Related papers: Abelian varieties with quaternion and complex mult…
We study the distribution of algebraic points on curves in abelian varieties over finite fields.
We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used…
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…
This is a survey paper on moduli spaces that have a natural structure of a (possibly incomplete) locally symmetric variety. We outline the Baily-Borel compactification for such varieties and compare it with the compactifications furnished…
We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…
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 $q$…
Heavenly abelian varieties are abelian varieties defined over number fields that exhibit constrained $\ell$-adic Galois representations for some rational prime $\ell$. At the ICMS Workshop held in November 2024, we presented evidence for…
Let K be a CM-field, i.e., a totally complex quadratic extension of a totally real field F. Let X be a g-dimensional abelian variety admitting an algebra embedding of F into the rational endomorphisms of X. Let A be the product of X and…
CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the…
In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is…
We say that an abelian variety $A_{/\mathbf Q}$ of dimension $g$ is {\em prosaic} if it is semistable, with good reduction at 2 and its points of order $2$ generate a $2$-extension of ${\mathbf Q}$. For $p \equiv 1 \bmod{8}$, let $M_u$ be…
For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the…
This is a slightly revised version of the author's 2010 diploma thesis. It is concerned with the interplay between real multiplication on Jacobian varieties, as the title suggests, and complex geodesics in the moduli space of curves. Large…
We formulate a conjecture on the finitude of rationality fields (i.e., Fourier coefficient fields) of newforms of bounded degree, and prove this for CM forms assuming a generalized Riemann hypothesis. Then we explicitly determine what…
By using the theory of Complex Multiplication for general Siegel modular varieties we construct arithmetic subalgebras for BC-type systems attached to number fields containing a CM field. Our approach extends the construction of Connes,…
An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian…
The paper presents a classification of quadratic extension algebras, also known as algebras of degree 2, as well as several characterizations of quaternion algebras over a field (of characteristic not 2). The presentation is not restricted…
Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with…
Given a polarized abelian scheme with action by a ring, and a projective finitely presented module over that ring, Serre's tensor construction produces a new abelian scheme. We show that to equip these abelian schemes with polarizations…
We prove (by a case-by-case analysis) a conjecture of Bernstein/Schwarzman to the effect that quotients of abelian varieties by suitable actions of (complex) reflection groups are weighted projective spaces, and show that this remains true…