated by M and N β isthe same group for every β . This proves the lemma.Apart from the powers of M , the dihedral group G from Lemma 5.2 consistsof the operators taking Ξ from Equation (7) to Q α,l D n − − lα,αk − − l for all k ∈ Z /n Z . Let N be the operator with the simplest choice k = 0, whose action is N (Ξ) = Q α,l D n − − lα,n − − l = Q α,l D lα,l , and consider G as generated by M and N .Moreover, given any β as above, the operator mapping Ξ to Q α,l D n − − lα,b β,α ( l ) liesin G (as N ◦ M k β or as M − k β ◦ N ). Let b T Q,R be the composition of T Q,R with N ◦ M k β = M − k β ◦ N . The operators T Q,R and b T Q,R are admissible on thesame divisors Ξ, but the action of b T Q,R is much simpler: It takes every such Ξto R Ξ /Q . Moreover, since we work with M -orbits rather than divisors, we canphrase the condition for admissibility of T Q,R (or of b T Q,R ) on some divisor Ξas the requirement that if Q ∈ D β,j for some j ∈ Z /n Z then the set containing29 is D γ,γk β ( j +1) . As the action of M k takes D β,j to D β,j − βk and D γ,γk β ( j +1) to D γ,γk β ( j +1) − γk , the congruence γk β ( j + 1) − γk ≡ γk β (cid:0) j − βk + 1 (cid:1) (mod n )shows that this requirement is well-defined on M -orbits. This condition actuallymeans that these operators are admissible on the (unique) divisor Ξ in thisorbit containing Q to the power n − 1. The action of b T Q,R on such divisorsis obtained using conjugation by the appropriate power of M . Explicitly, ittakes Ξ to R Ξ /Q , unless the index j equals n − Q n − Ξ /R n − . The operator b T R,Q is applicable precisely on those divisors whichare images of b T Q,R : Indeed, after this application Q is taken to D β,j +1 and R to D γ,γk β ( j +1) − , and if l = γk β ( j + 1) − βk γ ( l + 1) is congruent to j + 1modulo n . The operator b T R,Q is now seen to be the inverse of b T Q,R .The final step of the proof establishment of the Thomae formulae dependson the following Conjecture 5.3. The action of G and the admissible operators b T Q,R relate anytwo operators Ξ from Equation (7) satisfying the cardinality conditions. We remark that the ordering (4.9) of [FZ] was implicitly based on the factthat given a base point Q and an index γ ∈ N n which is prime to n , one of eachpair of the sets which are mixed by T Q,R is invariant under N β . This happensfor γ = β and for γ = n − β (the cases appearing in Chapters 4 and 5 andAppendix A of [FZ]), but in no other case. The cases studied in Chapter 6of that book considered a small number of divisors with a simple behavior, sothat ad-hoc considerations were sufficient to prove Conjecture 5.3 in these cases.Hence any proof of Conjecture 5.3 in generla must involve new considerations,and cannot resemble any of these special cases. In fact, as some Z n curves donot carry any such divisors Ξ, finding an entirely general argument might bedifficult.We now prove several assertions, which together with the special cases givenin [FZ], support Conjecture 5.3. Fix β ∈ N n which is prime to n , and take j ∈ Z /n Z . If j is neither 0 nor n − n , then consider the differencebetween the cardinality conditions corresponding to jk β and to ( j +1) k β (neitherelements of Z /n Z are 0 by our assumption on j , hence they both yield cardinalityconditions). This difference yields a relation of the form | D β,j | = | D n − β,n − − j | + u j , (13)where u j is t ( j +1) k β − t jk β plus the appropriate difference of the cardinalities ofthe sets D α,l with α different from β and n − β . For j = 0 we get Equation(13) by subtracting r n − β from the k β th cardinality condition (with u being t k β minus the appropriate cardinalities), and for j = n − − k β th cardinalitycondition minus r β yields Equation (13) as well (where u n − involves − t − k β andcertain cardinalities). We remark that this is the form in which the cardinalityconditions for the divisors ∆ in Theorems 2.6, 2.9, 2.13, 2.15, and A.1 of [FZ]are given. We adopt from [FZ] the useful notation ( D α,l ) Ξ for the sets D α,l appearing in Equation (7) for the divisor Ξ. Observe that in contrast to the30umbers t k , the numbers u j may be positive, negative or 0, and they depend onthe divisor Ξ (and on β ). In fact, given a divisor Ξ, the number u j arising fromthe index β is the additive inverse of the number u n − − j arising from n − β .We also remark that an argument similar to the proof of Proposition 5.1 showsthat replacing Ξ by M ik β (Ξ) takes the number u j to u j − i . Proposition 5.4. Let Ξ and Υ be two divisors such that ( D α,l ) Ξ = ( D α,l ) Υ for all l wherever α is neither β nor n − β . Assume that either ( i ) D β,j = ∅ and D n − β,j = ∅ for all j , or ( ii ) there exists some j such that both D β,j and D n − β,n − − j are non-empty. Then the operators b T Q,R , with Q and R being either P β,i or P n − β,i , are sufficient in order to reach from Ξ to Υ .Proof. Note that as the numbers u j depend on the sets D α,l for α being neither β nor n − β , the first hypothesis implies that they coincide for Ξ and Υ. Sincein case ( i ) we have r n − β = 0 and the sets D n − β,j are empty for every divisor,Equation (13) implies | ( D β,j ) Ξ | = | ( D β,j ) Υ | for all j . Moreover, Equation (13)shows that | D β,j | + | D n − β,n − − j | ≥ | u j | , with equality holding if and only if oneof the sets in question is empty. Summing the latter inequality over j ∈ Z /n Z shows that P n − j =0 | u j | ≤ r β + r n − β , and the hypothesis of case ( ii ) is satisfiedprecisely when this inequality is strict (in comparison, summing Equation (13)over j ∈ Z /n Z yields P n − j =0 u j = r β − r n − β ). Hence Ξ and Υ satisfy theassumption of case ( ii ) simultaneously, and we can indeed use this assumptionwithout referring to any of the divisors Ξ and Υ.The first observation we make is that for Q = P β,i lying in some set D β,j and R = P β,m , the operator b T Q,R can act on Ξ if and only if R ∈ D β,j +1 .In this case the operator interchanges these branch points. We can thus alsointerchange the point Q with a point S from D β,j +2 , provided that D β,j +2 isnot empty: Indeed, take R ∈ D β,j +1 , and the combination b T R,S ◦ b T Q,S ◦ b T Q,R is a composition of operators, each one applicable on the divisor on which it issupposed to act, which has the desired effect. Easy induction now shows thatif no set D β,l is empty then we can interchange any two points P β,i and P β,m with one another using these operators, regardless of the sets in which they lie.As in case ( i ) the divisor Υ can be obtained from Ξ by a finite sequence of suchtranspositions, this proves the assertion in this case.On the other hand, if R = P n − β,m (and Q is as above) then the admissibilitycondition is R ∈ D n − β,n − − j . Applying b T Q,R takes Q to D β,j +1 and R to D n − β,n − j . Hence b T Q,R is applicable again, so that we can move Q to D β,j + k and R to D n − β,n − − j + k for any k ∈ Z /n Z . Therefore if D β,j and D n − β,n − − j are both not empty then we can take an arbitrary point from each set and moveit to D β,l and D n − β,n − − l respectively for any l of our choice. We now claim thatalso in this case the operators b T Q,R allow us to interchange any two points P β,i and P β,m with one another. Indeed, assume that one point Q lies in C β,j and theother point S lies in C β,l , assume that both C β,k and C n − β,n − − k are non-empty,and take P ∈ C β,k and R ∈ C n − β,n − − k . Using the operations just described,we can take P and R to D β,j and D n − β,n − − j respectively, then Q and R to D β,l and D n − β,n − − l respectively, followed by transferring S and R to D β,j and31 n − β,n − − j again. Sending P and R back to their original sets completes acombination of admissible operations which acts as the asserted interchange. Ina similar manner we can replace any point from C n − β,j with any point from C n − β,l by admissible operations. Note that under the assumptions of case ( ii ),the divisor Υ can be reached from Ξ by a finite sequence of transfers of pointsfrom D β,j and D n − β,n − − j to D β,l and D n − β,n − − l followed by permutationsof the points P β,i , 1 ≤ i ≤ r β and of the points P n − β,i , 1 ≤ i ≤ r n − β . The proofof the proposition is therefore complete.The validity of Proposition 5.4 extends to divisors Ξ and Υ for which D β,j isempty and D n − β,j is not empty for all j ∈ Z /n Z . This is achieved either usingsymmetry or by a simple adaptation of the proof.We note that by taking β = 1 in a Z n curve of the form considered in SectionA.7 of [FZ], Proposition 5.4 can replace Theorem A.2 of that reference in theproof of the Thomae formulae for these Z n curves. The same statement holdsfor the special cases treated in Theorems 4.8 and 5.7 there. Indeed, if either r or r n − vanish then every two divisors satisfy the assumptions of case ( i )of Proposition 5.4. Otherwise the assumptions of case ( ii ) hold for any Ξ andΥ. The proof here is simpler than those given in [FZ] because of the freedomto move the base point. This removes a basic obstacle in the argument, anddeals with the non-singular case in a unified way, regardless of whether r = 1or r ≥ Lemma 5.5. Let Υ be a divisor satisfying the hypothesis of either case of Propo-sition 5.4, and let Ξ be a divisor satisfying the usual cardinality conditions. Let S be a branch point P γ,m for γ which is neither β nor n − β . Assume that theequality v P δ,i (Ξ) = v P δ,i (Υ) holds for every branch point P δ,i = S in which δ equals neither β nor n − β , and that the difference between v S (Ξ) and v S (Υ) is1. Then one can get from Ξ to Υ using the operators b T Q,R .Proof. Fix the sign of ± according to v S (Υ) = v S (Ξ) ± 1, and choose the element j ∈ Z /n Z such that the set containing S is ( D γ,γk β ( j +1) ) Ξ in case ± = + and is( D γ,γk β j − ) Ξ if ± = − . By denoting the difference | ( D β,i ) Υ | − | ( D n − β,n − − i ) Υ | by b u i we find that b u j = u j − b u j ± = u j ± + 1, and b u k = u k for any other k ∈ Z /n Z . Assume first that Ξ satisfies the hypothesis of case ( ii ) of Proposition5.4. Then the proof of this proposition shows that there exist points Q = P β,i and R = P n − β,p such that an appropriate power of b T Q,R takes Ξ to a divisor Σsuch that ( D β,j ) Σ contains Q . Otherwise the cardinalities of the sets ( D β,k ) Ξ (as well as ( D n − β,k ) Ξ ) are determined by the cardinalities of the other setscorresponding to Ξ. Indeed, the proof of Proposition 5.4 shows that in this casethe equality P j | u j | = r β + r n − β holds and at least one of D β,j and D n − β,n − − j is empty for all j ∈ Z /n Z . The cardinalities are thus determined by Equation(13). We claim that ( D β,j ) Ξ = ∅ in this case. Indeed, if D n − β,k = ∅ for all k (Υ satisfying the hypothesis of case ( i ) of Proposition 5.4) then b u j ≥ u j ≥ 1. On the other hand, the proof of Proposition 5.4 implies that the only32ase in which Υ satisfies the hypothesis of case ( ii ) of that proposition but Ξdoes not is when P k | u k | = r β + r n − β and P k | b u k | < r β + r n − β . Under the givenrelations between the u k s and the b u k s this can happen only if u j ± < < u j ,so that in particular ( D β,j ) Ξ = ∅ . Take Q ∈ ( D β,j ) Ξ , and choose Σ = Ξ inthis case. Now, the location of Q and S implies that the operator b T Q,S in case ± = + and b T S,Q if ± = − is admissible on Σ. Moreover, Υ and the image of Σunder this operator satisfy the hypothesis of Proposition 5.4. An application ofthe aforementioned proposition now completes the proof.Proposition 5.4 and Lemma 5.5 suggest another example for the validity ofConjecture 5.3. Consider a Z n curve X for which there is some β such thatone of the following is satisfied: ( i ) r β is much larger than r α for other α , or( ii ) r β + r n − β is a sum of positive integers which is relatively large. Then,Conjecture 5.3 holds for X , at least heuristically. This is so, since most divisorswill satisfy the assumptions of Proposition 5.4, hence can be related to many“neighboring” divisors.Combining Theorem 4.6 and Proposition 5.1 yields the main result of thispaper, which is the Thomae formulae for the general fully ramified Z n curve X : Theorem 5.6. Assume that X satisfies Conjecture 5.3. Then the value ofthe quotient θ en [Ξ](0 , Π) h Ξ from Theorem 4.6 is independent of the choice of thedivisor Ξ . Following [FZ] we would like to relate the characteristics to those arising fromthe non-special divisors ∆ from Theorem 1.6 with the choice of some branchpoint Q as base point. For every such Q and ∆ we define the denominator h Q ∆ to be h Ξ for Ξ = Q n − ∆. Theorem 5.6 then implies that if Conjecture 5.3 holdsfor the Z n curve X then the quotient θ en [ Q, ∆](0 , Π) h Q ∆ is independent of the choiceof the divisor ∆. Moreover, this quotient yields the same constant for everybase point Q . The formulation of Theorem 5.6 as appears here corresponds tothe “symmetric” Thomae formulae in [FZ]. The fact that [K] finds a constantfor every such Z n curve also suggests that Conjecture 5.3 might be true. f β,α In this Section we investigate the functions f β,α further, and obtain explicitexpressions to evaluate them in some cases. First observe that f β,α dependsonly on the number αk β modulo n (because a β,α and b β,α depend only on thisnumber). Fix d ∈ N n which is prime to n , and consider the functions f β,α forwhich α ≡ dβ (mod n ). Theorem 4.1 implies that all these functions coincide toa unique function, which we denote f ( n ) d . Lemma 4.5 expresses f ( n ) n − d in terms of f ( n ) d , so that we can restrict attention to those d ∈ N n satisfying d < n . More-over, Lemma 4.4 relates f ( n ) d to f ( n ) k d (recall that k d ∈ Z satisfies dk d ≡ n ),but here we do require k d ∈ N n ), or in fact shows that in the expression for h Ξ d = 1 and for d = n − k d = d (regardless of n ). Lemma 4.4 reduces toEquation (11) for d = 1 (since a α,α ( l ) = n − l − nδ l ) and is trivial for d = n − c ( δ, α ) such thatthey depend only on the number d ∈ N n satisfying α ≡ dδ (mod n ). Denot-ing the appropriate constant c ( n ) d , we must have c ( n ) d = c ( n ) k d by the assumptionmade when we defined h Ξ . The normalization c ( δ, α ) = max l ∈ N n f δ,α ( l ) satis-fies the first condition (so that c ( n ) d = max l ∈ N n f ( n ) d ( l )), and Lemma 4.4 showsthat it satisfies the second condition as well. These considerations allow us, insome cases, to write the full expression for h Ξ explicitly using just a few of thefunctions f ( n ) d .Now fix n and d ∈ N n which is prime to n . We begin our analysis of thefunctions f ( n ) d with Proposition 6.1. Assume that l ∈ N n equals pd + q with some q ∈ N d . Then f ( n ) d ( l ) is related to f ( n ) d ( q ) through the formula stating that f ( n ) d ( l ) equals f ( n ) d ( q ) + p ( n + d − − q − l ) = l ( n + d − − l ) d + f ( n ) d ( q ) − q ( n + d − − q ) d . Proof. Equation (12) translates to the equality f ( n ) d ( j + d ) = f ( n ) d ( j ) + n − − j for all j ∈ N n such that j + d < n . By applying this for j = di + q for all i ∈ N p (with j = q for i = 0 and j + d = l for i = p − 1) we obtain f ( n ) d ( l ) = f ( n ) d ( q ) + p − X i =0 [ n − − di + q )] = f ( n ) d ( q ) + p ( n − − q ) − dp ( p − 1) == f ( n ) d ( q ) + p ( n + d − − q − dp ) = f ( n ) d ( q ) + l − qd ( n + d − − q − l )(recall that l = pd + q hence p = l − qd ). This gives the first expression, and thesecond follows from simple arithmetics. This proves the proposition.Proposition 6.1 shows that knowing f ( n ) d ( q ) only for q ∈ N d is sufficient forevaluating f ( n ) d ( l ) for all l ∈ N n . It turns out useful to write n = sd + t forsome t ∈ N d in what follows. We now derive a few properties of the expression f ( n ) d ( q ) − q ( n + d − − q ) d with q ∈ N d appearing in Proposition 6.1. It turns outuseful to multiply this expression by − d , and call the result g ( d ) t ( q ) (this is anabuse of notation at this point, since we do not yet know that the value of g ( d ) t ( q )depends only on t and not on n = sd + t , but we use it nontheless). Recall thatEquation (11) for f ( n ) d compares the value which this function attains on q ∈ N d with the value it takes on d − − q , as well as the image of some d ≤ l ∈ N n under this function with the image of n + d − − l . The following Lemma givesa similar assertion for g ( d ) t ( q ): 34 emma 6.2. The expression g ( d ) t ( q ) is invariant under replacing q ∈ N t by t − − q and t ≤ q ∈ N d by d + t − − q .Proof. Express f ( n ) d ( l ) for l ≥ d in terms of the formula from Proposition 6.1,and compare it to f ( n ) d ( n + d − − l ). These values coincide by Equation (11), asremarked above. The terms involving l (but not the residue modulo d ) coincide,so that the remaining term (which is − g ( d ) t ( q ) /d by definition) must also givethe same value for q and for the residue of n + d − − l modulo d . As n = sd + t ,this residue is t − − q if q ∈ N t and equals d + t − − q for q ≥ t . This provesthe lemma.We remark that the proof of Lemma 6.2 implicitly uses the assumption s ≥ d < n ), since it requires the existence of d ≤ l ∈ N n which is congruentto q modulo d for every q ∈ N d . However, its assertion is true in general—seethe remark after Theorem 6.4 below. Lemma 6.3. ( i ) The equality f ( n ) d ( q + t ) = f ( n ) d ( q ) − s ( d − t − − q ) holdsfor every q ∈ N d − t . ( ii ) For q ∈ N d satisfying q ≥ d − t we have the equality f ( n ) d ( q + t − d ) = f ( n ) d ( q ) − ( s + 1)(2 d − t − − q ) .Proof. Set l = ( s − d + q + t = n − d + q ∈ N n in Equation (12), which nowtakes the form f ( l + d − n ) = f ( l ) + n − − l since l + d ≥ n . The left handside is just f ( q ), while for the right hand side we use the first expression fromProposition 6.1. Recall also that l = n − d + q and n = sd + t . In case ( i ) wehave p = s − q + t , hence the right hand side becomes f ( n ) d ( q + t )+( s − d − − q − t ) − − n +2 d − q = f ( n ) d ( q + t )+ s ( d − t − q − . This proves part ( i ). On the other hand, in case ( ii ) the parameter p is s andthe residue is q + t − d . The right hand side thus takes the form f ( n ) d ( q + t − d )+ s (3 d − − q − t ) − − n +2 d − q = f ( n ) d ( q + t − d )+( s +1)(2 d − − q − t )and part ( ii ) is also established. This proves the lemma.We now prove that the functions f ( n ) d can be evaluated using a recursiveprocess, based on Euclid’s algorithm for finding greatest common divisors usingdivision with residue: Theorem 6.4. If n = sd + t and l = pd + q for t and q in N d then we canwrite f ( n ) d ( l ) as l ( n + d − − l ) d − nd f ( d ) t ( q ) . The value of f ( n ) n − d ( l ) can be written as nd f ( d ) t ( q ) − l ( n − d − − l ) d .Proof. By Proposition 6.1, the first assertion boils down to the statement that g ( d ) t ( q ) = nf ( d ) t ( q ) for every q ∈ N d . Lemma 6.2 shows that g ( d ) t satisfies Equa-tion (11) for d and t . We wish to prove that it satisfies also the appropriateEquation (12), namely that g ( d ) t ( q ) + n ( d − − q ) gives us g ( d ) t ( q + t ) if q ∈ N d − t g ( d ) t ( q + t − d ) if q ≥ d − t . Recall that s = n − td . Take q ∈ N d − t and apply part( i ) of Lemma 6.3. This evaluates g ( d ) t ( q + t ) = ( q + t )( n + d − − q − t ) − df ( n ) d ( q + t )as q ( n + d − − q ) + t ( n + d − − q − t ) − tq − df ( n ) d ( q ) + ( n − t )( d − t − − q ) , which reduces to the asserted value g ( d ) t ( q ) + n ( d − − q ). For q ≥ d − t weuse part ( ii ) of Lemma 6.3 and write s + 1 = n + d − td in order to find that thedifference between g ( d ) t ( q + t − d ) = ( q + t − d )( n + 2 d − − q − t ) − df ( n ) d ( q + t − d )and g ( d ) t ( q ) = q ( n + d − − q ) − df ( n ) d ( q ) is( n + d − t )(2 d − t − − q ) − ( d − t )( n + 2 d − − q − t ) + q ( d − t ) = n ( d − − q ) , as desired. Since g ( d ) t (0) clearly vanishes, the desired equality g ( d ) t ( q ) = nf ( d ) t ( q )for all q ∈ N d follows from Theorem 4.1, which proves the formula for f ( n ) d . Theresult for f ( n ) n − d now follows from Lemma 4.5. This proves the theorem.The proof of Theorem 6.4 justifies the notation g ( d ) t a fortiori, since it indeeddepends only on the residue t of n modulo d . We remark that as the proof ofTheorem 4.1 requires only vanishing at 0 and Equation (12), Lemma 6.2 is notnecessary for the proof of Theorem 6.4. Hence Theorem 6.4 holds for every n and d (without the assumption d < n ), and Lemma 6.2 (for all n and d ) followsas a corollary of its proof. Lemma 6.2 in fact implies that the formula for f ( n ) d ( l )takes the same form for l ≡ q (mod d ) and for l which is equivalent to t − − q or to d + t − − q modulo d .For any y ∈ Z , the expression l ( n + d − − l ) − ny can be written as( l − y )( n + d − − y − l ) − y ( y − d + 1). Since for y = f ( d ) t ( q ) the number d − − y is f ( d ) d − t ( d − − q ), we can write f ( n ) d ( l ) = ( l − f ( d ) t ( q ))( n + f ( d ) d − t ( d − − q ) − l ) + f ( d ) t ( q ) f ( d ) d − t ( d − − q ) d . The latter formula presents an interesting symmetry between the formula for f ( n ) d ( l ) with n = sd + t and l = pd + q and the one for f ( m ) d ( j ) with m = rd − t and j = kd − − q . In particular, if f ( d ) t ( q ) = d − f ( n ) d ( l ) given in Theorem 6.4 becomes just ( l − d +1)( n − l ) d . Thus, Theorem 6.4 (orProposition 6.1) and Lemma 6.2 combine with Lemma 4.5 to give Corollary 6.5. If l is divisible by d , or if l is congruent to t − modulo d ,then f ( n ) d ( l ) = l ( n + d − − l ) d and f ( n ) n − d ( l ) = − l ( n − d − − l ) d . If d divides l + 1 , or if l ≡ t (mod d ) , then f ( n ) d ( l ) = ( l − d +1)( n − l ) d and f ( n ) n − d ( l ) equals ( d − n − l ( n − d − − l ) d ,or equivalently d − − ( l − d +1)( n − d − l ) d . f ( n ) d for some values of d . For d = 1 all the assertions of that Corollary yield the formulae f ( n )1 ( l ) = l ( n − l ) and f ( n ) n − ( l ) = − l ( n − − l ) for all l ∈ N . Equation (11) for these cases correspond tothe fact that the value of f ( n )1 is invariant under sending l to n − l − nδ l and f ( n ) n − attains the same value on l and on n − − l + nδ l,n − . Observe that for even n thefunction f ( n )1 attains its maximal value n at l = n , while for odd n the maximalvalue is n − , attained on l = n − and on l = n +12 . Hence the expression for thefunction f ( n )1 does not depend on the parity of n , but its normalizing constant c ( n )1 does depend on the parity of n . This is the reason for the different formulaefor odd and even n given in Chapter 4 of [FZ]. f ( n ) n − attains its maximal value n − l = n − n ), and n − − f ( n ) n − ( l ) equals( l + 1)( n − − l ) for l ∈ N n . Note that the expression [ C i , D i + k ] from Chapter5 of [FZ] becomes [ D ,i +1 , D n − ,n − − i − k ] in our notation, and the second indexof the latter set is i + 1 times αk β ≡ − n ) plus n − − k . Since k ( n − k )agrees with our n − − f ( n ) n − ( l ) for l = n − − k , we verify the results of thisChapter (and of Section A.7 there) as well.Corollary 6.5 also yields the full formula for the case d = 2 (for odd n ): f ( n )2 takes even l ∈ N to l ( n +1 − l )2 and odd l ∈ N to ( l − n − l )2 , while f ( n ) n − ( l ) is − l ( n − − l )2 if l is even and equals 2 − ( l − n − − l )2 if l is odd. Equation (11) issatisfies since the involution for d = 2 takes l ≥ n + 1 − l (and interchanges0 and 1) while the one with d = n − l ≤ n − n − − l (andinterchanges n − n − f ( n )2 ( l ) is obtained foreven l , and depends on whether n is equivalent to 1 or to 3 modulo 4: It equals n +2 n − (for l being n − or n +32 ) in the former case and n +2 n +18 (for l = n +12 )in the latter case. f ( n ) n − attains its maximal value n − l = n − l = n − n , and the difference n − − f ( n ) n − ( l ) equals ( l +2)( n − − l )2 if l is even and ( l +1)( n − − l )2 if l is odd. Lemma 4.4 now shows that f ( n ) n +12 takes l ∈ N n +12 to l ( n − − l ) and maps n +12 ≤ l ∈ N n to ( n − l )(2 l + 1 − n ), whilethe function f ( n ) n − sends l ∈ N n +12 to − l ( n − − l ) and takes n +12 ≤ l ∈ N n to 2 − ( n − − l )(2 l − − n ). These values suffice to determine h Ξ in the firstexample in Section 7, and in fact we can use the values of f ( n )2 and f ( n ) n − alonefor this purpose.For d = 3 we encounter the dependence of the form of f ( n ) d on the residue t of n modulo d . Corollary 6.5 again gives us the full answer: If t = 1 then f ( n )3 ( l )is l ( n +2 − l )3 and f ( n ) n − ( l ) equals − l ( n − − l )3 if 3 divides l , while otherwise f ( n )3 and f ( n ) n − attain ( l − n − l )3 and 4 − ( l − n − − l )3 on l respectively. On the other hand,for t = 2 we find that f ( n )3 ( l ) = l ( n +2 − l )3 and f ( n ) n − = − l ( n − − l )3 if l is congruentto 0 or 1 modulo 3, while the values ( l − n − l )3 and 4 − ( l − n − − l )3 are attainedonly on l ≡ t as well as on the parity of n . For t = 2 the maximal value of f ( n )3 is n +4 n +312 for odd n (with l being n +12 or n +32 ) and n +4 n (arising from l = n or from l = n +42 ) if n is even. f ( n ) n − attains its maximal value n − l = n − l = n − t = 1 then themaximal value which f ( n )3 attains is n +4 n +412 (on l = n +22 ) if n is even and is n +4 n − for odd n (with the value of l being either n − or n +52 ). The function f ( n ) n − then attains n − s = 4 s on l = 1 (and no larger values). The functionsfor which we can now apply Lemma 4.4 depend themselves on the value of t : k is s + 1 if t = 2 and is 2 s + 1 if t = 1, while k n − equals s for t = 1 and is 2 s + 1for t = 2. Hence we shall not write the formulae for these functions, but onlymention that their value on l depends on whether 0 ≤ l ≤ s , s +1 ≤ l ≤ n − s − n − s ≤ l ≤ n − 1. Moreover, the formulae for these functions take the sameform for t = 1 and for t = 2 except for l in the middle interval. In any case, thevalues of f ( n )3 and f ( n ) n − are sufficient to determine h Ξ explicitly in the secondexample below.To give the flavor of how the functions f ( n ) d look like for larger values of d , weuse our knowledge of the function f ( d )1 and f ( d ) d − from above in order to extractfrom Theorem 6.4 and Lemma 4.5 the following Corollary 6.6. In the case t = 1 (which means that d divides n − ) f ( n ) d ( l ) equals l ( n + d − − l ) − nq ( d − q ) d , or alternatively [ l − q ( d − q )][ n − ( q − d − − q ) − l ] − q ( q − d − q )( d − − q ) d , for every l ∈ N n which is congruent to q ∈ N n modulo d . The function f ( n ) n − d attains on such l the value nq ( d − q ) − l ( n − d − − l ) d , which can also be written as q ( q + 1)( d − q )( d + 1 − q ) − [ l − q ( d − q )][ n − ( q + 1)( d + 1 − q ) − l ] d . For t = d − (namely, if d divides n + 1 ) and l ∈ N n of the form pd + q we findthat f ( n ) d ( l ) is l ( n + d − − l )+ nq ( d − − q ) d , namely [ l + q ( d − − q )][ n + ( q + 1)( d − − q ) − l ] − q ( q + 1)( d − − q )( d − − q ) d . An element l ∈ N n of this form is taken by the function f ( n ) n − d to the number − nq ( d − − q ) − l ( n − d − − l ) d , which also equals q ( q +1)( d − − q )( d − − q ) − [ l + q ( d − − q )][ n +( q − d − − q ) − − l ] d − q ( d − − q ) . f ( n )2 , f ( n ) n − , f ( n )3 , and f ( n ) n − . It alsogives the formula for f ( n )4 and f ( n ) n − for all odd n and for f ( n )6 and f ( n ) n − for everyodd n which is not divisible by 3, since for both d = 4 and d = 6 (as well asfor d = 3 considered above) the only two elements of N d which are prime to d are 1 and d − 1. For example, if n = 4 s + 1 then f ( n )4 takes l which is divisibleby 4 to l ( n +3 − l )4 , other even l to ( l − n − − l )4 − 1, and odd l to ( l − n − l )4 . Inthis case if 4 divides l then f ( n ) n − ( l ) = − l ( n − − l )4 , the f ( n ) n − -image of other even l is 9 − ( l − n − − l )4 , and f ( n ) n − sends any odd l to 6 − ( l − n − − l )4 . On theother hand, for n = 4 s + 3 the function f ( n )4 takes any even l to l ( n +3 − l )4 , l which is congruent to 3 modulo 4 to ( l − n − l )4 , and l satisfying l ≡ ( l +1)( n +4 − l )4 − 1. The function f ( n ) n − here sends even l to − l ( n − − l )4 , while l whichsatisfies l ≡ − ( l − n − − l )4 and l which is congruent to 1modulo 4 is sent to − ( l +1)( n − − l )4 − 1. For d = 5 the formulae are based on thevarious functions f (5) t for 0 = t ∈ N . The fact that f (5)1 takes the numbers 0,1, 2, 3, and 4 to 0, 4, 6, 6, and 4 respectively and f (5)4 sends them to 0, − − d = 5 of Corollary 6.6. The values of f (5)2 are 0, 0, 4, 2, and 4 respectively, while f (5)3 takes the values 0, 2, 0, 4, and 4respectively. The values for f ( n )5 ( l ) and f ( n ) n − ( l ) thus agree with the values givenin Corollary 6.5. The additional values which we obtain are for l ≡ n = 5 s + 2 and for l ≡ n = 5 s + 3, where f ( n )5 ( l ) and f ( n ) n − ( l ) equal ( l − n +2 − l )+45 and − ( l − n − − l )5 respectively. In this Section we give examples of Thomae formulae for two families of Z n curves, and discuss the divisors on Z n curves belonging to a third family.We begin with the Z n curves associated with equations of the form w n = r Y i =1 ( z − λ i ) p Y i =1 ( z − σ i ) q Y i =1 ( z − τ i ) n − m Y i =1 ( z − µ i ) n − for odd n , where r + 2 p − q − m is divisible by n hence equals nu for some u ∈ Z . The number t k is ku + q + m if k ≤ n − = s and equals ks − p + 2 q + m if k ≥ n +12 = n − s . The only non-trivial index d which we shall need here is d = 2,so that in n = sd + t we have t = 1 and s = n − . The expressions using s aregiven here in order to emphasize the similarity with the results of the followingexample. We switch to a notation similar to that of [FZ], so that the branchpoints over λ i , µ i , σ i , and τ i are denoted P i , Q i , R i , and S i respectively. Adivisor Ξ in Equation (7) takes the form Q n − l =0 C n − − ll D ll E n − − ll F ll , where C l contains points of the sort P i , D l contains points of the type Q i , E l containspoints of the type R i , and F l contains points of the sort S i . Hence C l stands39or D ,l , D l is the set D n − ,n − − l , E l represents D ,l , and F l is our notation for D n − ,n − − l . Subtracting m = P l | D l | and q = P l | F l | from the first cardinalitycondition in Theorem 1.6 and subtracting the k th cardinality condition from the( k + 1)st condition yields the equality | C k | + | E k | + | E k +1 | = | D k | + | F k | + | F k +1 | + u for all 0 ≤ k ≤ n − = s − 1. A similar argument gives | C k | + | E k − n | + | E k +1 − n | = | D k | + | F k − n | + | F k +1 − n | + u for all n − s = n +12 ≤ k ≤ n − 2, while subtracting m = P l | C l | and p = P l | E l | from the ( n − nu , the same equality alsofor k = n − 1. Finally, the difference between the s th and ( s − | C n − | + | E | + | E n − | = | D n − | + | F | + | F n − | + u, corresponding to the remaining index k = n − = s = n − s − 1. Indeed the lefthand sides sum to r + 2 p and the sum of the right hand sides is nu + 2 q + m ,in agreement with the value of u .In order to evaluate h Ξ , we define ε to be 1 if n ≡ n ≡ f ( n )2 ( l ) and f ( n ) n − ( l ) depend on the parityof l , we separate the corresponding products into the product over l = 2 k with0 ≤ k ≤ n − = s and the product over l = 2 k +1 for 0 ≤ k ≤ n − = s − 1. Takingthe notational differences, the index shifts, Lemma 4.5, and the symmetry of f ( n )2 under l n + 1 − l into consideration, we obtain h Ξ = n − Y r =0 " n − − r Y l =0 (cid:0) [ C r , C r + l ][ D r , D r + l ][ E r , E r + l ][ F r , F r + l ] (cid:1) n [( n − / − l ( n − l )] ×× n − − r Y l =0 (cid:0) [ C r , D r + l ][ D r , C r + l ][ E r , F r + l ][ F r , E r + l ] (cid:1) nl ( n − l ) ×× s Y k =0 (cid:0) [ C r , E r +2 k − ρn ][ D r , F r +2 k − ρn ] (cid:1) n [( n +2 n +1 − ε ) / − k ( n +1 − k )] ×× s − Y k =0 (cid:0) [ C r , E r +2 k +1 − ρn ][ D r , F r +2 k +1 − ρn ] (cid:1) n [( n +2 n +1 − ε ) / − k ( n − − k )] ×× s Y k =0 (cid:0) [ C r , F r +2 k − ρn ][ D r , E r +2 k − ρn ] (cid:1) nk ( n +1 − k ) ×× s − Y k =0 (cid:0) [ C r , F r +2 k +1 − ρn ][ D r , E r +2 k +1 − ρn ] (cid:1) nk ( n − − k ) . ρ is the appropriate multiple of n which one needs for the resulting indexto be in N n .Take r = p = q = m = 1 (hence u = 0). Considering only divisors Ξ forwhich v Q (Ξ) = n − Q (i.e., Ξ = Q n − ∆ for some base point Q and divisor ∆ of degree g = n − | C l | = | D l | and | E l | = | F l | for all l ∈ N n ,representing the divisors P n − R n − − j S j , Q n − R n − − j S j , R n − P n − − j Q j , or S n − P n − − j Q j , with j ∈ N n (assuming the existence of order n − | C | = 1 these other solutions are with F , E n − and D n − being thenon-empty sets (and Ξ = P n − Q n − ) or with the points being in F , E and D (and Ξ is P n − Q R n − S ). For | D n − | = 1 we obtain again the solutionin which E n − , F , and C are non-empty and Ξ = P n − Q n − , and anothersolution is with the non-trivial sets E n − , F n − , and C n − and the divisor Q n − P R S n − . If | E | = 1 then we obtain one possibility in which F n − , D ,and C n − are the non-trivial sets (hence Ξ is R n − S n − ) and another one withnon-trivial F , D n − , and C n +12 and divisor R n − P n − Q n − S . Finally, bytaking | F n − | = 1 we get again the solution in which E , C n − , and D containpoints and Ξ = R n − S n − , as well as the solution in which the points lie in E n − , C n − , and D n − , representing the divisor S n − P n − Q n − R . This givesus all the possibilities of Ξ = Q n − ∆ with some branch point Q and divisor∆ from Theorem 6.3 of [FZ] not containing Q . Observe that the remainingpossibility for ∆ in that theorem, namely P n − Q n − R S , does not representany theta constant since θ [ Q, ∆](0 , Π) = 0 with this ∆ for every possible Q .We now consider the form which h Ξ takes in this case. If Ξ = P n − R n − − l S l or Ξ = Q n − R l S n − − l then h Ξ becomes[( λ − σ )( µ − τ )] n [( n +2 n +1 − ε ) / − l ( n +1 − l ) / [( λ − τ )( µ − σ )] nl ( n +1 − l ) / for even l and[( λ − σ )( µ − τ )] n [( n +2 n +1 − ε ) / − ( l − n − l ) / [( λ − τ )( µ − σ )] n ( l − n − l ) / if l is odd (the contribution of the first two lines is an empty product). In caseΞ takes the form R n − P n − − l Q l or S n − P l Q n − − l the denominator h Ξ takesthe form[( λ − σ )( µ − τ )] n [( n +2 n +1 − ε ) / − l ( n − − l )] [( λ − τ )( µ − σ )] nl ( n − − l ) if l satisfies 0 ≤ l ≤ n − and[( λ − σ )( µ − τ )] n [( n +2 n +1 − ε ) / − ( n − l )(2 l +1 − n )] [( λ − τ )( µ − σ )] n ( n − l )(2 l +1 − n ) for n +12 ≤ l ≤ n − P n − Q n − , R n − S n − ,41 n − Q R n − S , Q n − P R S n − , R n − P n − Q n − S , and S n − P n − Q n − R called “of second type” above, the associated denominator h Ξ equals[( λ − σ )( µ − τ )] n ( n − n +9 − ε ) / [( λ − µ )( σ − τ )] n ( n − . Stating these results with the notation and index conventions of [FZ], theseformulae reproduce the results of Section 6.1 and Appendix B of that reference.Take now n which is not divisible by 3, and write n = 3 s + t . As a secondexample we consider the family of Z n curves whose defining equation is w n = r Y i =1 ( z − λ i ) p Y i =1 ( z − σ i ) q Y i =1 ( z − τ i ) n − m Y i =1 ( z − µ i ) n − , where n divides r + 3 p − q − m with quotient u . The points P i , Q i , R i ,and S i have the same meaning as above, and the same statement holds forthe sets C l , D l , E l , and F l . Thus the divisor Ξ from Equation (7) again be-comes Q n − l =0 C n − − ll D ll E n − − ll F ll , but now E l stands for D ,l and F l denotes D n − ,n − − l . Repeating the operations of the previous example on the cardinal-ity conditions from Theorem 1.6 now yields the equalities | C k | + | E k | + | E k +1 | + | E k +2 | = | D k | + | F k | + | F k +1 | + | F k +2 | + u for every 0 ≤ k ≤ s − | C k | + | E k − n | + | E k +1 − n | + | E k +2 − n | = | D k | + | F k − n | + | F k +1 − n | + | F k +2 − n | + u if s + 1 ≤ k ≤ n − s − 2, and | C k | + | E k − n | + | E k +1 − n | + | E k +2 − n | = | D k | + | F k − n | + | F k +1 − n | + | F k +2 − n | + u in case n − s ≤ k ≤ n − 1. For the values k = s and k = n − s − t = 1 or t = 2. If t = 1 then the equalities become | C s | + | E | + | E | + | E n − | = | D s | + | F | + | F | + | F n − | + u and | C n − s − | + | E | + | E n − | + | E n − | = | D n − s − | + | F | + | F n − | + | F n − | + u, while for t = 2 we get | C s | + | E | + | E n − | + | E n − | = | D s | + | F | + | F n − | + | F n − | + u and | C n − s − | + | E | + | E | + | E n − | = | D n − s − | + | F | + | F | + | F n − | + u. In any case, summing all the equalities yields r + 3 p = m + 3 q + nu , as indeedfollows from the current definition of u .42n this case n can be either odd or even, and the maximal value of f ( n )1 canbe written uniformly as n +1 − e for both cases (recall that e is 1 for even n and2 for odd n ). The maximal value of f ( n )3 can be written uniformly in terms of t and e , but as the expression for h Ξ depend on t in other terms as well, weshall avoid doing so. In the expressions involving f ( n )3 or f ( n ) n − we separate theproduct into l = 3 k for 0 ≤ k ≤ s , l = 3 k + 1 for k between 0 and s + t − k = s if t = 2 since then 3 s + 1 = n − ∈ N n , but excludesthis value of k if t = 1 as 3 s + 1 = n no longer lies in N n in this case), and l = 3 k + 2 for 0 ≤ k ≤ s − 1. The maximal value of f ( n )3 is n +4 n +4 − e − if t = 1 and n +4 n +3( e − if t = 2. Using Lemma 4.5 we can write c ( n ) n − − f ( n ) n − ( l )as f ( n )3 ( n − − l ) + s if t = 1 and as f ( n )3 ( n − − l ) when t = 2. The valueof f ( n )3 (3 k + 1) can be written as k ( n − k ) − n − (i.e., k ( n − k ) − s ) if t = 1and as k ( n − k ) + n +13 (namely k ( n − k ) + s + 1) when t = 2. The constant n ± is absorbed into the preceding coefficients in the powers appearing in thecorresponding line. In total, h Ξ is given by n − Y r =0 " n − − r Y l =0 (cid:0) [ C r , C r + l ][ D r , D r + l ][ E r , E r + l ][ F r , F r + l ] (cid:1) en [( n +1 − e ) / − l ( n − l )] ×× n − − r Y l =0 (cid:0) [ C r , D r + l ][ D r , C r + l ][ E r , F r + l ][ F r , E r + l ] (cid:1) enl ( n − l ) ×× s Y k =0 (cid:0) [ C r , E r +3 k − ρn ][ D r , F r +3 k − ρn ] (cid:1) en [( n +4 n +13 − e ) / − k ( n +2 − k )] ×× s − Y k =0 (cid:0) [ C r , E r +3 k +1 − ρn ][ D r , F r +3 k +1 − ρn ] (cid:1) en [( n +8 n +9 − e ) / − k ( n − k )] ×× s − Y k =0 (cid:0) [ C r , E r +3 k +2 − ρn ][ D r , F r +3 k +2 − ρn ] (cid:1) en [( n +4 n +13 − e ) / − k ( n − − k )] ×× s Y k =0 (cid:0) [ C r , F r +3 k − ρn ][ D r , E r +3 k − ρn ] (cid:1) en [ k ( n +2 − k )+( n − / ×× s − Y k =0 (cid:0) [ C r , F r +3 k +1 − ρn ][ D r , E r +3 k +1 − ρn ] (cid:1) enk ( n − k ) × s − Y k =0 (cid:0) [ C r , F r +3 k +2 − ρn ][ D r , E r +3 k +2 − ρn ] (cid:1) en [ k ( n − − k )+( n − / for t = 1 and n − Y r =0 " n − − r Y l =0 (cid:0) [ C r , C r + l ][ D r , D r + l ][ E r , E r + l ][ F r , F r + l ] (cid:1) en [( n +1 − e ) / − l ( n − l )] × n − − r Y l =0 (cid:0) [ C r , D r + l ][ D r , C r + l ][ E r , F r + l ][ F r , E r + l ] (cid:1) enl ( n − l ) ×× s Y k =0 (cid:0) [ C r , E r +3 k − ρn ][ D r , F r +3 k − ρn ] (cid:1) en [( n +4 n +3 e − / − k ( n +2 − k )] ×× s − Y k =0 (cid:0) [ C r , E r +3 k +1 − ρn ][ D r , F r +3 k +1 − ρn ] (cid:1) en [( n +3 e − / − k ( n − k )] ×× s − Y k =0 (cid:0) [ C r , E r +3 k +2 − ρn ][ D r , F r +3 k +2 − ρn ] (cid:1) en [( n +4 n +3 e − / − k ( n − − k )] ×× s Y k =0 (cid:0) [ C r , F r +3 k − ρn ][ D r , E r +3 k − ρn ] (cid:1) enk ( n +2 − k ) ×× s − Y k =0 (cid:0) [ C r , F r +3 k +1 − ρn ][ D r , E r +3 k +1 − ρn ] (cid:1) en [ k ( n − k )+( n +1) / × s − Y k =0 (cid:0) [ C r , F r +3 k +2 − ρn ][ D r , E r +3 k +2 − ρn ] (cid:1) enk ( n − − k ) if t = 2, where subtracting ρn takes the index to N n .The numerical example which we now consider is with r = 3, p = m = 0,and q = 1 (and again u = 0). The divisors Ξ which are of the form S n − ∆,namely with | F n − | = 1, must satisfy also | C n − | = | C s | = | C n − − s | = 1. Thisyields the 6 divisors S n − P n − − si P sj , with i , j , and k being some choice for theindices 1, 2, and 3. On the other hand, if ord P k (Ξ) = n − | C | mustbe 1, so that the point S must lie in either F , F , or F . If it lies in F then the other non-empty sets must be C s and C n − − s , yielding the divisors P n − k P n − − si P sj . Having S in F assigns the other points to C s +1 and C n − s and produces the divisors P n − k P n − − si P s − j S . On the other hand, in the casewhere F contains S we obtain non-trivial sets which are C s and C n − s for t = 1and C s +1 and C n − s − if t = 2. The divisors can thus be written uniformly as P n − k P si P n − − sj S . Replacing n by 3 s + t shows that these results reproducethe divisors from Theorem 6.13 of [FZ], where each divisor from that theoremcomes multiplied by the ( n − h Ξ , we have contributions only from expressionsof the form [ C r , C r + l ] or [ C r , F r + l − ρn ]. We substitute n = 3 s + t , and writeour expressions in terms of s . We distinguish the cases t = 1 and t = 2, henceuse the value of t explicitly in each formula. For t = 1 we find that if Ξ is S n − P n − − si P sj , P n − i P n − − sj P sk , P n − j P sk P n − − si S , or P n − k P n − − si P s − j S ,then h Ξ takes the form[( λ i − λ j )( λ j − λ k )] en ( s +2 s +2 − e ) / ( λ i − λ k ) en ( s − s +2 − e ) / [( τ − λ i )( τ − λ k )] ens . t = 2 and Ξ being S n − P n − − si P sj , P n − j P n − − sk P si , P n − i P sj P n − − sk S ,or P n − k P n − − si P s − j S , the expression for h Ξ becomes( λ j − λ k ) en ( s +4 s +5 − e ) / [( λ i − λ j )( λ i − λ k )] en ( s +1 − e ) / ( τ − λ i ) en ( s +1) . These expressions are not reduced, since the combination of [ C r , C r + l ] havingpower 0 do not appear. The common factor is [( λ i − λ j )( λ i − λ k )( λ j − λ k )] en taken to a power which is s − s +2 − e for t = 1 and is s +1 − e if t = 2. Aftereliminating this common factor we obtain again the results of Section 6.2 of[FZ]. This shows that our general formula can indeed reproduce all the specialcases considered in [FZ].The first numerical example presented here involves a non-special divisor ofdegree g which is supported on all the branch points, namely P n − R S Q n − (see Theorem 6.3 of [FZ]). In this spirit we now give examples of fully ramified Z n curves admitting no M -orbits, though they do carry non-special divisors ofdegree g which are supported on the branch points. This means that all thenon-special degree g divisors supported on the branch point on these curvescontain all the branch points in their support. These examples relate to thefamilies of curves considered in Section 3 of [GDT], and have only 3 branchpoints. Let n be an odd number which is not divisible by 3, and consider the Z n curve defined by the equation w n = ( z − λ )( z − σ ) ( z − τ ) n − . As above, we denote the branch points lying over λ , σ , and τ by P , R , and S respectively. In this case we can find the non-special divisors directly, ratherthan using Theorem 1.6: Indeed, we have only 3 points, the degree g is n − ,and the rational equivalence of P R and S implies that in no such divisor canbe a multiple of either of these degree 3 divisors (as in the proof of Lemma1.5). For n ≥ 11 this leaves us with only 9 divisors to consider: P n − , P n − S , P n − S , P n − R , P n − RS , P n − RS , R n − , R n − S , and R n − S . For n = 7we have only 8 divisors, since the 6th and 8th divisors on this list coincide.Moreover, for n = 5 these are only 6 divisors, as the 6th divisor is not integral,the 3rd and 9th divisors coincide, and the 5th and 8th divisors coincide. Thebasis for the holomorphic differentials consists of those ω k with k satisfyingeither n < k < n or n < k < n . Substituting n = 3 s + t , these boundsbecome either s + 1 ≤ k ≤ n − or n − s ≤ k ≤ n − 1. Now, the holomorphicdifferentials ω n − s and ω s +1 have the (canonical) divisors P s − R s − S t − and P n − − s R s + t − S − t respectively. Moreover, for s ≥ n ≥ 7) the (integral)divisor P s − R s − S t +2 of ω n +1 − s is also canonical, and in case s + t ≥ n ≥ 11) the same assertion applies for the divisor P n − − s R s + t − S − t of ω s +2 .The inequality s + t ≥ n − − s ≥ n − , and since s ≥ t + 1 forodd n ≥ s − ≥ n − as well in this case. Hence for n ≥ P n − , R n − and R n − S are special. Moreover, if n ≥ 11 then s + t ≥ 5, and we have either s ≥ t + 3 (for n ≥ 13) or t = 2 (for n = 11).45his implies that P n − S , P n − S , P n − R , and R n − S are also special. Theremaining divisors P n − RS and P n − RS are special for all n ≥ 17 (since s + t ≥ n ≥ 17 admit no non-special divisors ofdegree g . In addition, for n = 13 we have t = 1, so that the divisor P n − RS (namely P RS ) is special also for this value of n . On the other hand, for n = 13the divisor P n − RS = P RS is non-special: The divisors P R , P R S , P R S , and P S of ω , ω , ω , and ω already considered are not multiplesof this divisor, and the same applies for the divisors P R S of ω and RS of ω . Similarly, in the case n = 11 the two divisors P RS and P RS arespecial: The differentials ω , ω , ω , and ω have the divisors P R S , P R S , P R , and P S considered above, the remaining differential ω has divisor RS , and none of these canonical divisors is a multiple of any of the divisors inquestion. As in these examples the divisors contain all the branch points in theirsupport, there are no M -orbits and non-vanishing characteristics, even thoughthere exist non-special divisors of degree g . This example completes, in somesense, the families presented in [GDT], and shows that for r = 1 the bound p > r is not sufficient (see also another example discussed briefly in Section8). Interestingly, for n = 7 there exists precisely one M -orbit (or characteristic)arising from the non-special divisors P R , P S , and RS , and the divisor P RS is also non-special. Hence the Thomae formulae are trivial also in this case.For n = 5 we have the 4 non-special P S , P R , R , and S , yielding 2 M -orbitswhich are related by N . We close this paper by a brief discussion of a few questions which arise fromour considerations.First, proving Conjecture 5.3. As mentioned above, [K] establishes Thomaeformulae for our general setting, which points to the direction of the validity ofConjecture 5.3. Therefore, even if Conjecture 5.3 does not hold, the Thomaequotient should give the same value on every orbit of this action. Therefore,in case some counter-example to Conjecture 5.3 arises, one might look for ad-ditional operations which extend the action of G and the operators b T Q,R andleaving the Thomae quotient invariant.Second, in the case n = 2 our divisors capture all the order n = 2 pointsin J ( X ). As already remarked in the introduction to [FZ], this statement doesnot extend to any case with n > 2. Indeed, the set of n -torsion points in J ( X ) is a free module of rank 2 g = ( n − (cid:0) P α r α − (cid:1) over Z /n Z , and thepoints P i,α generate (with the fixed base point Q ), a subgroup of rank at most P α r α − div ( w )). Infact, the observation that the only relations between the divisors Ξ in Section5 are given by powers of the operator M (where a relation means that twodivisors represent the same characteristic) suggests that these are the only rela-tions holding between the points ϕ Q ( P α,i ) in J ( X ). Hence the rank is precisely46 α r α − 2. Therefore it is interesting to ask what kind of divisors represent theother n -torsion points in J ( X ), and whether they are special or not. For thelatter characteristics, we ask whether denominators like our h Ξ (or h Q ∆ ), whichextends the Thomae formulae to these characteristics as well, might exist.Next, one may wish to consider the dependence of the Thomae formulae onthe actual choice of the function z (rather than just a projection from X tothe quotient under the Galois action). Since the denominators h Ξ are basedonly on differences between z -values, replacing z by z − ζ for some ζ ∈ C leaves the Thomae constant invariant. On the other hand, multiplying z bysome number u ∈ C ∗ multiplies h Ξ by u raised to the power deg h Ξ , whichchanges the constant. This shows, by the way, that all the denominators h Ξ must have the same degree. One might search for Thomae constants which areindependent of the choice of z . For example, multiplying all the constants bysome global product of differences, which would render the denominators h Ξ rational functions, should yield an expression which is independent of dilationsof z as well. If this is indeed possible, one needs only to consider replacing z by z and allowing branching at ∞ . In addition, assuming that such an invariantconstant exists, we observe that some Riemann surfaces may be given severalstructures of Z n curves. For example, the Z n curves from Section 3.3 of [FZ] areall hyper-elliptic, hence carry an additional structure of a Z curve. In this caseit is natural to ask whether connections between the Thomae constants arisingfrom the different Z n structures on X can be found.It may also be of interest to study the dependence on n of the number ofnon-special divisors of degree g or of M -orbits in families of Z n curves withrelated equations. More precisely, consider the Z n curve associated with anequation of the sort w n = p Y i =1 ( z − λ i ) c i q Y i =1 ( z − µ i ) n − d i , (14)where P pi =1 c i = P qi =1 d i (hence the sum of powers is divisible by n ), q ≤ p (otherwise replace w by w n − ), and n is large enough (i.e., n → ∞ ). Forsufficiently large n this yields t k = q for small k and t k = p for large k (here k isconsidered to be in N n and not in Z /n Z ). Note that the non-singular Z n curvesdo not describe such a family, since the corresponding number of points dependson n . On the other hand, the singular curves from Chapter 5 of [FZ] do lie insuch families for every choice of the parameter m . We have seen in the secondnumerical example in Section 7 (or in Section 6.2 of [FZ]) that these numbers areconstant (18 divisors for n ≥ M -orbits for n ≥ n ≥ 17. Considerations similar to thatexample show that w n = ( z − λ )( z − σ ) ( z − τ ) n − displays a similar behavior.Furthermore, the choice of r = 2, p = m = 0, and q = 1 in the first example inSection 7 yields the curves of the form w n = ( z − λ )( z − λ )( z − τ ) n − . Thesecurves have 4 non-special divisors (for n ≥ 5) and 2 M -orbits which are relatedvia N . Note that in all these cases we have q = 1, namely t k = 1 for small k .On the other hand, the singular curves from Section 3.3 of [FZ] (with p = q = 247nd the indices d i and e i being 1) admit 2 n − P intheir support (hence 2 n − M ). One easily sees that the total numberof divisors in this case is 4 n − 4. In the first numerical example of Section 7(considered in Section 6.1 of [FZ]) we have, for n ≥ n + 2 characteristics and2 n + 5 divisors (see Theorem 6.3 of [FZ]). Similar considerations show that bytaking r = p = q = m = 1 in the second family described in Section 7, the Z n curves of the form w n = ( z − λ )( z − σ ) ( z − τ ) n − ( z − µ ) n − carry n + 2characteristics for n = 3 s + t ≥ n + 4 divisors in total for n ≥ 8. In allthese cases we have q = 2, and the numbers of divisors and M -orbits are bothlinear in n . As for an example with q = 3, we state that for a singular curve ofthe form w n = ( z − λ )( z − λ )( z − λ )( z − µ ) n − ( z − µ ) n − ( z − µ ) n − (the Z n curves appearing in Chapter 5 of [FZ] with m = 3) the total number ofsuch divisors of degree g is 18 n − n + 33. Of these divisors, 3 n + 6 n − 14 donot contain a pre-fixed branch point and correspond to orbits of M (the latternumber indeed becomes 10 = for the non-singular Z n curve arising from n = 2, and is equal to the number 31 from Section 3.2 of [FZ] for n = 3). Theseobservations lead us to formulate the following Conjecture 8.1. The number of non-special divisors of degree g on the Z n curve associated to Equation (14) is described, for large enough n , by a polyno-mial of degree q − in n . The same assertion holds for the number of M -orbitson such a Z n curve. In fact, taking a closer look into the results of these examples leads to a finerform of Conjecture 8.1: Conjecture 8.2. Let X be a Z n curve described by Equation (14) , and let c and d be positive integers. Define x c = |{ i | c i = c }| and y d = |{ i | d i = d }| . Then p = P c x c and q = P d y d form partitions of p and q respectively. The leadingcoefficients of the two polynomials appearing in Conjecture 8.1 depend only onthese partitions of p and q . Both Conjectures 8.1 and 8.2 can be formulated in terms of assertions aboutthe numbers of solutions of combinatorial equations (Theorem 1.6 again), whereeach solution is assigned a multiplicity according to the number of divisors itrepresents (another combinatorial expression). References [BR] Bershadski, M., Radul, A., Conformal Field Theories with Addi-tional Z n Symmetry , Int. J. Mod. Phys. A, vol 2 issue 1, 165–171 (1987).[EbF] Ebin, D., Farkas, H. M., Thomae Formulae for Z n Curves , J. Anal.Math., vol 111, 289–320 (2010). 48EiF] Eisenmann, A., Farkas, H. M., An Elementary Proof of Thomae’sFormulae , OJAC, vol 3, (2008).[EG] Enolski, V., Grava, T. Thomae Formulae for Singular Z n Curves ,Lett. Math. Phys., vol 76, 187–214 (2006).[FK] Farkas, H. 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