On reduction of moduli schemes of abelian varieties with definite quaternion multiplications

TL;DR

This paper investigates the structure and reduction of moduli spaces of abelian varieties with quaternion multiplication in positive characteristic, classifying isogeny classes of related p-divisible groups.

Contribution

It provides a classification of isogeny classes of p-divisible groups with additional structures and studies the reduction of type D moduli spaces in positive characteristic.

Findings

Classification of isogeny classes of p-divisible groups with additional structures

Analysis of the reduction of type D moduli spaces of minimal rank

Extension of moduli space understanding to ramified primes in quaternion algebra

Abstract

In this paper we make an initial study on type D moduli spaces in positive characteristic $p\neq 2$, where we allow $p$ ramified in the definite quaternion algebra. We classify the isogeny classes of $p$-divisible groups with additional structures in question. We also study the reduction of the type $D$ moduli spaces of minimal rank.