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On the supersingular locus of the Siegel modular variety of genus 3 or 4

Ryosuke Shimada, Teppei Takamatsu

TL;DR

The paper analyzes the supersingular locus of Siegel modular varieties for genus $3$ and $4$, providing an explicit, perfection-after-descent description as a union of Deligne–Lusztig varieties of Coxeter type times affine spaces, with the index and closure relations controlled by the rational Bruhat–Tits building of an inner form $J$ of $ ext{GSp}_{2n}$. Using Ekedahl–Oort stratification, the $J$-stratification, and the Deligne–Lusztig reduction method, the authors show that genus $3$ is of positive Coxeter type and obtain a detailed, projective-description of the strata and their closures, including equidimensionality inputs. In genus $4$, a similar, though non-positive-Coxeter-type description holds with explicit decompositions into DL-type pieces and affine-space fibrations, while for genus $ \ge 5$ such a simple description fails, evidenced by nontrivial fibrations along reduction paths. These results refine the understanding of the basic locus, with potential implications for Kudla–Rapoport programs and related arithmetic questions, and illustrate the delicate dependence on genus in the geometry of supersingular loci.

Abstract

We study the supersingular locus of the Siegel modular variety of genus 3 or 4. More concretely, we decompose the supersingular locus into a disjoint union of the product of a Deligne-Lusztig variety of Coxeter type and a finite-dimensional affine space after taking perfection.

On the supersingular locus of the Siegel modular variety of genus 3 or 4

TL;DR

The paper analyzes the supersingular locus of Siegel modular varieties for genus and , providing an explicit, perfection-after-descent description as a union of Deligne–Lusztig varieties of Coxeter type times affine spaces, with the index and closure relations controlled by the rational Bruhat–Tits building of an inner form of . Using Ekedahl–Oort stratification, the -stratification, and the Deligne–Lusztig reduction method, the authors show that genus is of positive Coxeter type and obtain a detailed, projective-description of the strata and their closures, including equidimensionality inputs. In genus , a similar, though non-positive-Coxeter-type description holds with explicit decompositions into DL-type pieces and affine-space fibrations, while for genus such a simple description fails, evidenced by nontrivial fibrations along reduction paths. These results refine the understanding of the basic locus, with potential implications for Kudla–Rapoport programs and related arithmetic questions, and illustrate the delicate dependence on genus in the geometry of supersingular loci.

Abstract

We study the supersingular locus of the Siegel modular variety of genus 3 or 4. More concretely, we decompose the supersingular locus into a disjoint union of the product of a Deligne-Lusztig variety of Coxeter type and a finite-dimensional affine space after taking perfection.
Paper Structure (10 sections, 13 theorems, 60 equations)

This paper contains 10 sections, 13 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Affine Deligne-Lusztig Varieties
  5. The $\mathbb J$-stratification
  6. Deligne-Lusztig Reduction Method
  7. Length Positive Elements
  8. The Case of Genus 3
  9. The Case of Genus 4
  10. The Case of Genus $\geq 5$

Key Result

Theorem A

Let $n=3$. The variety $X_\mu(\tau)$ is universally homeomorphic to a union of the product of a Deligne-Lusztig variety of Coxeter type and a finite-dimensional affine space. The index set of this stratification and the closure relations between strata can be described in terms of the rational Bruha

Theorems & Definitions (25)

  • Theorem A: Theorem \ref{['main theo']}
  • Theorem B: Theorem \ref{['main theo2']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Remark 2.6
  • ...and 15 more