Jeng-Daw Yu
We study the cohomological properties of quasi-canonical lifts of an ordinary K3 surface over a finite field. As applications, we prove a Torelli type theorem for ordinary K3 surfaces over finite fields and establish the Hodge conjecture for arbitrary self products of certain K3 surfaces over the complex numbers.
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We extend the Serre-Tate theory of canonical lifts of ordinary abelian varieties to arbitrary unpolarised families of ordinary abelian varieties parameterised by a $p$-adic formal scheme $S$. We show that the canonical lift is the unique lift to $W(S)$ which admits a $δ$-structure in the sense of Joyal, Buium, and Bousfield. We prove analogous statements for families of ordinary $p$-groups and $p$-divisible groups.
We show that the canonical-lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti an…
We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.
Double Artin-Schreier extensions of rational function fields with many lifted automorphisms
本文通过双Artin-Schreier扩展构造新的普通函数域，并确定其大自同构群。
Fourier-Mukai partners of abelian varieties and K3 surfaces in positive and mixed characteristics
本文研究了在正特征和混合特征下阿贝尔簇和K3曲面的傅里叶-穆卡伊伙伴，证明了其提升性质并恢复了原有结果。
The Tate Conjecture for Ordinary Calabi-Yau Threefolds
该研究利用典范提升理论，证明有限域上普通卡拉比-丘三维簇的积分几何塔特猜想及部分普通卡拉比-丘簇除子类的对应猜想。
Deformations and Lifts of Calabi-Yau Varieties in Characteristic $p$
本文用导出代数几何研究特征 $p$ 中 Calabi-Yau 簇的形变，证明 Bogomolov-Tian-Todorov 定理的混合特征类比，并给出普通 Calabi-Yau 簇到特征 $0$ 的典范提升，推广 Serre-Tate 定理。
We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .
EPW varieties as moduli spaces on ordinary GM surfaces and special GM threefolds
本文证明与强光滑 Gushel-Mukai 曲面相关的双对偶 EPW 六次曲面可实现为其有界导出范畴上半稳定对象的模空间，并观察到与特殊 Gushel-Mukai 三维族相关的双对偶 EPW 曲面可实现为其 Kuznetsov 分量上半稳定对象的模空间，进而讨论了主要结果到双 EPW 六次曲面和双 EPW 曲面的推广，以及在温和假设下对 Bayer 和 Perry 关于具有等价 Kuznetsov 分量的 Gushel-Mukai 三维族的一个陈述的改进。
Quintic surfaces with 18 cusps
该研究在$\boldsymbol{P}^3$中构造出含18个普通尖点的五次曲面，基于Barth–Rams分解，经有限域与特征零提升验证，为代数几何中奇异五次曲面的研究提供了新实例。
Intersections of special cycles on Shimura curves and Siegel Maass forms
本文研究Shimura曲线上的特殊循环交集与Siegel Maass形式的关系，通过非全纯模形式和theta提升方法，推广了Rickards的计数结果，并给出了傅里叶系数的几何解释。
For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characte…
S-integrality for families of ordinary K3 surfaces and algebraicity theorems
本文证明正特征GSpin Shimura簇上特殊除子的S-整性定理，通过形式特殊自同态的代数性定理结合丢番图逼近技术，并利用点态单演定理及Mumford-Tate猜想的正特征类比完成证明。
We define a regularized theta lift from SL_2 to orthogonal groups over totally real fields. It takes harmonic `Whittaker forms' to automorphic Green functions and weakly holomorphic Whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.
Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null…
Lifting Elementary Abelian Covers of Curves
本文针对特征p域上曲线的初等阿贝尔p-群伽罗瓦覆盖的提升问题，给出依赖p-循环子覆盖提升分支轨迹的组合判据，研究分支点在特殊纤维的合并规律，并分析多类(ℤ/2)³-覆盖的提升情形。
For a K3 surface of finite height over a field of odd characteristic, there exists a smooth lifting to the ring of Witt vectors such that the reduction map from the Picard group of the generic fiber to the Picard group of the special fiber is isomorphic. In this paper, using this result, we give a criterion for a K3 surface of finite height over a field of odd characteristic to be an Enriques K3 surface or to be a K3 surface in terms of the Neron-Severi lattice. Then we show every Kummer surface has an Enriques involution. We also give a classification of K3 surfaces of Picard rank of 20 over …
Generalizing work of Gross--Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to certain quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the norms of the CM values of those meromorphic modular forms arising as regularized theta lifts of holomorphic Whittaker forms.
Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields
本文在全局函数域上证明了普通半稳定椭圆曲线的Iwasawa主猜想，通过引入χ公式和μ不变量假设，展示了相关特征理想与p进L-函数的联系。
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