Thomae's formula and a discussion of using it for understanding levels of a Modular Tower

Riemann was well aware, that looking at one curve covering of the sphere did not give much insight into the nature of families of covers. He knew the major value of his work on theta functions should be to studying various spaces of covers of the sphere. He died too young to fulfill this idea.

For most mathematicians the idea of a family is visually quite simple: A polynomial in two variables with some parameters as coefficients. That may be an easy-to-look-at family and doing so allows you to claim you are considering families. Yet, it captures nothing intrinsic about understanding families as a collection of covers of a given type. Investigations on the moduli space of curves of genus g generalize to investigations of the Regular Inverse Galois Problem. All those have phrasings as to the nature of the space parametrizing families of covers.

Ambiguities in Riemann's generalization of Abel's Theorem: The generalization is based on trying to fix one θ function — an odd nongenerate function (multiplicity one at most points of its theta divisor) on the universal cover of a complex torus defined by a normalized period matrix. This still left many ambiguities.

θ changes if you change the homology basis determining the period matrix (fundamental lattice), though its oddness does not.
Excluding the case of curves of genus 1, there is usually more than one such odd nongenerate θ.
Coordinates for forming the θ don't vary naturally in the parameters defining the family.
Evaluating the θ at divisors on its corresponding curve is often tricky.

Comments on items in this list: Item #1 involves a symplectic group of transformations in the equivalence class of possible θ choices. Much of the transformation formula is "automorphic" in character, depending only on the determinant of the denominator of symplectic transformation. Still, if we denote the fundamental lattice by L, there is a mysterious character when a transformation doesn't induce the identity on L/2L. The rest of the items #2-#4 are even more demanding. Especially since one goal is to use the variation of the thetas to describe properties of a serious family — say, a Hurwitz family defined by a Nielsen class. If you don't interpret the variation of the thetas by expressions in the algebraic parameters for the family (as in item #3), then you are really back at considering the variation of thetas on the whole moduli space of curves of genus g. That is not likely to be helpful for understanding a particular family. Explanation of item #4: The θ lives on the universal cover of the Jacobian and rarely is there a canonical embedding of a curve in its Jacobian that respects the family coordinates.

Yet, there are reasons to be positive: Some Hurwitz families defined by covers with nonabelian monodromy naturally come with thetas already partially normalized. For example, by having a canonical theta attached to them (as in Hurwitz spaces of covers with odd order branching [Altgps, Lem. 5.15]). I write this to encourage investigating the theta variation of these by considering what we can learn from variation of thetas along families of cyclic covers. Even θs (even as functions on the complex torus) also appear below. Any Riemann surface has 2^2g-1+2^g-1 even thetas and 2^2g-1-2^g-1 odd thetas up to an exponential, nowhere zero, factor. I also use C_wbelow to denote Complex affine space with coordinates (w₁, ,w_g).

Early in the game — by the 1870s — researchers did try to investigate families of genus g cyclic covers through variation of thetas along them. The archetype is Thomae's for hyperelliptic curves. I view Thomae's formula as a way to produce θ functions evaluated at specific types of division points so that you see (say for explicit powers of them) their transformational behavior.

Why Thomae fits with Generalizing Abel's Theorem: That says Riemann's theta to the fourth power evaluated at certain explicit two division points — made explicit by forming them from explicit expressions in the branch points of the cover — appears in terms of discriminant polynomials in the branch points of the hyperelliptic cover. Evaluating θs at torsion points to draw conclusions is built into Riemann's original goal: To generalize Abel's Theorem. That is, to express any function on a Riemann surface as a ratio of "translates of the theta by points constituent in the divisor of the function." To get the meaning of this consider how each function — with divisor w₁'+ +w_t' - w₁''- - w_t'' on an elliptic curve C_w/L can be expressed as

θ(w-w₁') … θ(w-w_t')/ θ(w-w₁'') … θ(w-w_t'').

Here the logarithmic derivative of the θ (in w) is an antiderivative of the Weierstrass π-function [A, p.274]. Actually evaluating the θs at torsion points, assuming you can express the torsion points in the coordinates of the family, is giving something close to automorphic functions along the family [Altgps, Prop. 5.18].

Work of Nakayashiki and Kopeliovic: Nakayashiki has generalized Thomae to other cyclic covers. Then, Kopeliovic has used this to form identities in period matrices over these locii. Kopeliovic's idea is to form relations in these θ-characteristics based on the right sides of these formulae. They form "discriminant" polynomials generating a basis for an irreducible representation of the symmetric group on the number of branch points of the cover. Since the dimension of the space they generate (by a famous hook formula of Frobenius) is far less than the number of expressions, there must be many relations in the thetas evaluated at these carefully chosen division points. Those relations define, in turn, relations in the periods. This "Schottky Goal" for families of cyclic covers is time-honored, though it is not mine.

Nonabelian goals: My interest is in related situations where the covers are not cyclic (or even abelian). My first example is where the monodromy of the covers in the families at level k is (Z/p^k+1)²x^s Z/3 (Z/3 action faithful so the group has no Z/p quotient, p is not 3; when p=2 and k=0 the group is A₄). For these covers there are four branch cycles, two represented by +1, two by -1, in Z/3 [Lum, §6.3].

In this case, I take abelianized MTs. The levels are reduced Hurwitz spaces. They each have dimension 1, and they are — as are all reduced Hurwitz spaces defined by four conjugacy classes — upper half plane quotients, j-line covers. Further, like all reduced Hurwitz spaces they have cusps on their boundaries. [Lum, §6.4.2] gives a complete description of the cusps when p=2 and k=0, and uses them to show that neither component is a modular curve, despite many modular curve-like properties.

Continuing with the case p=2: At level 0 there are two components. As an example of using a formula of [Se] you can tell one supports an even θ and the other an odd θ. We care about the even one, because that component has components above it at level 1, while the other does not. At level 1 there are six components. Three of them support an even θ-null, and two of those are H(arbater)-M(umford) components. We know some form of this behavior on components replicates at all higher levels of the abelianized MT [Lum, Ex. 4.13].

The Central Question about the H-M components: Is the θ-nulls — θs evaluated at 0 on one of these components— nonzero almost everywhere? If so, the next question is how to express differentials of 3rd kind supported on the cusps of the family using this θ-null.

From the MT view, if you replace Z/3 by Z/2 in my examples (where the action of Z/2 is by multiplication by -1) you have modular curves [Lum, §6.1-2]. This idea that the two examples are seemingly related belies that the moduli spaces are astoundingly different. Of course, as I often try to get people to hear: modular curves for the prime p are to Modular Towers (with p defining the tower) as the dihedral group D_p is to all p-perfect finite groups (including all simple groups of order divisible by p). For modular curves in this analogy I exclude p=2 because D₂is not 2-perfect (it has a Z/2 quotient), though p=2 is actually the hardest case if, for example, you take a simple group.

[A] L. Ahlfors, Complex Analysis, International Series in Pure and Applied Mathematics, 1979.
[N] A. Nakayashiki, On the Thomae formula for Z/N curves, Publ. Res. Inst. Math Sci. 33 (6) (1997) 987—1015
[Se] J.P. Serre, Revêtements a ramification impaire et thêta-caractèristiques, C.R.Acad. Sci. Paris 311 (1990), 547—552.