Non-Abelian L Function for Number Fields

Co-abelian toroidal compactifications of torsion free ball quotients

Let X' be the toroidal compactification of the quotient of the complex 2-ball by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is an abelian surface, birational to X'. The present work can be viewed as an illustration for the presence of a plenty of non-compact co-abelian torsion free toroidal compactifications. More precisely, it shows that all the admissible values for the volume of a torsion free quotient of the complex 2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers. The article provides three types of infinite series of finite unra…

Curves of genus g on an abelian variety of dimension g

In this paper we prove a general theorem concerning the number of translation classes of curves of genus $g$ belonging to a fixed cohomology class in a polarized abelian variety of dimension $g$. For $g = 2$ we recover results of Göttsche and Bryan-Leung. For $g = 3$ we deduce explicit numbers for these classes.

Fixed points and entropy of endomorphisms on simple abelian varieties

In this paper we investigate fixed-point numbers and entropies of endomorphisms on abelian varieties. It was shown quite recently that the number of fixed-points of an iterated endomorphism on a simple complex torus is either periodic or grows exponentially. Criteria to decide whether a given endomorphism is of the one type or the other are still missing. Our first result provides such criteria for simple abelian varieties in terms of the possible types of endomorphism algebras. The number of fixed-points depends on the eigenvalues and we exactly show which analytic eigenvalues occur. This ins…

由阿贝尔 n 重上的割线层产生的 Weil 型阿贝尔 2n 重上的闭链

Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

本文利用阿贝尔 n 重上的割线层证明判别式 -1 的 Weil 型阿贝尔六重上 Weil 类的代数性，并经退化论证推出阿贝尔四重情形及 Hodge 猜想。

On a numerical characterization of non-simple principally polarized abelian varieties

To every abelian subvariety of a principally polarized abelian variety $(A, \mathcal{L})$ we canonically associate a numerical class in the Néron-Severi group of $A$. We prove that these classes are characterized by their intersection numbers with $ \mathcal{L}$; moreover, the cycle class induced by an abelian subvariety in the Chow ring of $A$ modulo algebraic equivalence can be described in terms of its numerical divisor class. Over the field of complex numbers, this correspondence gives way to an explicit description of the (coarse) moduli space that parametrizes non-simple principally pola…

On Abelian subvarieties of bounded degree in a polarized Abelian variety

If A is an Abelian variety, endowed with a polarization L, we study the function N_A(t) which counts the number of Abelian subvarieties S in A such that for the induced polarization L|_S the Euler characteristic χ(L|_S) is bounded above by t. We give an estimate for the asymptotic order of growth of this function.

带群作用的阿贝尔簇上的极化计数

Counting polarizations on abelian varieties with group action

针对主要极化阿贝尔簇模空间$\mathcal{A}_g$奇点集正维分支的非常一般元素，研究模自同构群作用下主要极化的计数问题，证明其数量不总是1。

Automorphisms of Abelian Varieties and Principal Polarizations

The Narasimhan-Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of non-isomorphic principal polarizations on any given abelian variety. We show, for example, that the Jacobian of the genus four underlying curve of Schoen's I-WP minimal surface has at least 9 non-isomorphic principal polarizations. We also explore the Jacobians of Klein's quartic, Fermat's quartic, Bring's curve, and more.

Extensions of abelian varieties defined over a number field

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.

Asymptotics of class number and genus for abelian extensions of an algebraic function field

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.

Counting the number of the twists of certain polarized abelian varieties

We count the number of isomorphism classes of degree $d$-twists of some polarized abelian varieties over finite fields of odd prime dimension. This can be seen as a higher dimensional analogue of the counting problem for elliptic curves case.

阿贝尔簇的模性

The modularity of an abelian variety

本文研究了阿贝尔簇在有理数域上的模性，扩展了椭圆曲线的模性概念，并提出简单阿贝尔簇的模性猜想。

On the variation of the rank of Jacobian varieties on unramified abelian towers over number fields

Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(τ,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/τB(k)$ varies when we take an unramified abelian cover $π:C'\to C$ defined over $k$.

Cycles in the de Rham cohomology of abelian varieties over number fields

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on …

Shioda-Tate formula for an abelian fibered variety and applications

We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered hyperkähler manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.

The Existence of an Abelian Variety over the Algebraic Numbers isogenous to no Jacobian

We prove the existence of an Abelian variety $A$ of dimension $g$ over $\Qa$ which is not isogenous to any Jacobian, subject to the necessary condition $g>3$. Recently, C.Chai and F.Oort gave such a proof assuming the André-Oort conjecture. We modify their proof by constructing a special sequence of CM points for which we can avoid any unproven hypotheses. We make use of various techniques from the recent work of Klingler-Yafaev et al.

Salem Numbers and Abelian Surface Automorphisms

We classify two-dimensional complex tori admitting automorphisms with positive entropy in terms of the entropies they exhibit. For each possible positive value of entropy, we describe the set of two-dimensional complex tori admitting automorphisms with that entropy.

Dynamical degrees of automorphisms on abelian varieties

For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.

A short guide to p-torsion of abelian varieties in characteristic p

There are many equivalent ways to describe the p-torsion of a principally polarized abelian variety in characteristic p. We briefly explain these methods and then illustrate them for abelian varieties A of arbitrary dimension g in several important cases, including when A has p-rank f and a-number 1 and when A has p-rank f and a-number g-f. We provide complete tables for abelian varieties of dimension up to four.

Decompositions of singular abelian surfaces

Given an abelian surface, the number of its distinct decompositions into a product of elliptic curves has been described by Ma. Moreover, Ma himself classified the possible decompositions for abelian surfaces of Picard number $1 \leq ρ\leq 3$. We explicitly find all such decompositions in the case of abelian surfaces of Picard number $ρ= 4$. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group on the factors of a given decompos…

Elliptic curves of bounded degree in a polarized Abelian variety

For a polarized complex Abelian variety A, of dimension g>1, we study the function N_A(t) counting the number of elliptic curves in A with degree bounded by t. We describe elliptic curves as solutions of Diophantine equations which, at least for small dimensions g=2 and g=3, can actually be made explicit, and we show that computing the number of solutions is reduced to the classical topic in Number Theory of counting points of the lattice Z^n lying on an explicit bounded subset of R^n. We obtain, for Abelian varieties of small dimension, some upper bounds for the counting function.

Computing algorithm for reduction type of CM abelian varieties

Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction of $\mathcal{A}$ at the prime ideal. In this paper we derive an algorithm which allows to decompose the group scheme ${\rm A}[p]$ into indecomposable quasi-polarized ${\rm BT}_1$-group schemes. This can be done for the unramified $p$ on the basis of its decomposition into prime ideals in the endomorphism algebra of ${\rm A}$. We also compute all types of such…

Non-isogenous abelian varieties sharing the same division fields

Two abelian varieties $A_1, A_2$ over a number field $K$ are called strongly iso-Kummerian if the torsion fields $K(A_1[d])$ and $K(A_2[d])$ coincide for all $d \geq 1$. For all $g \geq 4$ we construct geometrically simple, strongly iso-Kummerian $g$-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair.

An explicit basis for the rational higher Chow groups of abelian number fields

We review and simplify A. Beilinson's construction of a basis for the motivic cohomology of a point over a cyclotomic field, then promote the basis elements to higher Chow cycles and evaluate the KLM regulator map on them.

Jordan constants of quaternion algebras over number fields and simple abelian surfaces over fields of positive characteristic

We compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative subgroup of the endomorphism algebras of simple abelian surfaces over fields of positive characteristic.

Polarizations on abelian varieties

The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F-polarization on every abelian variety in the isogeny class.

Isogeny classes of abelian varieties over function fields

We study isogeny classes of abelian varieties over a function field in one variable over the field of complex numbers.

Abelian Ideals and the Variety of Lagrangian Subalgebras

For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the variety of Lagrangian subalgebras of $\mathfrak g \ltimes \mathfrak g^{\ast}$ and the set of abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In particular, the number of such orbits equals $2^{\text{rk} \mathfrak g}$ by Peterson's theorem on abelian ideals.