HTML or PDF (or PPT) files in ./othlist-theta

HTML and/or PDF files in the folder othlist-theta
othlist-mt: Arithmetic and Homological Contributors to Modular Towers: Ties to the article called Modular Tower Time Line For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 4] items below.
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My own interest in this topic comes from its application to generalizations of towers of modular curves. These generalizations, like Shimura varieties, have levels, and primes attached to them, though those levels are Hurwitz spaces, and the group theory behind the structures is nonabelian rather than the abelian theory associated with abelian varieties: See Modular Tower Time Line. Still, this study has found Shimura varieties (and Shimura's books), as well as the classical theory of Θ-functions valuable.

✺ 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,…,z_r of points. That number is the same as there are a_j ∈ (Z/n)^*, j=1,…,r, that add to 0 mod n, modulo the semi-direct product of Z/n with (Z/n)^* on (a₁,…,a_r). 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 noted in theta-domain.html.

The author takes the case where the a_j s are all the same, so r=3m. He runs over the various partitions of z₁,…,z_r 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.html %-%-% yaac-cyc3thetaids04-08-07.pdf