Related papers: On CM abelian varieties over imaginary quadratic f…

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P(T) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a…

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is…

We show that a smooth proper weakly ordinary variety $X$ of maximal Albanese dimension satisfies $\chi(X, \omega_X) \geq 0$. We also show that if $X$ is not of general type, then $\chi(X, \omega_X) = 0$ and the Albanese image of $X$ is…

We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\mathcal{H}_{\mathbf{v}}$, in terms of a canonically defined basis $\mathcal{B}_{gm}$ of a certain lattice of virtual…

We compare general inequalities between invariants of number fields and invariants of abelian varieties over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded…

Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of…

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that…

We present an algorithm that, on input of a CM-field $K$, an integer $k\ge1$, and a prime $r \equiv 1 \bmod k$, constructs a $q$-Weil number $\pi \in \O_K$ corresponding to an ordinary, simple abelian variety $A$ over the field $\F$ of $q$…

Let $R$ be a finite dimensional algebra of finite global dimension over a field $k$. In this paper, we will characterize a $k$-linear abelian category $\mathscr C$ such that $\mathscr C\cong \operatorname {tails} A$ for some graded right…

We characterize the abelian varieties arising as absolutely simple factors of GL2-type varieties over a number field k. In order to obtain this result, we study a wider class of abelian varieties: the k-varieties A/k satisfying that…

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a finite abelian group of odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ denote its square root of the inverse different, which exists by Hilbert's…

Given an abelian variety $A$ defined over a finite field $k$, we say that $A$ is "cyclic" if its group $A(k)$ of rational points is cyclic. In this paper we give a bijection between cyclic abelian varieties of an ordinary isogeny class…

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is constructive, building on the work of Madan--Pal in the case $m=1$ to produce an explicit…

The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterl\'e (1989), namely that if $E$ is an elliptic curve of prime conductor then…

We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and $G$ a finite group of odd order. If $K_h$ is a weakly ramified $G$-Galois $K$-algebra, then its square root $A_h$ of the inverse different is a locally free…

The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a…

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…

In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and…