Related papers: Height of varieties over finitely generated fields

We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing…

Given asymptotic counts in number theory, a question of Venkatesh asks what is the topological nature of lower order terms. We consider the arithmetic aspect of the inertia stack of an algebraic stack over finite fields to partially answer…

We give a formalism of mixed sheaves on varieties over a subfield of the complex number field.

In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as…

In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the…

Let $F$ be a global field of characteristic $p > 0$ and $G$ a finite abelian $p$-group. In this paper we treat the question of counting $G$-torsors over $F$ for certain heights developed in [DY25].

We prove that the infinitesimal variations of Hodge structure arising in a number of geometric situations are non-generic. In particular, we consider the case of generic hypersurfaces in complete smooth projective toric varieties, generic…

We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue…

Inspired by Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety X of relative…

In this paper we give an algorithm for computing equations of Brauer-Severi varieties over perfect fields of characteristic 0. As an example we show the equations of all Brauer-Severi surfaces defined over $\mathbb{Q}$.

This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.

Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the notion of compactified representation of $GL_N(\OK)$ and, we see how to associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a compactified…

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…

The main result of the work ``The nilpotence conjecture in K-theory of toric varieties'' is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties.

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2,2,2) whose Hilbert 2-class fields are finite.

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and…

An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic…