Abstract Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$ -dimensional abelian variety an $n$ -torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$ -dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.
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Factorization problems in complex reflection groups
We enumerate factorizations of a Coxeter element into arbitrary factors in
the complex reflection groups G(d, 1, n) (the wreath product of the symmetric group
with a cyclic group) and its subgroup G(d, d, n), applying combinatorial and algebraic
methods, respectively. After a change of basis, the coefficients that appear are the same
as those that appear in the corresponding enumeration in the symmetric group.
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- Award ID(s):
- 1855536
- PAR ID:
- 10172347
- Date Published:
- Journal Name:
- Séminaire lotharingien de combinatoire
- Issue:
- 82B
- ISSN:
- 1286-4889
- Page Range / eLocation ID:
- #57
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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