A note on supersingular abelian varieties

In this note we show that any supersingular abelian variety is isogenous to a superspecial abelian variety without increasing field extensions. The proof uses minimal isogenies and the Galois descent. We then construct a superspecial abelian variety which not directly defined over a finite field. This answers negatively to a question of the author [J. Pure Appl. Alg., 2013] concerning of endomorphism algebras occurring in Shimura curves. Endomorphism algebras of supersingular elliptic curves over an arbitrary field are also investigated. We correct a main result of the author's paper [Math. Res. Let., 2010]

Supersingular abelian varieties and quaternion hermitian lattices (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

This note gives a survey on relations between the theory of quaternion hermitian lattices and that of supersingular abelian varieties, including relations between polarizations, moduli loci, automorphism groups, curves with many rational points, and class numbers, type numbers, lattice automorphisms, algebraic modular forms. For readers' convenience, we give some explicit formulas for related numbers and give a slightly big list of related references. The last section is an announcement of new results on supersingular loci of low dimensions

Superspecial rank of supersingular abelian varieties and Jacobians

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines information about the decomposition of a supersingular abelian variety up to isomorphism; namely it is a bound for the maximal number of supersingular elliptic curves appearing in such a decomposition.Comment: V2: New coauthor, major rewrit

Endomorphism algebras of QM abelian surfaces

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

The Tate conjecture for K3 surfaces over finite fields

Artin's conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin's conjecture over fields of characteristic p>3. This implies Tate's conjecture for K3 surfaces over finite fields of characteristic p>3. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p>3.Comment: 20 pages, minor changes. Theorem 4 is stated in greater generality, but proofs don't change. Comments still welcom

Mass formula for supersingular abelian surfaces

We show a mass formula for arbitrary supersingular abelian surfaces in characteristic $p$.Comment: 10 page

Superspecial Abelian Varieties and the Eichler Basis Problem for Hilbert Modular Forms

Let $p$ be an unramified prime in a totally real field $L$ such that $h^+(L)=1$. Our main result shows that Hilbert modular newforms of parallel weight two for $\Gamma_0(p)$ can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This can be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.Comment: to appear in J.N.