A p-adic Height Function Of Cryptanalytic Significance

On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions

Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) $p$-adic heights on $B$ along anticyclotomic $\Z_{p}$-extensions of $E$. This provides evidence towards Schneider's conjecture on the non-vanishing of $p$-adic heights. For CM elliptic curves over $\Q$, the result was previously known as a consequence of work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based o…

Totally $T$-adic functions of small height

Let $\mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce the notion of a totally $T$-adic function: one that is algebraic over $\mathbb{F}_q(T)$ and whose minimal polynomial splits completely over the completion $\mathbb{F}_q(\!(T)\!)$. We give two proofs that the height of a nonconstant totally $T$-adic function is bounded away from zero, each of which provides a sharp lower bound. We spend the majority of the paper providing explicit constructions of totally $T$-adic functions of small height (via arithmetic dynamics) and minimum height (via …

$\mathcal{L}$-invariants, $p$-adic heights and factorization of $p$-adic $L$-functions

We continue with our study of the non-critical exceptional zeros of Katz' $p$-adic $L$-functions attached to a CM field $K$, following two threads. In the first thread, we redefine our (group-ring-valued) $\mathcal{L}$-invariant associated to each $\mathbb{Z}_{p}$-extension $K_Γ$ of $K$ in terms of $p$-adic height pairings and interpolate them as $K_Γ$ varies to a universal (multivariate) group-ring-valued $\mathcal{L}$-invariant. In the second thread, we use our results to study the exceptional zeros of the non-genuine Rankin--Selberg $p$-adic $L$-functions attached to the self-products of ne…

$p$-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进连分数的高度与超越性

Heights and transcendence of $p$--adic continued fractions

本文研究p进连分数的超越性，通过p进欧几里得算法分析Browkin算法，给出周期p进连分数的高度上界，并结合p进Roth型结果证明了两类p进连分数的超越性。

Constructing totally $p$-adic numbers of small height

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behaviour at finitely many places.

Heights and totally $p$-adic numbers

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totally $p$-adic field if the map $f$ has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset $X$ in the compositum of all number fields of degree at most $d$ such that $f(X)=X$ for some non-linear polynomial $f$. This answers a question of W. Narkiewicz from 1963.

Totally p-adic Numbers of Degree 3

The height of an algebraic number $α$ is a measure of how arithmetically complicated $α$ is. We say $α$ is totally $p$-adic if the minimal polynomial of $α$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic numbers. In this paper, we investigate what can be said about the smallest nonzero height of a degree $3$ totally $p$-adic number. In particular, we provide an algorithm to determine, given a prime $p$, the smallest height of a degree $3$ totally $p$-adic number.

The p-adic height pairings of Coleman-Gross and of Nekovar

We prove that the p-adic height pairing of Nekovar, considered for algebraic curves, gives the p-adic height pairing of Coleman and Gross, defined using Coleman integration.

p-Adic sigma functions and heights on Jacobians of genus 2 curves

Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $Θ$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\colon J(\overline{\mathbb{Q}})\to \mathbb{Q}_p$ by means of $p$-adic analogues of Néron functions of divisor $2Θ$. We define such Néron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic Néron function $λ_v$ at a…

Height pairings, exceptional zeros and Rubin's Formula: The multiplicative group

In this paper we prove a formula, much in the spirit of one due to Rubin, which expresses the leading coefficients of various p-adic L-functions in the presence of an exceptional zero in terms of Nekovar's p-adic height pairings on his extended Selmer groups. In a particular case, the Rubin-style formula we prove recovers a p-adic Kronecker limit formula. In a disjoint case, we observe that our computations with Nekovar's heights agree with the Ferrero- Greenberg formula (more generally, Gross' conjectural formula) for the leading coefficient of the Kubota-Leopoldt p-adic L-function (resp., th…

$p$-adic高度配对的差分公式：通过Bloch-Kato对数映射

A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map

本文研究$p$-adic高度配对对局部分裂的依赖性，通过Bloch-Kato对数给出差分公式，并证明在特定条件下至少一个分圆高度配对非平凡。

p-adic Heights of Heegner points on Shimura curves

Let $f$ be a primitive Hilbert modular form of parallel weight $2$ and level $N$ for the totally real field $F$, and let $p$ be a rational prime coprime to $2N$. If $f$ is ordinary at $p$ and $E$ is a CM extension of $F$ of relative discriminant $Δ$ prime to $Np$, we give an explicit construction of the $p$-adic Rankin-Selberg $L$-function $L_{p}(f_E,\cdot)$. When the sign of its functional equation is $-1$, we show, under the assumption that all primes $\wp\vert p$ are principal ideals of ${\mathcal O}_{F}$ which split in ${\mathcal O}_{E}$, that its central derivative is given by the $p$-adi…

关于塞勒复形、斯塔克系统和导出p-进高度

On Selmer complexes, Stark systems and derived $p$-adic heights

论文发展了Nekovář的塞勒复形理论，证明在温和假设下，塞勒复形与Poitou-Tate复形准同构，并应用此结果构建了控制塞勒群的Heegner点斯塔克系统，同时证明Bertolini-Darmon导出p-进高度配对与Nekovář的相同。

在双曲曲线上的p进高度计算

Computing p-adic heights on hyperelliptic curves

本文提出一种计算双曲曲线在p处的局部Coleman-Gross p进高度的算法，该算法比之前的算法更高效且适用于奇数和偶数次双曲曲线。

p-adic height pairings on abelian varieties with semistable ordinary reduction

We prove that for abelian varieties with semistable ordinary reduction the p-adic Mazur-Tate height pairing is induced by the unit root splitting of the Hodge filtration on the first deRham cohomology.

二 genus 曲线的 Jacobians 上的一个朴素 p-进高度

A naive p-adic height on the Jacobians of curves of genus 2

研究二 genus 曲线的 Jacobians 上的朴素 p-进高度构造，基于 affine 方程 y² = f(x) 和 p 进数理论，展示其二次性质并与 Bianchi 构造的 p-进高度相等。

The crystalline period of a height one $p$-adic dynamical system over $\mathbf{Z}_p$

Let $f$ be a continuous ring endomorphism of $\mathbf{Z}_p[[x]]/\mathbf{Z}_p$ of degree $p.$ We prove that if $f$ acts on the tangent space at $0$ by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group over $\mathbf{Z}_p.$ The proof relies on finding a stable embedding of $\mathbf{Z}_p[[x]]$ in Fontaine's crystalline period ring with the property that $f$ appears in the monoid of endomorphisms generated by the Galois group of $\mathbf{Q}_p$ and crystalline Frobenius. Our result verifies, over $\mathbf{Z}_p,$ the height one…

Computing local p-adic height pairings on hyperelliptic curves

We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage.

p-adic heights and Vologodsky integration

We extend the result of a previous work to the case of curves with semi-stable reduction. In this case, one can replace Coleman integration by Vologodsky integration to extend the Coleman-Gross definition of a $p$-adic height pairing. we show that this pairing still equals the one defined by Nekovar.

On the $p$-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded (twisted) height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the $p$-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is $p$-adic meromorphic.

Equidistribution and the heights of totally real and totally p-adic numbers

C.J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally p-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height to generalize both Bogomolov-type results to a wide variety of heights arising in arithmetic dynam…

Quadratic Chabauty and rational points I: p-adic heights

We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit nonabelian Chabauty.

The Chabauty--Coleman method and p-adic linear forms in logarithms

Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the $p$-adic distance between rational points in terms of the heights of a set of Mordell-Weil generators of the Jacobian. We also explain how, in some cases, a conjecture on the 'Wieferich statistics' of Jacobians of curves implies a bound on the height of rational points of curves of small rank, in terms of the usual invariants of the curve and the height of Mord…

阿贝尔簇上完全 $p$-adic 扩张的下界

Lower bound for the canonical height on abelian varieties over totally p-adic extensions

本文证明了阿贝尔簇在完全 $p$-adic 扩张中满足 Bogomolov 性质，首次在良好退化情况下获得此类结果。

超椭圆Mumford曲线的算法：$p$-adic 均匀化、$p$-adic 积分和 $p$-adic 高度

Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights

针对超椭圆Mumford曲线，提出计算$p$-adic Schottky群的算法，并利用该均匀化计算$p$-adic Abelian积分和Schneider高度，通过SageMath数值示例验证。

Coleman-Gross height pairings and the $p$-adic sigma function

We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Colema…

关于高度为一的p进动力系统的卢宾猜想

Lubin's conjecture for height-one $p$-adic dynamical systems

利用p进霍奇理论，证明了关于\(\mathbb{Q}_p\)有限扩域整数环\(\mathcal{O}\)上高度为一的交换形式幂级数对的卢宾猜想，表明这样的一对是\(\mathcal{O}\)上形式群的自同态对。

Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell-Weil group tensored with the rationals.

导出 p-adic 高度与 Bertolini–Darmon–Prasanna p-adic L 函数的首项系数

Derived $p$-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna $p$-adic $L$-function

本文研究 BDP p-adic L 函数，提出 p-adic BSD 猜想，借助代数对象与反环轮 p-adic 高度证明首项系数及消失阶结论，并在超奇异情形证明主猜想。

一阶p-进弦理论与Néron局部高度函数

One-loop $p$-adic string theory and the Néron local height function

研究p-进弦世界片动作在PGL(2,Q_p)布鲁hat-蒂茨树商下的双描述，揭示其与Tate曲线Néron-Tate局部高度函数的关系。

Coleman-Gross 高度与 $p$-adic Néron 函数在 genus 2 曲线的 Jacobians 上

Coleman-Gross Heights and $p$-adic Néron Functions on Jacobians of Genus $2$ Curves

本文研究了 genus 2 曲线的 Jacobians 上 $p$-adic Néron 函数的理论，将 Coleman-Gross 高度分解为局部高度函数，并扩展了 Balakrishnan 和 Besser 的工作。

p-进解析度量与二次查巴乌 I

p-adic adelic metrics and Quadratic Chabauty I

本文提出了一种基于p-进阿雷克洛夫理论的新方法，用于构造abelian簇上的p-进高度，并简化了二次查巴乌方法用于计算有理点。

Height one specializations of Selmer groups

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to four dimensional Galois representations coming from (i) the tensor product of two cuspidal Hida families $F$ and $G$, (ii) its cyclotomic deformation, (iii) the tensor product of a cusp form $f$ and the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k \geq 2$. We prove control theorems to relate (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to its cyclotomic deformation a…

几何二次Chabauty方法与$p$进高度

Geometric quadratic Chabauty and $p$-adic heights

本文针对亏格大于1的有理曲线，在满足秩条件下，将几何二次Chabauty方法转化为p进高度与p进积分的语言，给出对应算法，证明其生成的p进点集包含于上同调方法的结果并描述差异。

任意约化类型超椭圆曲线上 $p$-进高度的算法

Algorithms for $p$-adic Heights on Hyperelliptic Curves of Arbitrary Reduction

本文提出计算任意约化类型超椭圆曲线上 Coleman--Gross $p$-进高度的算法，涵盖局部高度计算、Blakestad 分裂算法及数值实例，包括首个坏约化曲线的二次 Chabauty 示例。

无限扩张中的小高度点

On points of small height in infinite extensions

本文提出渐近正无限扩张概念，证明此类扩张上对数Weil高度和椭圆曲线典范高度的下界，推广Bombieri-Zannier及Baker的Bogomolov性质结果。

Lubin的猜想对于高度为一的p进动力系统在循环 tame 扩张上的情况

Lubin's conjecture for height-one $p$-adic dynamical systems over cyclo-tame extensions

研究高度为一的非可逆与可逆形式幂级数对(f, u)在循环 tame 扩张上的性质，通过晶格特征和p进Hodge理论工具验证Lubin猜想的新情况。

Local and global canonical height functions for affine space regular automorphisms

Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using previous results, we recover results on arithmetic properties of f-periodic points and non f-periodic points. We…

Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the…

$p$-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…

Commutative algebraic groups and $p$-adic linear forms

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ to $\mathbb G_a$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic coefficients and an algebraic point $u$ in a $p$-adic neighbourhood of the origin with the condition that $l$ does not vanish at $u$. We give a lower bound for the $p$-adic absolute value of $l(u)$ which depends up to an effectively computable constant only on the height of the linear form, the height of the point $u$ and $p$.

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。

On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou

Our goal in this article is to prove a form of $p$-adic Birch and Swinnerton-Dyer formula for the second derivative of the $p$-adic $L$-function associated to a newform $f$ which is non-crystalline semistable at $p$ at its central critical point, by expressing this quantity in terms of a $p$-adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through $f$. This, among other things, leads us to a proof of an a…

A generalization of Beilinson's geometric height pairing

In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety $B$ over an algebraically closed field, with values in the second $\ell$-adic cohomology group of …

$p$进微分方程的收敛 Newton 多边形 IV：控制图

The convergence Newton polygon of a $p$-adic differential equation IV : controlling graphs

本文研究 $p$进解析曲线上带联络向量丛的收敛半径有限性，利用超和性与 Newton 多边形部分高度给出控制图大小的界，并将其与 de Rham Euler 示性数联系起来。

p-adic根分离与整数多项式的判别式

p-adic root separation and the discriminant of integer polynomials

研究固定次数和有界高度的整数多项式的p-adic根分离问题，以及判别式被素数幂整除的计数问题，证明了存在大量具有几乎素数幂判别式的不可约多项式。

全 $v$-进数域的某些无限非阿贝尔扩张的 Bogomolov 性质

Bogomolov Property of some infinite nonabelian extensions of a totally $v$-adic field

本文研究数域上椭圆曲线在由挠点生成的无限非阿贝尔扩张中的高度下界，证明了绝对对数 Weil 高度与 Néron-Tate 高度的 Bogomolov 性质，并在有理基域的环面情形给出显式下界。

Newton's Method Over Global Height Fields

Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton's method applied to a squarefree polynomial f with K-coefficients will succeed in finding a root of f in the v-adic topology for infinitely many places v of K. Furthermore, we show that if K is a finite extension of the rationals or of the rational function field over a fini…

Canonical heights on elliptic curves in characteristic p

We define a new canonical height pairing on the rational points of elliptic curves over global function fields which takes values in the multiplicative group of a completion of the function field. This height serves as an analogue of both the classical Neron-Tate height and analytic p-adic heights. Our main result is that under some general hypotheses this pairing is non-degenerate.

Rankin--Selberg 乘积的反分圆 $p$-进 $L$-函数

Anticyclotomic $p$-adic $L$-functions for Rankin--Selberg product

本文在两种根数下沿反分圆方向为 hermitian 型自守形式的 Rankin--Selberg 乘积构造了 $p$-adic $L$-functions，并提出相应的非消失猜想和主猜想。

有限高度 Galois 表示的潜在半稳定性：相对情形

Potential semistability of Finite height Galois representations: Relative case

本文定义光滑半稳定约化 adic 空间上平展 Z_p-局部系的有限高度概念，并利用解析棱镜 F-晶体和 Du-Liu-Moon-Shimizu 的纯性结果，证明有限高度局部系在有限平展覆盖下的拉回是半稳定的。

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…

The motivic anabelian geometry of local heights on abelian varieties

We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the cours…