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Quotients of abelian varieties by reflection groups

Eric M. Rains

Abstract

We prove (by a case-by-case analysis) a conjecture of Bernstein/Schwarzman to the effect that quotients of abelian varieties by suitable actions of (complex) reflection groups are weighted projective spaces, and show that this remains true after reduction to finite characteristic (including characteristics dividing the order of the group!). We also show that an analogous statement holds (with five explicitly enumerated exceptions) for actions of quaternionic reflection groups on supersingular abelian varieties.

Quotients of abelian varieties by reflection groups

Abstract

We prove (by a case-by-case analysis) a conjecture of Bernstein/Schwarzman to the effect that quotients of abelian varieties by suitable actions of (complex) reflection groups are weighted projective spaces, and show that this remains true after reduction to finite characteristic (including characteristics dividing the order of the group!). We also show that an analogous statement holds (with five explicitly enumerated exceptions) for actions of quaternionic reflection groups on supersingular abelian varieties.
Paper Structure (43 sections, 47 theorems, 115 equations, 8 tables)

This paper contains 43 sections, 47 theorems, 115 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Actions of reflection groups on abelian varieties
  3. Crystallographic reflection groups
  4. Checking polynomial invariants
  5. First steps towards classification
  6. Cohen-Macaulay invariants
  7. Imprimitive groups
  8. Strongly crystallographic actions in rank 1
  9. Classification in rank $\ge 3$
  10. Classification in rank 2
  11. Invariants in rank $1$
  12. Invariants in rank $\ge 3$
  13. Invariants in rank 2
  14. Classification of crystallographic actions of primitive groups
  15. Real reflection groups
  16. ...and 28 more sections

Key Result

Lemma 2.1

Let $g\in \mathop{\mathrm{Aut}}\nolimits(X)$ be a reflection and let $H\subset \mathop{\mathrm{Aut}}\nolimits(X)$ be a discrete subgroup normalized by but not containing $g$. Then the image of $g$ in $\mathop{\mathrm{Aut}}\nolimits(X/H)$ is a reflection.

Theorems & Definitions (157)

  • Remark
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 147 more