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Galois Realisations of $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ via non-unirational Hilbert Irreducibility

Julian Demeio, Damián Gvirtz-Chen

TL;DR

The paper advances the regular Inverse Galois Problem by proving non-thin, non-unirational Hilbert Property results for Hilbert modular surfaces of K3 type and then constructing geometrically integral $PSL_2({\mathbb {F}}_{p^2})$-covers over these surfaces. The approach combines Shimura theory with a multiple-fibration method to produce abundant genus $1$ fibrations and rational curves, enabling the specialization of covers to yield regular $PSL_2({\mathbb {F}}_{p^2})$-extensions when there exists a Kronecker symbol $\genfrac(){}{}{\ell}{p}=-1$ for some $\ell\le 41$, $\ell\neq 31$. A core part of the work is a detailed arithmetic of algebraic cycles on low-discriminant Hilbert modular surfaces, including cusp, elliptic, and Hirzebruch–Zagier divisors, whose intersection graphs underpin the non-thinness conclusions. The results thus provide new, geometry-driven instances of the regular inverse Galois realizations over $\mathbb{Q}$ and Hilbertian fields, with implications for the distribution of rational points and families of Galois covers on Shimura varieties.

Abstract

We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ subject to congruence conditions on $p$.

Galois Realisations of $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ via non-unirational Hilbert Irreducibility

TL;DR

The paper advances the regular Inverse Galois Problem by proving non-thin, non-unirational Hilbert Property results for Hilbert modular surfaces of K3 type and then constructing geometrically integral -covers over these surfaces. The approach combines Shimura theory with a multiple-fibration method to produce abundant genus fibrations and rational curves, enabling the specialization of covers to yield regular -extensions when there exists a Kronecker symbol for some , . A core part of the work is a detailed arithmetic of algebraic cycles on low-discriminant Hilbert modular surfaces, including cusp, elliptic, and Hirzebruch–Zagier divisors, whose intersection graphs underpin the non-thinness conclusions. The results thus provide new, geometry-driven instances of the regular inverse Galois realizations over and Hilbertian fields, with implications for the distribution of rational points and families of Galois covers on Shimura varieties.

Abstract

We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups subject to congruence conditions on .
Paper Structure (28 sections, 24 theorems, 46 equations, 10 figures, 1 table)

This paper contains 28 sections, 24 theorems, 46 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation and terminology
  4. Genus $1$ fibrations and the Hilbert Property
  5. A criterion for the non-unirational Hilbert Property
  6. Regular variants of the Hil-bert Property
  7. Preliminaries on Hilbert modular surfaces
  8. Hilbert modular surfaces and rational models
  9. Arithmetic models.
  10. Connected components.
  11. Galois action on $\pi_0$.
  12. Examples.
  13. Construction of the $\operatorname{PSL}_2({\mathbb {F}}_{p^2})$-cover
  14. Arithmetic of algebraic cycles on Hilbert modular surfaces
  15. Cusp divisors.
  16. ...and 13 more sections

Key Result

Theorem 1

For any prime $p$ for which there exists an $\ell \in {\mathcal{L}}$ with the Kronecker symbol $\genfrac(){}{}{\ell}{p}$ equal to $-1$, the group $\mathop{\mathrm{PSL}}\nolimits_2({\mathbb {F}}_{p^2})$ is a regular Galois group over ${\mathbb {Q}}$.Concretely, this implies that $\mathop{\mathrm{PSL}

Figures (10)

  • Figure 1: Extended simply-laced Dynkin diagrams. $\tilde{A}_n$ and $\tilde{D}_n$ have $n+1$ nodes.
  • Figure 2: The convex hull of $(M_s\otimes {\mathbb {R}})_+$ and two of the (infinitely many) maximal cones of the fan $\Sigma$ highlighted, for $K={\mathbb {Q}}(\sqrt 5)$, $s=$ principal cusp.
  • Figure 3: Exceptional divisors on the blowup $\mathop{\mathrm{Bl}}\nolimits_{\tilde{\tau}_1,\tilde{\tau}_2,\tau}U$ of a $3^-$ elliptic point.
  • Figure :
  • Figure :
  • ...and 5 more figures

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5: Demeio
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 39 more