formula and a discussion of using it for understanding levels of a
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.
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.
- 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
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
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 22g-1+2g-1 even thetas and 22g-1-2g-1
odd thetas up to an exponential, nowhere zero, factor. I also use Cw below
to denote Complex affine space with coordinates (w1, ,wg).
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
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'+ +wt' - w1''- - wt'' on an elliptic
curve Cw/L can be expressed as
θ(w-w1') … θ(w-wt')/
θ(w-w1'') … θ(w-wt'').
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
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
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
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
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 Dp
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
D2 is 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,
[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.