Heights of Heegner points on Shimura curves

$p$-adic Gross-Zagier Formula for Heegner Points on Shimura Curves over Totally Real Fields

The main result of this text is a generalization of Perrin-Riou's p-adic Gross-Zagier formula to the case of Shimura curves over totally real fields. Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of parallel weight $2$, which is a new form and is ordinary at $p$. Let $E$ be a totally imaginary quadratic extension of $F$ of discriminant prime to $p$ and to the conductor of $f$. We may construct a $p$-adic $L$ function that interpolates special values of the complex $L$ functions associated to $f$, $E$ and finite order Hecke characters of $E$. The $p$-adic Gross-Zag…

Ribet bimodules and the specialization of Heegner points

We describe the specialization of Heegner points on Shimura curves at primes of bad reduction. Moreover, we give some reciprocity laws relating the Galois action on these points to natural actions on the set of singular points and the set of connected components of the fiber.

On Heegner points of large conductors

Given a parametrisation of an elliptic curve over Q by a Shimura curve, we show that the images of almost all Heegner points are of infinite order. For parametrisations of elliptic curves by modular curves this was proven earlier by Nekovar and Schappacher by a different method.

Vertical Distribution Relations For Special Cycles on Unitary Shimura Varieties

We consider cycles on a 3-dimensional Shimura varieties attached to a unitary group, defined over extensions of a CM field $E$, which appear in the context of the conjectures of Gan, Gross, and Prasad \cite{gan-gross-prasad}. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of \cite{jetchev:unitary}, and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $Λ$-module constructed from Heegner points.

lectic Heegner 类

Plectic Heegner classes

本文引入了新的部分全局Galois上同调类，涵盖lectic Heegner点和mock plectic不变量。通过克服先前构造的限制，lectic Heegner类有望提供对高秩椭圆曲线算术更精细的控制。

Shimura 曲线上 Heegner 循环的不可分性与 Selmer 群

Indivisibility of Heegner cycles over Shimura curves and Selmer groups

本文证明 Shimura 曲线上 Heegner 循环与 theta 元素构成二分 Euler 系统，并由此推导高权模形式的 Gross-Zagier-Kolyvagin 逆定理，表明剩余 Selmer 群秩为一时 Heegner 循环像非零。

Big Heegner points and special values of $L$-series

In \cite{LV}, Howard's construction of big Heegner points on modular curves was extended to general Shimura curves over the rationals. In this paper, we relate the higher weight specializations of the big Heegner points of \emph{loc.cit.} in the definite setting to certain higher weight analogues of the Bertolini-Darmon theta elements. As a consequence of this relation, some of the conjectures formulated in \cite{LV} are deduced from recent results of Chida-Hsieh.

Heegner points on Hijikata-Pizer-Shemanske curves

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic or- ders closely related to those introduced by Hijikata{Pizer{Shemanske in the 80's. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In par- ticular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.

A p-adic Shimura-Maass operator on Mumford curves

We study a p-adic Shimura-Maass operator in the context of Mumford curves defined by C. Franc is his Ph.D. Thesis. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in Franc. We also study the relation of this operator with generalized Heegner cycles, in the spirit of Bertolini-Darmon-Prasanna and Brooks.

$p$-adic heights of Heegner points and Beilinson-Flach elements

We give a new proof of Howard's $Λ$-adic Gross-Zagier formula, which we extend to the context of indefinite Shimura curves over $\mathbf{Q}$ attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two-variable $p$-adic $L$-function restricted to the anticyclotomic line to the cyclotomic $p$-adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the original approaches of Gross-Zagier and Perrin-Riou, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and their expli…

$p$-adic heights of generalized Heegner cycles

We relate the $p$-adic heights of generalized Heegner cycles to the derivative of a $p$-adic $L$-function attached to a pair $(f, χ)$, where $f$ is an ordinary weight $2r$ newform and $χ$ is an unramified imaginary quadratic Hecke character of infinity type $(\ell,0)$, with $0 < \ell < 2r$. This generalizes the $p$-adic Gross-Zagier formula in the case $\ell = 0$ due to Perrin-Riou (in weight two) and Nekováur (in higher weight).

全实域上的四元数大Heegner点

Quaternionic big Heegner points over totally real fields

本研究将Howard的相容Heegner点族构造推广到全实域上的Gross曲线与志村曲线塔，同时处理全定与不定四元数代数，定义平行权希尔伯特模形式族的大Heegner点，构建对应p进L函数类似物与大Heegner类系统。

A Gross--Kohnen--Zagier Type Theorem for Higher-Codimensional Heegner Cycles

We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.

中心 $L$-导数的极值与 Heegner 点的高度

Extreme values of central $L$-derivatives and heights of Heegner points

本文用共振方法求中心L-导数的极值，并通过Gross-Zagier公式得到Heegner点高度及Beilinson-Bloch高度的相应大值。

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and CM points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin-Selberg convolution associated with a cusp form of half-integral weight $n/2 +1 $, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shi…

Hauteur asymptotique des points de Heegner

The asymptotic behaviour of the Neron-Tate height of Heegner points on a rational elliptic curve attached to an arithmetically normalized new cusp form f of weight 2, level N and trivial character is studied in this paper. By Gross-Zagier formula, this height is related to the special value at the critical point for the derivative of the Rankin-Selberg convolution of f with a certain weight one theta series attached to some ideal class of some imaginary quadratic field. Asymptotic formula for the first moments asociated to these Dirichlet series are proved and experimental results are carefull…

Heegner points and the rank of elliptic curves over large extensions of global fields

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our…

CM cycles on Kuga-Sato varieties over Shimura curves and Selmer groups

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a…

The p-adic Gross-Zagier formula on Shimura curves

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin-Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obta…

Stark-Heegner points and diagonal classes

Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field $K$. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime $p$ that remains inert in $K$, but are conjectured to be rational over ring class fields of $K$ and to satisfy a Shimura reciprocity law describing the action of $G_K$ on them. The main conjectures of \cite{darmon-hpxh} predict that any linear combination of Stark-Heegner points weighted by the values o…

志村曲线上的p进Gross-Zagier公式，II：非分裂素数

The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes

本文将p进Gross-Zagier公式推广到Hilbert模阿贝尔簇，并去除了Perrin-Riou原有证明中要求p在相关二次扩张中分裂的限制条件。

Shimura 簇中“混合轨道”的高度

Heights on 'Hybrid orbits' in Shimura varieties

本文在 abelian 型 Shimura 簇情形下证明“混合猜想”，统一推广了 Andreé-Oort 与 André-Pink-Zannier 两个猜想。

Computation of ATR Darmon points on non-geometrically modular elliptic curves

ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon--Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner points, thanks to results of Zhang and Darmon--Rotger--Zhao. In this paper we compute for the first time an algebraic ATR point on a curve which is not uniformizable by any Shimura curve, thus providing…

非阿基米德 plectic 雅可比簇

Non-Archimedean plectic Jacobians

本文通过高维四元数 Shimura 簇的非阿基米德一致化，为 plectic Heegner 点提供几何解释，并定义从 Mumford 簇范畴到拓扑群的 plectic 雅可比函子，推广了经典 Mumford 曲线雅可比函子。

酉 Shimura 簇的模高 II: 除子的算术生成级数

Modular Heights of Unitary Shimura Varieties II: Arithmetic Generating Series of Divisors

本文通过定义并计算酉 Shimura 簇上除子的算术生成级数及其算术相交数，推导了酉 Shimura 曲线的模高公式及其上 CM 点的高度公式。

Heights of pre-special points of Shimura varieties

Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding Z-Hodge structure. Our bound is the final step needed to complete a proof of the Andre-Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.

酉 Shimura 簇的模高度 III：主定理的证明

Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem

通过比较混合 Eisenstein-theta 级数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。

四元数志村簇上特殊点的高度

Heights of Special Points on Quaternionic Shimura Varieties

本文用 CM 阿贝尔簇的 Faltings 高度给出四元数志村簇上特殊点高度的显式公式，并证明其与部分 CM 型的典范高度相容，从而重新证明部分 CM 型高度关于判别式具有次多项式上界。

Rankin-Selberg L-函数的导数与广义Heegner圈的高度

Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles

本文证明权2k新形式与无穷型(2t,0)非分歧虚二次Hecke特征的Rankin-Selberg L-函数中心导数等于广义Heegner圈的Beilinson-Bloch高度，统一推广了Gross-Zagier公式和Zhang的高权公式。

酉 Shimura 簇的模高度 III：主定理的证明

Modular Heights of Unitary Shimura Varieties I: Derivatives of Eisenstein Series

本文通过比较混合 Eisenstein-theta 级数导数的全纯投影与除子生成级数的算术度，证明了 CM 数域上酉 Shimura 簇的模高度公式。

Quaternion algebras, Heegner points and the arithmetic of Hida families

Given a newform f, we extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovář's extended Selmer groups attached to suitable bi…

On rigid analytic uniformizations of Jacobians of Shimura curves

The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over the field of rational numbers at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of Cerednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenberg's construction of local points on …

Horizontal non-vanishing of Heegner points and toric periods

Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular $\GL_2$-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $χ$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,χ)$ is self-dual with root number $-1$. We show that the number of class group characters $χ$ with bounded ramification such that $L'(1, A, χ) \neq 0$ increases with the absolute value of the discriminant of $K$. We also consider a rather general rank zero situation. Let $π$ be a cuspidal cohomological automorphic representation over $\GL_{2}(\BA_{F})$. Let $χ$ be a …

更高海根点非 torsion 性的马祖尔猜想加强

Strengthenings of Mazur's Conjecture for Higher Heegner Points

本文通过伽罗瓦轨道与海根轨道的相互作用，定量加强了马祖尔猜想关于更高海根点非 torsion 性的结论，避免了对水平结构的严格限制。

The geometric and arithmetic volume of Shimura varieties of orthogonal type

We apply the theory of Borcherds products to calculate arithmetic volumes (heights) of Shimura varieties of orthogonal type up to contributions from very bad primes. The approach is analogous to the well-known computation of their geometric volume by induction, using special cycles. A functorial theory of integral models of toroidal compactifications of those varieties and a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics are used. We obtain some evidence in the direction of Kudla's conjectures on relations of heights of special cycles on these v…

阿基米德算术西格尔-韦尔公式：关于 Shimura 曲线上的一般权重特异除子的平均公式

Archimedean arithmetic Siegel--Weil formula for general weights over Shimura curves

本文基于阿基米德算术理论，研究Shimura曲线上的特异除子权重问题，通过Whittaker函数的导数推导出平均公式，揭示了特殊除子与算术高度配对的关系。

Comportement asymptotique des hauteurs des points de Heegner

The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner points on a rational elliptic curve E has been determined in [12]. In addition, the second order term has been conjectured. In this paper, we prove that this conjectured second order term is the right one; this yields a power saving in the remainder term. Cancellations of Fourier coefficients of GL(2)-cusp forms in arithmetic progressions lie in the core of the proof.

Degree and height estimates for modular equations on PEL Shimura varieties

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.

Faltings heights of CM cycles and derivatives of L-functions

We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.

A $p$-adic construction of ATR points on $\mathbb{Q}$-curves

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit $p$-adic analytic formula which allows for the effective, algorithmic calculation of such points.

计算某些零维 Shimura 曲线上的点数

Counting points on some genus zero Shimura curves

本文利用零维 Shimura 曲线上有界高度点的计数，研究数域 $K$ 上具潜在四元数乘法的阿贝尔曲面数量。

Shimura 簇上的典范高度与 André-Oort 猜想

Canonical Heights on Shimura Varieties and the André-Oort Conjecture

本文旨在完全一般性地证明 André-Oort 猜想，该猜想涉及 Shimura 簇上特殊子簇的分布。

The rationality of quaternionic Darmon points over genus fields of real quadratic fields

Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner poi…

四元数 Shimura 曲线的模高

Modular Heights of Quaternionic Shimura Curves

本文证明了一个公式，将全实数域上四元数 Shimura 曲线的模高表示为 Dedekind zeta 函数对数导数，基于 Yuan-Zhang-Zhang 和 Yuan-Zhang 的工作，受 Gross-Zagier 和 Kudla 纲领启发。

实数域上具有四元数乘法的阿贝尔曲面的超特殊素数

Superspecial primes for QM abelian surfaces over real number fields

本文将Baba和Granath关于特定四元数乘法阿贝尔曲面超特殊素数的结果，扩展到模域为任意实嵌入数域且弱化局部条件的情形，证明依赖志村曲线Heegner除子相交理论。

Heights of Kudla-Rapoport divisors and derivatives of L-functions

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…

Kuga-Sato 簇的 Griffiths 群中的 Heegner 圈

Heegner cycles in Griffiths groups of Kuga-Sato varieties

本文利用复 Abel-Jacobi 映射证明 Kuga-Sato 簇 Griffiths 群中由固定虚二次域的 Heegner 圈生成的子群具有无限秩，推广了 Schoen 的经典结果，并基于尖点形式线积分公式给出新证明。

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

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…

Plectic Jacobians

Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties - called plectic Jacobians - using the middle degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths' intermediate Jacobians and rests on Nekovar-Scholl's notion of plectic Hodge structures. Moreover, we construct exotic Abel-Jacobi maps sending certain zero-cycles on QSVs to plectic Jacobians.

Derivatives of Eisenstein series and Faltings heights

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated to a Shimura curve and the second term in the Laurent expansion at s=1/2 of an Eisenstein series of weight 3/2 for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions.…

Kolyvagin's result on the vanishing of $\sha(E/K)[p^\infty]$ and its consequences for anticyclotomic Iwasawa theory

Let $E$ be an elliptic curve defined over ${\bf Q}$ and $K$ an imaginary quadratic field satisfying the Heegner hypothesis. A classical result of Kolyvagin states that, under suitable assumptions, if the basic Heegner point $y_K \in E(K)$ is not divisible by an odd prime $p$, then the groups $E(K)/{\bf Z} y_K$ and $\sha(E/K)$ are finite and their orders are prime to $p$. In this article we develop the following themes: firstly, we discuss improvements of Kolyvagin's result, following Cha (2005) and Lawson and Wuthrich (2016). Secondly, we prove an abstract Iwasawa-theoretical result which allo…

CM-points and Lattice counting on arithmetic compact Riemann surfaces

Let $X(D,1) =Γ(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant $d <0$ on $X(D,1)$ as $d \to -\infty$. We prove that if $|d|$ is sufficiently large compared to the radius $r \approx \log X$ of the circle, we can improve on the classical $O(X^{2/3})$-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.

阿贝尔簇的同时超奇异约化

Simultaneous supersingular reductions of abelian varieties

本文研究了 Shimura 簇中阿贝尔概形的 $p$-adic Hecke 轨道在多个素数下的同时超奇异约化，精确刻画了映射像，并以此更概念化地证明了 Mazur 关于高阶 Heegner 点非挠性的猜想。

Faltings heights of abelian varieties with complex multiplication

Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain $L$-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.