Related papers: On superspecial abelian varieties over finite fiel…

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over…

In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $\lvert \mathrm{SSp}_2(\mathbb{F}_q)\rvert$ of…

In this paper we determine the number of isomorphism classes of superspecial abelian varieties $A$ over the prime field $\Fp$ such that the relative Frobenius morphism $\pi_A$ satisfying $\pi_A^2=-p$.

The characteristic polynomials of abelian varieties over the finite field $\mathbb{F}_q$ with $q=p^n$ elements have a lot of arithmetic and geometric information. They have been explicitly described for abelian varieties up to dimension 4,…

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

In this article, we give a complete description of the characteristic polynomials of supersingular abelian varieties over finite fields. We list them for the dimensions upto 7.

We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding…

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…

Extending the results of [Asian J. Math. 2019], in [Doc. Math. \textbf{21}, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of \textit{odd} degree over the…

Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z. We are interested in knowing…

In this note, we propose the modular height of an abelian variety defined over a field of finite type over Q. Moreover, we prove its finiteness property.

The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian…

In this paper, we give an equivalent condition for an abelian variety over a finite field to have multiplication by a quaternion algebra over a number field. We prove the result by combining Tate's classification of the endomorphism…

In this article, we derive a list of possible characteristic polynomials of the Frobenius of simple supersingular abelian varieties of dimension 1, 2, 3, 4, 5, 6, 7 over $\mathbb{F}_q$ where $q = p^n$, n odd.

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…

Let $A$ be an abelian variety over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by the Weil polynomial $f_A$. We assume that $f_A$ is separable. For a given prime number $\ell\neq\mathrm{char}\, k$ we give a…

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations we define, for each prime $\ell$, a local factor which measures the relative frequency with which $f\bmod \ell$ occurs as the characteristic polynomial…

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let ${a_1,\ldots,a_r}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique…

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When…