On a generalization of the support problem of Erdos and its analogues for abelian varieties and K-theory

On the $k$-resultant modulus set problem on varieties over finite fields

Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erdős-Falconer distance problem: If $|A|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then we have \[Δ_{k}(A)=\{|x_1+\cdots+x_k|\colon x_i\in A\}\supseteq \mathbb{F}_q^*.\] In this paper, we provide improvements and extensions of their result.

An effective local-global principle for algebraic varieties and the sum product problem in finite fields

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products and reciprocals of linear forms. This allows us to make some progress towards a question of B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev and I. D. Shkredov (2019) on an extreme case of the Erdős-Szemerédi conjecture in finite fields.

On a torsion analogue of the weight-monodromy conjecture

We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.

On the Shafarevich conjecture for irreducible symplectic varieties

Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the Shafarevich conjecture for irreducible symplectic varieties of fixed deformation class. We also observe that the second cohomological generalization of the Shafarevich conjecture does not hold in general, and discuss another formulation of a cohomological generalization.

The Iwasawa Main conjecture of constant ordinary abelian varieties over function fields

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places.

A proof of Néron--Ogg--Shafarevich criterion via its archimedean analogue

In this short note, we deduce the classical Néron--Ogg--Shafarevich criterion on good reduction of abelian varieties from its archimedean analogue: a holomorphic family of abelian varieties over a punctured disc extends to the whole unit disc if and only if the topological monodromy representation is trivial.

Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties

We prove finite field analogues of integral representations of Appell- Lauricella hypergeometric functions in many variables. We consider certain hypersurfaces having a group action and compute the numbers of rational points associated with characters of the group, which will be expressed in terms of Appell-Lauricella functions over finite fields.

A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the …

Kummer theory of abelian varieties and reductions of Mordell-Weil groups

Let A be an abelian variety over a number field F with End(A/F) commutative. Let S be a subgroup of A(F) and let x be a point of A(F). Suppose that for almost all places v of F the reduction of x modulo v lies in the reduction of S modulo v. In this paper we prove that x must then lie in S + A(F)_tors. This provides a partial answer to a generalization (due to W. Gajda) of the support problem of Erdos.

Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.

Brauer-Siegel theorem for elliptic surfaces

We consider higher-dimensional analogues of the classical Brauer-Siegel theorem focusing on the case of abelian varieties over global function fields. We prove such an analogue in the case of constant families of elliptic curves and abelian varieties.

Integral Frobenius for Abelian Varieties with Real Multiplication

In this paper we introduce the concept of $\mathit{integral}$ $\mathit{Frobenius}$ to formulate an integral analogue of the classical compatibility condition linking the collection of rational Tate modules $V_λ(A)$ arising from abelian varieties over number fields with real multiplication. Our main result gives a recipe for constructing an integral Frobenius when the real multiplication field has class number one. By exploiting algorithms already existing in the literature, we investigate this construction for three modular abelian surfaces over $\mathbf{Q}$.

André--Pink--Zannier猜想的特征p类似物

A characteristic p analogue of the André--Pink--Zannier conjecture

该研究针对特征p下的André--Pink--Zannier猜想，证明了Hodge型Shimura簇中具大单值群的常值函数域点情形及相应Hecke等分布代数类似物，通过Hecke对应分析与Weyl特殊点充足性证明完成论证。

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 …

The median Genocchi numbers, Q-analogues and continued fractions

The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag varieties. Second, we prove that the generating function of the Poincar\' e polynomials of the degenerate flag varieties can be written as a simple continued fraction. As an application we prove that the Poincar\' e polynomials coincide with the $q$-version of the normalized median Genocchi numbers introduced by Han and Zeng.

Homomorphisms of abelian varieties over geometric fields of finite characteristic

We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial endomorphisms.

A Note on the Specialization Theorem for Families of Abelian Varieties

In this note, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's Specialization Theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over ${\Q}$.

Non-archimedean entire curves in closed subvarieties of semi-abelian varieties

We prove a non-archimedean analogue of the fact that a closed subvariety of a semi-abelian variety is hyperbolic modulo its special locus, and thereby generalize a result of Cherry.

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.

Good reduction of Fano threefolds and sextic surfaces

We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the Shafarevich conjecture for smooth sextic surfaces, which appears to be the first non-trivial result in the literature on the arithmetic of such surfaces. Moreover, we exhibit certain moduli stacks of Fano varieties which are not hyperbolic, which allows us to show that the analogue of the Shafarevich conjecture does not always hold for Fano varieties. Our results al…

On the $\ell$-adic Galois representations attached to nonsimple abelian varieties

We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).

Erdős-Kac 定理的推广及其应用

Generalizations of Erdős-Kac theorem with applications

本文将 Erdős-Kac 定理从阿贝尔幺半群推广到满足附加条件的任意子集，统一研究 omega 函数与除数函数等算术函数，并应用于数域、整体函数域和几何不可约射影簇。

On the Selmer groups of abelian varieties over function fields of characteristic p>0

In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a …

非阿基米德平衡度量及其在完全退化阿贝尔簇上的应用

Non-Archimedean balanced metrics and their application to totally degenerate abelian varieties

本文研究了非阿基米德平衡度量，证明了在极化完全退化阿贝尔簇上该度量存在，并应用于退化族中Calabi-Yau度量的一致估计。

Twists of Drinfeld-Stuhler modular varieties

Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.

Toward Dirichlet's unit theorem on arithmetic varieties

In this paper, we would like to propose a fundamental question about a higher dimensional analogue of Dirichlet's unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.

Notes on an analogue of the Fontaine-Mazur conjecture

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.

Towards a theory of local Shimura varieties

This is a survey article that advertizes the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also review their theory in the light of this idea. We also discuss conjectures on the $\ell$-adic cohomology of local Shimura varieties.

分裂酉Shimura簇的Oort猜想

Oort's conjecture for split unitary Shimura varieties

该研究证明分裂酉Shimura簇基本层上通用阿贝尔簇的自同构群为{±1}（除少数退化情形），类比Oort猜想并计算了通用p-可除群的一般自同构群。

On the image of l-adic Galois representations for abelian varieties of type I and II

In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.

Abeliana变种上积分 torsion 点

Integral torsion points on abelian varieties over function fields

本文在全局函数域上证明了关于阿贝尔变种上 torsion 点非 Zariski 密度的猜想，并建立了相关的 Tate-Voloch 类型定理，从而得到 Galois 轨道 torsion 点的对数等分布结果。

函数域上阿贝尔簇的GCD与ABC不等式

GCD and ABC inequalities for abelian varieties over function fields

该研究证明了特征0函数域上阿贝尔簇有理点的GCD不等式与几何abc猜想的特殊情形，采用三步证明模式，借助Xie-Yuan高度等工具解决了Buium的相关猜想。