Graham Everest
A new proof is given for the explicit formulae for the non-archimedean canonical height on an elliptic curve. This arises as a direct calculation of the Haar integral in the elliptic Jensen formula.
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To compute generators for the Mordell-Weil group of an elliptic curve over a number field, one needs to bound the difference between the naive and the canonical height from above. We give an elementary and fast method to compute an upper bound for the local contribution to this difference at an archimedean place, which sometimes gives better results than previous algorithms.
We solve an energy minimization problem for local fields. As an application of these results, we improve on lower bounds set by Bombieri and Zannier for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In the case of fields with mixed archimedean and non-archimedean splitting conditions, we are able to combine our bounds with similar bounds at the archimedean places for totally real fields.
We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.
Degeneration of the archimedean height pairing of algebraically trivial cycles
研究代数上平凡循环在退化族中阿基米德高度配对的极限行为，通过非阿基米德几何高度配对控制，并提出对Beilinson--Bloch高度配对正性的新见解。
We generalize results about local heights previously proved in the case of discrete absolute values to arbitrary non-archimedean absolute values of rank 1. First, this is done for the induction formula of Chambert-Loir and Thuillier. Then we prove the formula of Burgos--Philippon--Sombra for the toric local height of a proper normal toric variety in this more general setting. We apply the corresponding formula for Moriwaki's global heights over a finitely generated field to a fibration which is generically toric. We illustrate the last result in a natural example where non-discrete non-archime…
A tropical formula for non-Archimedean local heights
本文提出了一种基于热带几何的非阿基米德局部高度计算方法，通过定义delta形式和星积来实现。
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。
We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C} ((t))$, in analogy to results by Pila and Wilkie, resp. by Bombieri…
On the non-Archimedean metric Mahler measure
本文提出了度量Mahler测度的非阿基米德版本$M_\infty$，证明了其取值为1当且仅当$α$为单位根，并验证了其在相应向量空间上定义了射影高度，最后给出了根式情形下的计算方法。
A dynamical system over a non-archimedean field
本文从复与非阿基米德势理论视角综述非阿基米德动力系统，梳理 Berkovich 射影直线、动力模空间与高度函数，并总结 Rumely 等变量及 Silverman 猜想等成果。
We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field F_q and the degree, generalizing work by Sedunova for fixed q. We also deduce a uniform non-Archimedean Pila-Wilkie theorem, generalizing work by Cluckers-Comte-Loeser.
Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-…
In this article we introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irra…
Using a `height-to-radical' identity, we define the archimedean contribution to the radical, $r_\arch$, and we give a new proof of the abc theorem for the field of meromorphic functions. The first step of the proof is completely formal and yields that the height is bounded by the radical, $h\leq r$, where $r=r_\na+r_\arch$ is the radical completed with the archimedean contribution. The second step is analytic in nature and uses the lemma on the logarithmic derivative to derive a bound for $r_\arch$.
We prove a formula, which, given a principally polarized abelian variety $(A,λ)$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron--Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,λ)$.
Let $B$ be a half-integral symmetric matrix of size $n$ defined over $\mathbb{Q}_p$. The Gross-Keating invariant of $B$ was defined by Gross and Keating, and has important applications to arithmetic geometry. But the nature of the Gross-Keating invariant was not understood very well for $n\geq 4$. In this paper, we establish basic properties of the Gross-Keating invariant of a half-integral symmetric matrix of general size over an arbitrary non-archimedean local field of characteristic zero.
Let $F$ be the function field of a curve over a non-archimedean local field. Let $m \geq 2$ be an integer coprime to the characteristic of the residue field of the local field. In this article, we show that every element in $H^{3}(F, μ_{m}^{\otimes 2})$ is of the form $χ\cup (f) \cup (g)$, where $χ$ is in $H^{1}(F, \mathbb{Z}/m\mathbb{Z})$ and $(f)$, $(g)$ in $H^{1}(F, μ_{m})$. This extends a result of Parimala and Suresh, where they show this when $m$ is prime and when $F$ contains $μ_{m}$.
Canonical local heights and Berkovich skeleta
本文从Berkovich解析空间视角研究非阿基米德域上阿贝尔簇的典范局部高度，细化了Néron关于其与相交重数关系的经典结果，并将Tate的显式公式推广到高维。
We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.
Dynamical Canonical Heights and Finite Trees
本文针对素数幂次的特定多项式，基于Dimitrov的思路，通过构建有限树并结合核心熵分析，建立了游荡点典范高度的平方级衰减下界。
Green's function on the Tate curve
本文研究了Tate曲线上的非阿基米德Green函数，其在p→∞极限下恢复了Néron局部高度函数，并赋予了非阿基米德共形场论两点函数的物理意义。
We prove a local converse theorem for $GL_n$ over the archimedean local fields which characterizes an infinitesimal equivalence class of irreducible admissible representations of $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ in terms of twisted local gamma factors.
The Local Gross-Prasad Conjecture over Archimedean Local Fields
该研究采用C. Moeglin和J.L.-Waldspurger的方法，完成了阿基米德局部域上局部Gross-Prasad猜想一般情形的证明，推进了相关数论问题的研究。
Counting roots of unity on the graphs of Laurent series over non-Archimedean local fields
该研究在非阿基米德局部域上对洛朗级数的单位根映射进行分类，并通过有效界限探讨了其与曼因-穆福德猜想的关系。
Let $F$ be a non-archimedean local field. The classification of the irreducible representations of $GL_n(F)$, $n\ge0$ in terms of supercuspidal representations is one of the highlights of the Bernstein--Zelevinsky theory. We give an analogous classification for metaplectic coverings of $GL_n(F)$, $n\ge0$.
Multiplicity formula for induced representations: Bessel and Fourier-Jacobi models over Archimedean local fields
本文推广Moeglin-Waldspurger方法，证明了阿基米德局部域上Bessel与Fourier-Jacobi模型的诱导表示重数公式，并据此对所有经典群给出局部Gan-Gross-Prasad猜想的一致证明。
Let $F$ be a local field over $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$, and let $D$ be a central simple division algebra over $F$ of degree $d$. In the $p$-adic case, we assume $p>de+1$ where $e$ is the ramification degree over $\mathbf{Q}_p$; otherwise, we need only assume $p$ and $d$ are coprime. For the subgroup $I_1=1+\varpi_D \mathcal{O}_D$ of $D^\times$ we determine the structure of $\mathrm{H}^1(I_1, π)$ as a representation of $D^\times/I_1$ for an arbitrary smooth irreducible representation $π$ of $D^\times$. We use this to compute the group $\mathrm{Ext}^1_{D^\times}(π,π')$ for arbitrary…
We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $ε>0$ such that only finitely many $P\in K^{\text{ab}}$ have canonical height less than $ε$ with respect to $f$. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the $K$-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in $K$. We further prove uncond…
Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap to complete the analytic theory of the complete $L$-function. In this paper, we will establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburg's global integral for the adjoint representation of $\mathrm{GL}_3$. Via the local functional equation, the local gamma factor $Γ(s,π,\…
We derive an upper bound on the support of matrix coefficients of suprecuspidal representations of the general linear group over a non-archimedean local field. The results are in par with those which can be obtained from the Bushnell--Kutzko classification of supercuspidal representations, but they are proved independently.
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。
We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose entropy is the global canonical height and whose periodic points are counted asymptotically by t…
We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.
Schwartz spaces on L-monoids: non-Archimedean
该研究在非阿基米德局部域上，结合相关假设完成了Braverman-Kazhdan-Ngô纲领，且该纲领对一般线性群可无条件成立。
Asymptotics of local height pairing
研究局部高度对的渐进行为，通过同调代数和Deligne上同调方法比较不同双延伸构造，扩展了Beilinson的Hodge上同调理论。
$L^2$-property for algebraic stacks over local non-archimedean fields
本文研究了局部非阿基米德域上代数堆叠的$L^2$-范数性质，证明了在特定条件下这些范数的有限性，并与分析Langlands对应中的猜想相关。
In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.
We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form $π\otimesσ$ where $π$ is a ladder representation and $σ$ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.
Local information of ADC quadratic lattices over algebraic number fields
本文研究了ADC二次格在代数数域上的局部结构，确定了二元和三元格的表示集、计数公式及密度，并探讨了其在非阿基米德局部域上的性质与应用。
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。
We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.
Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean
本文作为系列论文首篇，提出新的局部极限方法，证明所有非阿基米德位的关键局部定理，为证明余秩1算术Siegel–Weil公式奠定基础。
In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in non-Archimedean local fields.
We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, th…
In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over $\bbR$, $\bbC$ or $\bbH$ of independent interest.
Non-Archimedean Kelvin Transformation
本文在非阿基米德局部域上引入并研究开尔文变换的类比，将其与作用于K^n上实值或复值函数的Vladimirov-Taibleson算子联系起来。
Finite-Sum Realization of Archimedean Asai and Exterior-Square $L$-Factors
该研究在数论领域，将 Archimedean Asai L-因子与外平方 L-因子分别表示为特定局部 ζ 积分的有限和，完善了 L-因子的有限和实现相关理论。
Co-rank $1$ Arithmetic Siegel--Weil II: Local Archimedean
本文作为系列论文第二篇，证明余秩1情形下关键阿基米德局部定理及纯阿基米德相交数下的局部算术Siegel–Weil公式，通过新极限方法将Green流与局部Whittaker函数离中心导数相联系。
In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(π)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields and thus are only new when $F=\mathbb{R}$.
We prove a variant of a square function estimate for the extension operator associated to the moment curve in non-archimedean local fields. The arguments rely on a structural analysis of congruences (sublevel sets) of univariate polynomials over field extensions of the base field. Our analysis can be adapted to the archimedean setting as well.
We calculate the asymptotic behavior of the dimension of the fixed vectors of $π$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $π$ an admissible representation of $\mathrm{GL}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $π$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.
In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schröer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.
In this paper, we partially complete the local Rankin-Selberg theory of Asai $L$-functions and $ε$-factors as introduced by Flicker and Kable. In particular, we establish the relevant local functional equation at Archimedean places and prove the equality between Rankin-Selberg's and Langlands-Shahidi's $ε$-factors at every place. Our proofs work uniformly for any characteristic zero local field and use as only input the global functional equation and a globalization result for a dense subset of tempered representations that we infer from work of Finis-Lapid-Müller. These results are used in an…
Inspired by Manin's approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in $P^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to PGL(n).
Non-tempered Gan-Gross-Prasad conjecture for Archimedean general linear groups
本文针对阿基米德局部域上的一般线性群，证明了非tempered Gan-Gross-Prasad猜想中“周期蕴含相关”的方向，结合已有结果解决了该猜想。
It is proved that certain types of modular cusp forms generate irreducible automorphic representation of the underlying algebraic group. Analogous archimedean and non-archimedean local statements are also given.
Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from previous papers. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-…
We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.
A Mixed Robinson--Schensted--Knuth Construction in the Representation Theory of $\mathrm{GL}_n$ over Local non-Archimedean Fields
针对局部非阿基米德域上GL_n的表示，提出混合-RSK这一新型RSK变体的组合方法，计算其根基，解决了Erez Lapid的猜想。
Non-Archimedean Neumann problem: weak and strong solutions
本文研究非阿基米德局部域上带Vladimirov-Taibleson分数阶微分算子的诺伊曼问题，分别用Dipierro等人的方法和超度量恒等式分析弱解与强解。
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