Adrian Vasiu
We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over $\dbF_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.
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We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. We also verify most of the He-Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are non-empty for these models.
By the Jacquet-Langlands correspondence and Faltings' isogeny theorem, it is known that the abelian variety J_0^{Dpq}(N) (the Jacobian of a Shimura curve of discriminant Dpq with Gamma_0(N) level structure) is isogenous to the pq-new subvariety of J_0^D(pqN), whenever p and q are distinct primes not dividing D or N. However, this approach is entirely nonconstructive, and gives no information at all about any particular isogeny between the two varieties. In this paper, we determine the kernels of all possible maps from one variety to the other up to support on a small finite set of maximal idea…
CM points on Shimura curves via QM-equivariant isogeny volcanoes
本文利用 QM 等变同源火山描述一般层 Shimura 曲线上指定序的 CM 点轨迹，完成计数、本原剩余域与次数判定，并据此研究零星 CM 点。
Finiteness of function field-valued points on exceptional Shimura varieties
研究证明了例外Shimura品种上函数域值点的有限性，并推广到更一般的Shimura品种和素数条件。
We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prov…
We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the q-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprised of unitary divisors, which arose in recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geome…
This paper concerns two families of divisors, which we call the `orthogonal' and `unitary' special cycles, defined on integral models of Shimura curves. The orthogonal family was studied extensively by Kudla-Rapoport-Yang, who showed that they are closely related to the Fourier coefficients of modular forms of weight 3/2, while the `unitary' divisors are analogues of cycles appearing in more recent work of Kudla-Rapoport on unitary Shimura varieties. Our main result shows that these two families are related by (a formal version of) the Shimura lift.
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。
Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)
本文研究 parahoric 层 Hodge 型 Shimura 簇积分模型的模 $p$ 点，通过归约到非常特殊 parahoric 情形证明 CM 点约化猜想及 Langlands–Rapoport 预测，并确定 Ekedahl–Oort 层连通分支。
In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization, in terms of geometry mod p, of curves in positive characteristics which are reduction of Shimura curves over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian varieties over k, char k=p. We show that some conditions on the l-adic monodromy over such a curve imply that this curve can be lifted to a Shimura curve.
We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds' lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.
Ekedahl-Oort strata in the $\mathsf{GU}(q-2,2)$ Shimura variety
本文针对$\textsf{GU}(q-2,2)$志村簇的特征p纤维，提出若干新颖技术，以深化对一般签名下Ekedahl-Oort分层及其与Newton分层相互作用的理解。
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。
Shimura lift of Rankin-Cohen brackets of eigenforms and theta series
本研究将Hecke本征形式与θ级数乘积的志村提升结论拓展至Rankin-Cohen括号情形，证明二者的Rankin-Cohen括号的志村提升等于该本征形式自身的Rankin-Cohen括号。
Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian variety over $\mathbb{Q}$. For an odd prime $\ell$, we provide sufficient conditions for the non-existence of rational points of order $\ell$ on $J$. As an application, we find some non-trivial subgroups of the kernel of an isogeny from the new quotient of $J_0(pq)$ to $J$.
Let $J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $65$. We study the isogenies $J_0(65)\rightarrow J^{65}$ defined over $\mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7$, as is predicted by a conjecture of Ogg.
The Ekedahl-Oort and Newton stratification of the $\mathsf{GU}(3,2)$ Shimura variety
本文研究$\textsf{GU}(3,2)$志村簇的特征$p$纤维，完整刻画了Ekedahl-Oort分层与Newton分层的相互作用，明确了两类分层中相交的层。
On a conjecture of Pappas and Rapoport
本文证明了帕帕斯和拉波波特关于Shimura型变种的 canonical 整体模型存在性的猜想，并展示了通过整体局部Shimura变种统一同源类，以及证明了Kisin和帕帕斯关于局部模型图的猜想。
In this paper we use automorph class theory formalism to construct a lifting of similitudes of quadratic Z-modules of arbitrary ternary nondegenerate quadratic forms to morphisms between certain subrings of associated Clifford algebras. The construction explains and generalizes Shimura's correspondence in the case of theta-series of positive-definite quadratic forms. Relations between associated zeta-functions are considered.
The Shimura Lift and Hilbert Modular Forms
本文汇集半整数权模形式等相关公式，给出志村提升的四种形式并应用于实二次域的希尔伯特模形式，未宣称原创性。
We describe the construction and properties of a singular theta lift for the orthogonal group $\SO(2,1)$. We obtain locally harmonic Maass forms in the sense of Bringmann-Kane-Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura-Shintani correspondence. This work provides extensions of the theta lifts considered by Hoevel and Bringann-Kane-Viazovska.
Integral canonical models of exceptional Shimura varieties
本文证明例外 Shimura 簇在足够大的素数处存在积分典范模型，并推广至周期映射像，进而得到若干算术推论。
Polarizations of abelian varieties over finite fields via canonical liftings
本文通过典范提升和范畴等价，显式计算了有限域上同源类中阿贝尔簇的极化同构类，并实现了相应算法。
We study the formal neighbourhood of a point in $μ$-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a shifted cascade. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms in analogy with the Serre-Tate canonical lift of an ordinary abelian variety.
We introduce a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we prove that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In the second part to this paper, we will show how to evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts.
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n-2, 2) with n>3. We prove a conjecture of Gross on the parametrizations of the superspecial locus in the special fiber of an orthogonal type Shimura variety and its lift in the integral model by certain homogeneous spaces. As applications, we provide a mass formula for the superspecial locus.
We seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics via characterizing curves in positive characteristics which are reduction of Shimura curve over $\mathbb{C}$. In this paper, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties over $k, \text{char} k=p$. We show that in the generic ordinary case, some tensor decomposition of the isocrystal associated to the family imply that this curve can be lifted to a Shimura curve.
When $p>2$, we construct a Hodge-type analogue of Rapoport-Zink spaces under the unramifiedness assumption, as formal schemes parametrising "deformations" (up to quasi-isogeny) of $p$-divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of "local Shimura varieties" conjectured by Rapoport and Viehmann.
The Shimura lift and congruences for modular forms with the eta multiplier
本文证明了带Dedekind eta乘子与Dirichlet特征扭曲的半整权尖点形式的Shimura型对应，确定提升的权、级及Atkin-Lehner特征值，并由此证明一大族半整权模形式空间存在无穷多个素数给出任意幂模的二次同余。
The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties. We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can b…
In a previous paper (arxiv:1409.7353), we introduced a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we proved that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In this paper, we will evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts. Using the Siegel-Weil formula …
Kummer-Artin-Schreier-Witt Theory
本文提出通过库默-阿廷-施雷尔-维特理论，将正特征中的阿廷-施雷尔-维特同态显式提升到零特征，并探讨其在不分离扩张和斯万展开式研究中的应用。
We generalize the $Λ$-adic Shintani lifting for $GL_2(Q)$ to indefinite quaternion algebras over $Q$.
We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny…
Let (G,X) be a Shimura datum and K a neat open compact subgroup of $G(\mathbb{A}_f)$. Under mild hypothesis on (G,X), the canonical construction associates a variation of Hodge structure on $\textrm{Sh}_K(G,X)(\mathbb{C})$ to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over $\textrm{Sh}_K(G,X)$ and is functorial in (G,X). Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representation…
For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational points. In other words, the main result asserts that there is a finite set $P(k)$ of prime numbers depending on $k$ such that: if there is a prime divisor of the discriminant of $B$ which is not in $P(k)$, then $M^B$ has no $k$-rational points. Moreover, we can take $P(k)$ to satisfy the following: There is an effectively computable constant $C(k)$ depending…
This paper is a continuation of [G-dS1]. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call "tautological foliations", are defined on Hilbert modular varieties, and lift to characteristic $0$. The second, the "$V$-foliations", are defined on unitary Shimura varieties in characteristic $p$ only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are $p$-closed, and the locus where they are smooth. Where not smooth, we construct a "successive blow up" …
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…
We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.
On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma
本文研究了从SL₃三重覆盖到PGL₃的Shimura提升，并通过全球周期非消失性刻画提升像，从而建立基本引理。
We study "quadratic periods" on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products of automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is that this integral refinement is implied by another conjecture (Conjecture D below) regarding integrality of theta lifts between certain quaternionic unitary groups.
We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s$-conjugacy classes. As an application, we verify the Axioms in \cite{HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s$-conjugacy class in a residually split group, the set of connected components is "controlled" by the set of straight elements associated to the $\s$-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with \…
This work is the first part in a series of three dedicated to the foundations of integral aspects of Shimura varieties and of Fontaine's categories. It deals mostly with the unramified context of (arbitrary) mixed characteristic (0,p). Among the topics covered we mention: the generalization of the classical Serre-Tate theory of ordinary p-divisible groups and of their canonical lifts, the generalization of the classical Serre-Tate-Dwork-Katz theory of (crystalline) coordinates for ordinary abelian varieties, the strong form of the generalized Manin problem, global deformations in the generaliz…
Explicit images for the Shimura Correspondence
本文为Yang关于Shimura对应的结果提供了构造性证明，得到了在特定条件下与eta-乘子关联的第r次Shimura提升的显式公式，并推广至Hecke特征形式相关运算的提升。
Unitary Shimura varieties at ramified primes and arithmetic transfer
本文研究分歧素数处酉 Shimura 簇，构造局部模型比较同构，提升至 p-可除群与 Rapoport-Zink 空间，并完全证明算术转移猜想。
On the cubic Shimura lift to $PGL(3)$: Hecke correspondences
本文为 Hecke 对应建立新基本引理，证明 $PGL_3(F)$ 与 $SL_3(F)$ 三次覆盖的球 Hecke 代数同构并匹配分布，为新的整体 Shimura 提升及周期刻画奠定关键基础。
The geometric Sen morphism is the unique lift of the Kodaira--Spencer morphism
本文证明几何Sen映射是Kodaira-Spencer映射的唯一提升，并展示了其作为晶格Hodge周期映射导数的性质，同时拓展了非minuscule周期域的应用。
An adelic approach to the Kudla-Millson lifts over totally real number fields
本文通过adele方法计算全实域上正交Shimura簇的Kudla-Millson theta提升的Fourier展开，证明其单射性并给出上同调维数下界。
The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n-1) over Q is uniformized by a formal scheme \Cal N. In the case when p is inert, we define special cycles Z(x) in \Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional -- in which case it reduce…
On canonicity for integral models of Shimura varieties with hyperspecial level
本文提出新的积分 canonical 模型定义，扩展了Kisin、Kim-Madapusi和Imai-Kato-Youcis的工作，同时解决了例外Shimura簇的非空性问题，并统一证明了所有牛顿层和Ekedahl-Oort层的非空性。
We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over finite fields. We prove every K3 surface of finite height over a finite field admits a characteristic 0 lifting whose generic fiber is a K3 surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a K3 surface over a finite field. To obtain these results, we construct an analogue of Kisin's algebraic group for a K3 surface of finite height, and construct characteristic 0 lif…
We generalize the definition of CM cycles beyond the small and big CM ones studied by various authors and give a uniform formula for the CM values of Green functions associated to these special cycles in general using the idea of regularized theta lifts. Finally, as an application to Siegel 3-fold $X_2(2)$, we can compute special values of theta functions and Rosenhain $λ$-invariants at a CM cycle, which is useful for genus two curve cryptography.
Integral magneticity of the level-two K3 packet: CM theta lifts and a 2-isogeny trace contraction
研究二级K3包中三个亚纯模形式\(C_4\)、\(C_{6a}\)、\(C_{6b}\)的积分磁性。通过超几何变换、形式等同及相关公式，证明其分母为1的性质及奇素数可除性，处理2的情况，得出二级K3包具有全局分母为1的磁性这一主要贡献。
We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight 2-n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of f. Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the cen…
Recent progress on Langlands reciprocity for $\mathrm{GL}_n$: Shimura varieties and beyond
本文综述了过去十五年数域上 $\mathrm{GL}_n$ 的全局 Langlands 互反性猜想进展，重点探讨了 Shimura 簇及局部对称空间的相关结果，并介绍了 Calegari-Geraghty 方法在超越 Taylor-Wiles 经典设定的模提升定理证明中的应用。
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.
The motivic Hecke algebra for PEL Shimura varieties
本研究构建了作用于PEL志村簇上同调的动机Hecke代数，定义其为动机三角范畴中可构造动机自同态代数的子代数，结合内部动机理论得到带该代数作用的Chow动机，并列出适用情形。
The connected components of affine Deligne--Lusztig varieties
本文通过p-adic Hodge理论方法，全面解决了affine Deligne-Lusztig变种和局部Shimura变种的连通分量问题，并推导了关于Shimura变种积分模型的新几何结果。
We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory an…
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