The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of “resolvent problems” formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension at 𝑝=2 for the symmetric groups 𝑆𝑛 is equal to the essential dimension at 2 of certain 𝑆𝑛-coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic p, specifically Serre-Tate theory, as well as a family of remarkable mod 2 symplectic 𝑆𝑛-representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the 𝑝=2 case. In the second half of this paper we introduce the notion of ℰ-versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are ℰ-versal. We use these ℰ-versality result to deduce the equivalence of Hilbert’s 13th Problem (and related conjectures) with problems about congruence covers.
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The essential dimension of congruence covers
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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- PAR ID:
- 10330092
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 157
- Issue:
- 11
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 2407 to 2432
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X21007594
