HTML and/or PDF files in the folder othlist-cov
othlist-mt: Arithmetic and Homological Contributors to Modular Towers: Ties to the article called Modular Tower Time Line For an html and pdf (or rtf, or ppt) file with the same name, the html is an exposition. Click on any of the [ 9] items below.
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Papers on families of genus 0 covers of the sphere
These papers use techniques suitable for low genus covers. They come especially from problems on variables separated equations.
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1. Arithmétique et Espaces de Modules de Revêtements, Pierre Dèbes Debes-HurwSpaces-SchinCf.pdf

2. Peter Muller, Cofinite Integral Hilbert Sets, Habilitation at Heidelberg, 1999 mueller-habil.pdf

Papers below deal with the moduli of Riemann surfaces presented as covers of other Riemann surfaces
This introduces invariants of families of covers interpreted as components of Hurwitz spaces. We now know of many separators of components – including those generalizing the separation of θ-functions into even and odd types – based on aspects of the group theory attached to sequences of covers.
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3. Francesca Vetro, On Hurwitz spaces of coverings with one special fiber: Vol. 240 (2009), No. 2, 383–398 DOI: 10.2140/pjm.2009.240.383. I match my notation to that of papers on my web site, not exactly the same as hers. The major work on explicit detection of connected components of Hurwitz spaces starts from constructing Hurwitz spaces of covers of the classical Riemann z-sphere P1. Vetro's papers are motivated by that case, especially the situation where the sphere covers themselves are a sequence W →h X →f P1z, with h a cyclic degree u cover, and f has simple branching. Her papers, however, with u = 2, exchange P1z with these conditions:
  • Y has genus g,
  • f has degree n, is simply-branched at r-1 points, and has one more branch point with a fixed cycle type and index e, and
  • f has monodromy group Sd.
These last two assumptions – except when Y is the sphere – are independent.

The monodromy group of hof in this case is a Weyl group, and this paper treats the case when it is either Bn or the subgroup of the case Dn generated by certain long roots, and when a certain normalizer condition holds. The html file explains these groups. The main result indicates precisely the connected components in both cases under the additional condition that r-1+e ≥ 2n. Although this is a special case, one general direction would be to compare the components in more general Nielsen classes with a fixed Y with the precise count of connected components from a Schur multiplier in inv_gal.pdf when Y = P1, and the branch cycles have large multiplicity. Note: Conway-Fried-Parker-Voelklein requires all conjugacy classes to appear with high multiplicity, while Vetro keeps one fixed. Vetro2009OneSpecFiber.html %-%-% Vetro2009OneSpecFiber.pdf

Papers below concentrate on finding non-trivial θ-nulls on Hurwitz spaces
Often θ-functions are about objects on Riemann surfaces related to integrals, so abelian covering theory. Yet, in their use in understanding families of covers they have often been restricted to surfaces presented as abelian covers of the sphere. Going beyond that that severe limitation is my main research objective.
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4. Θ-functions on Hurwitz space loci: This topic comes from its application to generalizations of towers of modular curves. Generalizations, like Shimura varieties, with levels, and primes attached to them, though those levels are Hurwitz spaces, and the group theory behind the structures has a nonabelian piece extending the pure abelian theory associated with torsion points on abelian varieties: See Modular Tower Time Line. There is a discussion of Thomae's formula and how to use it for understanding levels of a Modular Tower in Thomae.html. The html file explains the θ-goals of this part of the web site. theta-domain.html

5. M. Artebani and P. Pirola, Algebraic functions with even monodromy, to appear PAMS April 2004. A3rP1-artibani-pirola.pdf

6. Yaacov Kopeliovic, Theta Constant Identities at Periods of Coverings of Degree 3, International Journal of No. Theory 4, World Sci. Publishing, No. 5 (2008), 1–9. In the world around Riemann, groups were just coming into vogue, though abelian groups were well-understood, and as even today, much easier to work with. Denote the invertible integers mod n by (Z/n)*. The first loci to interest those who inspected what they could learn from Θ functions were the abelian covers of the sphere. It is easy to count the number of cyclic, totally ramified, covers of the sphere branched at a given set, z1,…,zr of points. That number is the same as there are aj ∈ (Z/n)*, j=1,…,r, that add to 0 mod n, modulo the semi-direct product of Z/n with (Z/n)* on (a1,…,ar). Here, the goal is to find relations among θ-values where the evaluation is at divisor classes of order 6 defined on certain such cyclic covers with n=3. Any cover having branch cycles of odd order has a uniquely defined half-canonical class attached to the cover, and from that a uniquely defined &Theta-divisor, as in theta-domain.html.

The author takes the case where the aj s are all the same, so r=3m. He runs over the various partitions of z1,…,zr into subsets of order m, and uses the work of the work of Thomae (1870) and Nakayashiki (1997) to evaluate the 6th power of the Θ at a 6-division point on the Jacobian that depends only on the partition. From the form of this evaluation, he can run over all the partitions and take the span of these expressions to conclude from, say, the hook formulas for representations of the symmetric group, many among them are identically zero. From these, in special cases he considers what are these θ-identities, comparing with Matsumoto (2008). yaac-cyc3thetaids04-08-07.pdf

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