skip to main content

Explore Research Products in the NSF-PAR

Title: 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.  more » « less
Award ID(s):
1944862 1811846
NSF-PAR ID:
10330094
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Mathematische Annalen
ISSN:
0025-5831
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ( , Compositio Mathematica)
    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. 
    more » « less
  2. ; ; ( , Mathematische Annalen)
    Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation ρ of a profinite group on a local pro- p domain A contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We enlarge Bellaïche’s ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of ρ, symmetries that constrain its image, and in terms of the adjoint trace ring of ρ, which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for GL2-Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms. 
    more » « less
  3. ; ; ; ; ( , SciPost Physics)

    We present microscopic, multiple Landau level, (frustration-free and positive semi-definite) parent Hamiltonians whose ground states, realizing different quantum Hall fluids, are parton-like and whose excitations display either Abelian or non-Abelian braiding statistics. We prove ground state energy monotonicity theorems for systems with different particle numbers in multiple Landau levels, demonstrate S-duality in the case of toroidal geometry, and establish complete sets of zero modes of special Hamiltonians stabilizing parton-like states, specifically at filling factor\nu=2/3ν=2/3. The emergent Entangled Pauli Principle (EPP), introduced in [Phys. Rev. B 98, 161118(R) (2018)] and which defines the “DNA” of the quantum Hall fluid, is behind the exact determination of the topological characteristics of the fluid, including charge and braiding statistics of excitations, and effective edge theory descriptions. When the closed-shell condition is satisfied, the densest (i.e., the highest density and lowest total angular momentum) zero-energy mode is a unique parton state. We conjecture that parton-like states generally span the subspace of many-body wave functions with the two-bodyMM-clustering property within any given number of Landau levels, that is, wave functions withMMth-order coincidence plane zeroes and both holomorphic and anti-holomorphic dependence on variables. General arguments are supplemented by rigorous considerations for theM=3M=3case of fermions in four Landau levels. For this case, we establish that the zero mode counting can be done by enumerating certain patterns consistent with an underlying EPP. We apply the coherent state approach of [Phys. Rev. X 1, 021015 (2011)] to show that the elementary (localized) bulk excitations are Fibonacci anyons. This demonstrates that the DNA associated with fractional quantum Hall states encodes all universal properties. Specifically, for parton-like states, we establish a link with tensor network structures of finite bond dimension that emerge via root level entanglement.

     
    more » « less
  4. ; ; ( , Proceedings of the American Mathematical Society)

    We introduce the notions of symmetric and symmetrizable representations ofSL2⁡<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}. The linear representations ofSL2⁡<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations ofSL2⁡<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}. By investigating aZ/2Z\mathbb {Z}/2\mathbb {Z}-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations ofSL2⁡<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations ofSL2⁡<#comment/>(Z){\operatorname {SL}_2(\mathbb {Z})}that are subrepresentations of a symmetric one.

     
    more » « less
  5. ; ; ( , Mathematics of computation)
    We study the essential minimum of the (stable) Faltings' height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the Néron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email