A family of varieties with exactly one pointless rational fiber

Rigid-analytic varieties with projective reduction violating Hodge symmetry

We construct examples of smooth proper rigid-analytic varieties admitting formal model with projective special fiber and violating Hodge symmetry for cohomology in degrees $\geq 3$. This answers negatively a question raised by Hansen and Li.

Shimura 簇上特殊圈的相关连通分支

Connected components of special cycles on Shimura varieties

本文利用 Chai-Hida 方法和普通 p-Hecke 对应研究 GSpin Shimura 簇中低余维特殊圈特殊纤维的不可约分支，证明极化 K3 曲面模空间特殊纤维的不可约性，并应用于与 Howard 合作的高余维闭链生成级数模性研究。

Torsion points on isogenous abelian varieties

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau-Martínez to obtain uniform b…

On $p$-adic differential equations on semistable varieties II

This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequ…

Non-simple abelian varieties in a family: geometric and analytic approaches

Let $A_t$ be a family of abelian varieties over a number field $k$ parametrized by a rational coordinate $t$, and suppose the generic fiber of $A_t$ is geometrically simple. For example, we may take $A_t$ to be the Jacobian of the hyperelliptic curve $y^2 = f(x)(x-t)$ for some polynomial $f$. We give two upper bounds for the number of $t \in k$ of height at most $B$ such that the fiber $A_t$ is geometrically non-simple. One bound comes from arithmetic geometry, and shows that there are only finitely many such $t$; but one has very little control over how this finite number varies as $f$ change…

On rigid varieties with projective reduction

In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that p-adic Hopf varieties will never have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves.

$U(2)$ Shimura 簇上自守线丛的丰富性

Ampleness of Automorphic Line Bundles on $U(2)$ Shimura Varieties

本文证明了 $U(2)$ Shimura 簇上自守线丛丰富性的猜想，即特殊纤维上丰富性等价于系数满足特定不等式，并由此推导出 Hilbert 模簇的类似结论。

Kudla's modularity conjecture on integral models of orthogonal Shimura varieties

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.

Cycles on Shimura varieties via geometric Satake

We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set…

Central leaves on Shimura varieties with parahoric reduction

We investigate the geometry of the special fiber of the integral model of a Shimura variety with parahoric level at a given prime place. To be more precise, we deal with the definition of central leaves in this situation, their local closedness, and the relationship between the folations for varying parahoric level. This is connected to the verification of axioms for integral models formulated by He and Rapoport.

Specialization of canonical heights on abelian varieties

Given a family of abelian varieties over a quasiprojective smooth curve $T^0$ over a global field and a point $P$ on the generic fiber, we show that the Néron-Tate canonical height $h_{X_t}(P_t)$ of $P_t$ along each fiber is exactly equal to a Weil height $h_{\overline M}(t)$ given by an adelic metrized line bundle $\overline M$ on the unique smooth projective curve $T$ containing $T^0$. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of $P$ is equivalent to $\overline M$ being big.

族中的非单阿贝尔簇：算术方法

Non-simple abelian varieties in a family: arithmetic approaches

本文受Ellenberg等人工作启发，采用算术方法将其关于曲线雅可比行列式特定单参数族的结果推广到所有具有大几何单模的阿贝尔簇族，通过应用Heath-Brown型界和优化覆盖推导结论。

Shimura 变种的分裂模型上的循环

Cycles on splitting models of Shimura varieties

本文通过分裂模型构造了 Shimura 变种的 exotic Hecke 对应关系，并验证了 Tate 猜想的通用实例。

Linnik's problem in fiber bundles over quadratic homogeneous varieties

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-…

Hurwitz Number Fields

The canonical covering maps from Hurwitz varieties to configuration varieties are important in algebraic geometry. The scheme-theoretic fiber above a rational point is commonly connected, in which case it is the spectrum of a Hurwitz number field. We study many examples of such maps and their fibers, finding number fields whose existence contradicts standard mass heuristics.