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Oort's conjecture on automorphisms of generic supersingular abelian varieties

Eva Viehmann

Abstract

We prove Oort's conjecture that generically on the supersingular locus of the moduli space of principally polarized abelian varieties of genus g and in characteristic p, the automorphism group of the universal principally polarized abelian variety consists only of $\pm 1$, unless g=2 or 3 and p=2. On the way, we provide an explicit description of the a=1-locus in the Rapoport-Zink space of principally polarized supersingular p-divisible groups of any dimension g. We also prove analogous results for generic automorphism groups on moduli spaces of supersingular p-divisible groups with and without polarization.

Oort's conjecture on automorphisms of generic supersingular abelian varieties

Abstract

We prove Oort's conjecture that generically on the supersingular locus of the moduli space of principally polarized abelian varieties of genus g and in characteristic p, the automorphism group of the universal principally polarized abelian variety consists only of , unless g=2 or 3 and p=2. On the way, we provide an explicit description of the a=1-locus in the Rapoport-Zink space of principally polarized supersingular p-divisible groups of any dimension g. We also prove analogous results for generic automorphism groups on moduli spaces of supersingular p-divisible groups with and without polarization.
Paper Structure (8 sections, 15 theorems, 69 equations)

This paper contains 8 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Reformulation
  3. Coordinates for one irreducible component
  4. Proof of Theorem \ref{['thmmain']}
  5. Claim \ref{['keyclaim']} holds for $s=1$.
  6. Claim \ref{['keyclaim']} holds for $s=2$
  7. Claim \ref{['keyclaim']} holds for $s=3$ and $p=2$
  8. The non-polarized case

Key Result

Theorem 1.1

Let $(g,p)\neq (2,2),(3,2)$. Then there is a dense open subscheme $Y$ of $\mathscr S_g$ such that for every $x\in Y(\overline{\mathbb{F}}_p)$, we have ${\rm Aut}(A_x,\lambda_x)=\{\pm 1\}$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.4: Zink's lemma
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Lemma 3.2
  • ...and 26 more