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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Modular functions and resolvent problems: With an appendix by Nate Harman
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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- PAR ID:
- 10330094
- Date Published:
- Journal Name:
- Mathematische Annalen
- ISSN:
- 0025-5831
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension
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Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of
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null (Ed.)Let $A_1$ and $A_2$ be abelian varieties over a number field $K$. We prove that if there exists a non-trivial morphism of abelian varieties between reductions of $A_1$ and $A_2$ at a sufficiently high percentage of primes, then there exists a non-trivial morphism $A_1\to A_2$ over $\bar K$. Along the way, we give an upper bound for the number of components of a reductive subgroup of $GL_n$ whose intersection with the union of $Q$-rational conjugacy classes of $GL_n$ is Zariski-dense. This can be regarded as a generalization of the Minkowski-Schur theorem on faithful representations of finite groups with rational characters.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00208-022-02395-8
