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Infinitely many supersingular primes for some Mumford's abelian fourfolds

Fangu Chen

Abstract

Elkies proved the infinitude of supersingular primes for elliptic curves over real number fields. We generalize Elkies' result to some abelian fourfolds in Mumford's families, and more generally, to certain families of Kuga-Satake abelian varieties. The proof relies on the study of local deformation spaces at closed points of the integral model of a Hodge-type Shimura variety, based on the work of Madapusi, and on the analysis of real points of a Shimura curve, based on the work of Shimura.

Infinitely many supersingular primes for some Mumford's abelian fourfolds

Abstract

Elkies proved the infinitude of supersingular primes for elliptic curves over real number fields. We generalize Elkies' result to some abelian fourfolds in Mumford's families, and more generally, to certain families of Kuga-Satake abelian varieties. The proof relies on the study of local deformation spaces at closed points of the integral model of a Hodge-type Shimura variety, based on the work of Madapusi, and on the analysis of real points of a Shimura curve, based on the work of Shimura.

Paper Structure

This paper contains 32 sections, 34 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The main result
  4. Related works and heuristics
  5. The strategy of the proof
  6. Reduction of the CM cycles $\mathcal{P}_{\lambda}, \mathcal{P}_{4\lambda}$ modulo $\lambda$
  7. Real CM points on $\mathcal{P}_{\lambda}, \mathcal{P}_{4\lambda}$
  8. Organization of the paper
  9. Notation and conventions
  10. Preliminaries
  11. The Shimura curves
  12. Integral model
  13. Special endomorphisms
  14. Intersection number
  15. Quadratic reciprocity
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $F$ be a totally real number field with narrow class number $1$ and $B$ be a quaternion algebra over $F$ unramified at all finite places and exactly one of the real places of $F$. Let $\mathcal{O}$ be a maximal order of $B$ and $\mathcal{O}^1$ be the group of units of $\mathcal{O}$ of reduced no Let $A$ be an abelian variety parametrized by the Hodge type Shimura datum $(G,X)$ (see sec:Shimura

Theorems & Definitions (70)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1: MR3484114*5.4, 5.5
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6: MR638719*Theorem 167
  • ...and 60 more