On reduction of Hilbert-Blumenthal varieties

Let $O_F$ be the ring of integers of a totally real field $F$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_F$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by $a$-numbers on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [GO, J. Alg. Geom. 2000] on the stratifications when $p$ is unramified in $O_F$. We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when $p$ is totally ramified in $O_F$.Comment: A shortened revised versio

Variations of mass formulas for definite division algebras

The aim of this paper is to organize some known mass formulas arising from a definite central division algebra over a global field and to deduce some more new ones.Comment: 17 page

A note on the Mumford-Tate Conjecture for CM abelian varieties

The Mumford-Tate conjecture is first proved for CM abelian varieties by H. Pohlmann [Ann. Math., 1968]. In this note we give another proof of this result and extend it to CM motives.Comment: 10 pages, to appear in Taiwanese J. Mat

Endomorphism algebras of QM abelian surfaces

We determine endomorphism algebras of abelian surfaces with quaternion multiplication.Comment: 14 pages. Lemma 2.10 correcte

On the existence of maximal orders

We generalize the existence of maximal orders in a semi-simple algebra for general ground rings. We also improve several statements in Chapter 5 and 6 of Reiner's book concerning separable algebras by removing the separability condition, provided the ground ring is only assumed to be Japanese, a very mild condition. Finally, we show the existence of maximal orders as endomorphism rings of abelian varieties in each isogeny class.Comment: 22 pages. To appear in Int. J. Number Theor