On approximation of p-adic numbers by p-adic algebraic numbers

On a new type of the l-adic regulator for algebraic number fields

For an algebraic number field K such that prime l splits completely in K we define a regulator R(K) that characterize the subgroup of universal norms from the cyclotomic extension of K in the completed group of S-units of K, where S consists of all prime divisors of l. We prove that inequality R(K) not equals 0 follows from the l-adic Schanuel conjecture and holds true for some Abelian extensions of imaginary quadratic fields.

p-adic Eisenstein-Kronecker series and non-critical values of p-adic Hecke L-functions of an Imaginary Quadratic Field when the conductor is divisible by p

We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.

$p$-进代数数的超代数不变量

Hyper-algebraic invariants of $p$-adic algebraic numbers

本文引入超代数元素的超驯指数和超惯性指数，阐明$p$-进Mal'cev-Neumann域$\mathbb{L}_p$中超代数子域$\mathbb{L}_p^{\operatorname{ha}}$与$\overline{\mathbb{Q}}_p$、$\mathbb{C}_p$的关系，并给出超代数元素在$\mathbb{Q}_p$上驯分歧的判别准则。

Parametric rectilinearization for p-adic semi-algebraic sets and rationality of p-adic integrals

We present a rectilinearization theorem for p-adic semi-algebraic sets depending on parameters. As an application of our main theorem we present an alternative proof of a rationality result for parametric p-adic inte- grals, due to Denef.

p进霍奇理论导论

An Introduction to p-adic Hodge Theory

本综述介绍启发p进霍奇理论的相关数学领域，呈现多位学者的核心成果，说明其所需的前置数学背景。

权2新形式的二次扭曲的代数中心$L$-值的2-adic赋值

The 2-adic valuations of the algebraic central $L$-values for quadratic twists of weight 2 newforms

本文在温和假设下，给出了权2新形式的二次扭曲的代数中心$L$-值的2-adic赋值的尖锐下界，并对无穷多个导子计算了该赋值。

On multiple polylogarithms in characteristic $p$: $v$-adic vanishing versus $\infty$-adic Eulerianness

In this paper, we give a simultaneous vanishing principle for the $v$-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points, where $v$ is a finite place of the rational function field over a finite field. This principle establishes the fact that the $v$-adic vanishing of CMPLs at algebraic points is equivalent to its $\infty$-adic counterpart being Eulerian. This reveals a nontrivial connection between the $v$-adic and $\infty$-adic worlds in positive characteristic.

p进代数Hahn级数的支撑集的序型

The order types of the supports of $p$-adic algebraic Hahn series

本文解决了Lampert关于p进代数Hahn级数支撑集序型的问题，证实Kedlaya的猜想，刻画了支撑集序型至多为ω的Q_p代数数，并确定了p进阴影映射的双代数集。

关于代数整数幂的β进制展开中排除某一位的β进制展开

On $β$-adic expansions of powers of algebraic integer omitting a digit

本文研究了代数整数幂的β进制展开中排除特定数字的性质，证明了其数量随N的变化规律。

p进估值生成函数族的超越性与代数独立性

Transcendence and algebraic independence of a family of $p$-adic valuation generating functions

本文研究了p进估值生成函数族的超越性与代数独立性，证明了其在单位圆内非零代数点处的超越性，并得出了乘法独立代数参数下的代数独立性结果。

Modularity of p-adic Galois representations via p-adic approximations

We give a direct approach to recover some of the results of Wiles and Tayor on modularity of certain 2-dimensional p-adic representations of the absolute Galois group of Q.

An explicit theory of $π_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,μ_{N},\infty\})$ - II-2 : From standard algebraic relations of weighted multiple harmonic sums to those of cyclotomic $p$-adic multiple zeta values

Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmoni…

Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, III

All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(τ)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers, divisible by the square of a prime divisor of 5, in the field $K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I is prov…

Congruences of algebraic $L$-functions of motives

We develop a framework to investigate conjectures on congruences between the algebraic part of special values of $L$-functions of congruent motives. We show that algebraic local Euler factors satisfy precise interpolation properties in $p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in quite large generality for $p$-adic families of automorphic motives. We formulate two conjectures refining (and correcting) the currently existing formulation of the Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebras and pointing out the links between conjectu…

Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, II: The Rogers-Ramanujan continued fraction

In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction $r(τ)$, and are shown to be periodic points of an algebraic function.

G-伴随在代数堆上的应用以及对Shimura堆上 canonical ℓ-adic 局部系统的兼容性研究

G-companions on algebraic stacks and applications to canonical $\ell$-adic local systems on Shimura stacks

本文扩展了Drinfeld的定理至一般reductive群上的光滑Artin堆，并应用于研究Shimura堆上canonical ℓ-adic局部系统的兼容性。

Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, I

Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 1$ (mod $8$) are shown to be periodic points of a fixed algebraic function $T(z)$ defined on the punctured disk $0< |z|_2 \le \frac{1}{2}$ of the maximal unramified, algebraic extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. All ring class fields of odd conductor over imaginary quadratic fields in which the prime $p=2$ splits are shown to be generated by complex periodic points of the algebraic function $T$, and conversely, all but two of …

On geometry of p-adic polynomials

An analogue of the Gauss-Lucas theorem for polynomials over the algebraic closure $\mathbb C_p$ of the field of $p$-adic numbers is considered.

Algebraic monodromy groups of $l$-adic representations of $\mathrm{Gal}({\overline{\mathbb Q}}/{\mathbb Q})$

In this paper we prove that for any connected reductive algebraic group G and a large enough prime $l$, there are continuous homomorphisms $$\mathrm{Gal}(\bar\mathbb Q/\mathbb Q) \to G(\bar\mathbb Q_l)$$ with Zariski-dense image, in particular we produce the first such examples for $SL_n, Sp_{2n}, Spin_n, E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a mod-$l$ representation of $\mathrm{Gal}(\bar\mathbb Q/\mathbb Q)$ related to the Weyl group of $G$ and use a variation of Stefan Patrikis' generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.

A generalization of Darmon's conjecture for Euler systems for general p-adic representations

Darmon's conjecture on a relation between cyclotomic units over real quadratic fields and certain algebraic regulators was recently solved by Mazur and Rubin by using their theory of Kolyvagin systems. In this paper, we formulate a "non-explicit" version of Darmon's conjecture for Euler systems defined for general $p$-adic representations, and prove it. In the process of the proof, we introduce a notion of "algebraic Kolyvagin systems", and develop their properties.

Hensel's Lemma, Backward Dynamics and p-adic Approximations

The problem of backward dynamics over the ring of p-adic integers is studied. It is shown that Inverse Limit Theory provides the right framework. Backward iterations of a polynomial with p-adic integer coefficients are constructed by solving congruences modulo powers of p, which inturn are solved by Hensel's lifting lemma.

Simultaneous p-adic Diophantine approximation

The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the Jarník-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. Th…

实数与p-进数同时乘法有理逼近

Simultaneous multiplicative rational approximation to a real and a $p$-adic numbers

本文构造了实数与p-进数的新例子，证明其满足Einsiedler和Kleinbock在2007年提出的同时乘法有理逼近猜想。

多重多对数的对偶-反射公式及其 ℓ-进伽罗瓦类似物

Duality-reflection formulas of multiple polylogarithms and their $\ell$-adic Galois analogues

本文利用射影直线去掉三点后的 S₃-对称性所诱导的关联子代数关系，证明了复数与 ℓ-进多重多对数在指标对偶和变量反射两方面的一致公式。

Small totally $p$-adic algebraic numbers

The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than $\log(3)/p$.

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 $p$.

混合特征中的代数类与André的p进周期

Algebraic classes in mixed characteristic and André's p-adic periods

本文定义André的p进周期代数，构建tannakian框架研究其超越度并类比Grothendieck周期猜想，展示经典p进函数特殊值作为周期出现的例子，联系代数类经典问题。

$p$-adic 性质的分裂多项式和代数 sigma 函数

$p$-adic properties of division polynomials and algebraic sigma functions

本文研究椭圆曲线分裂多项式序列的 $p$-adic 收敛性，在 ordinary 和 supersingular 情形下证明收敛并给出极限的显式代数公式，应用于椭圆可除序列。

$p$-adic $L$-functions on metaplectic groups

With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half-integral weight is lopsided; the analytic theory is strong whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture -- the $p$-adic $L$-function obtained by interpolating the complex $L$-function at special values. This is achieved through the Rankin-Selberg method and the explicit Fourier expansion of non-holomorphic Siegel Eisenstein series. The construction of the $p$-stabilisation in this …

Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)

Explicit solutions of the cubic Fermat equation are constructed in ring class fields $Ω_f$, with conductor $f$ prime to $3$, of any imaginary quadratic field $K$ whose discriminant satisfies $d_K \equiv 1$ (mod $3$), in terms of the Dedekind $η$-function. As $K$ and $f$ vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension $\textsf{K}_3$ of the $3$-adic field $\mathbb{Q}_3$. This is used to give a dynamical proof of a class number rela…

The p-adic closure of a subgroup of rational points on a commutative algebraic group

Let G be a commutative algebraic group over Q. Let Gamma be a subgroup of G(Q) contained in the union of the compact subgroups of G(Q_p). We formulate a guess for the dimension of the closure of Gamma in G(Q_p), and show that its correctness for certain tori is equivalent to Leopoldt's conjecture.

一个v-adic版本的安德森-布朗诺-帕纳诺斯线性独立性准则及其应用

A $v$-adic variant of Anderson-Brownawell-Papanikolas linear independence criterion and its application

本文研究了v-adic卡利茨多重polylogarithms在代数点上的线性关系，并证明了当v的度数为一时，这些关系可以由k-线性关系产生，同时提出了一个函数域版本的furusho-yamashita猜想。

Counting rational points close to $p$-adic integers and applications in Diophantine approximation

We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …

实二次域上扭三元积伽罗瓦表示的（代数）p-进阿廷形式化

(Algebraic) $ p $-adic Artin formalism of twisted triple product Galois representations over real quadratic fields

本文将Büyükboduk等人的代数分解框架推广到实二次域，研究扭三元积伽罗瓦表示的分解问题，并首次在秩(0,2)情形下取得分解结果，推动了全实域上三元积动机的研究。

代数Rankin-Selberg $p$-进 $L$ 函数的函数方程

Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions

本文为 Selmer 复形建立代数函数方程，从而得到代数 $p$-进 $L$ 函数的函数方程，并证明其完美性，应用于 Iwasawa 理论中 Rankin-Selberg 乘积的研究。

Algebraic functional equations and completely faithful Selmer groups

Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p \geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_\infty/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group $G=\operatorname{Gal}(K_\infty/K)$. Under some mild additional hypotheses this gives an algebraic functional equation of the conjectured $p$-adic L-function. As an application we construct comp…

On periods: from global to local

Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.

An Ultrametric lethargy result and its application to p-adic number theory

In this paper we show a lethargy result in the non-Arquimedian context, for general ultrametric approximation schemes and, as a consequence, we prove the existence of p-adic transcendental numbers whose best approximation errors by algebraic p-adic numbers of degree less than or equal to n decays slowly.

Height bounds for algebraic numbers satisfying splitting conditions

In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as totally real and totally p-adic numbers. In this paper, we extend these techniques and solve the corresponding minimization programs for real intervals and p-adic discs, obtaining several new lower bounds for the height of algebraic numbers all of whose conjugates lie in such sets.

从 $p$-进代数流形采样

Sampling from $p$-adic algebraic manifolds

受实代数流形采样研究启发，提出利用互补维数随机线性空间切割局部域上仿射或射影代数流形的采样方法，给出实现并验证其在多类代数结构采样场景的适用性。

On $p$-adic quaternionic Eisenstein series

We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a $p$-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of $p$-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).

关于非完整性、超越性与 $p$-进赋值

On non-holonomicity, transcendence and $p$-adic valuations

本文研究了 $p$-进赋值相关幂级数的性质，证明其在特征 0 下非完整且非代数，找到了使级数值超越的无穷多个有理数与代数无理数，并应用于 $p$-自动序列。

Deformation of $\ell$-adic sheaves with Undeformed Local Monodromy

Let $X$ be a smooth connected algebraic curve over an algebraically closed field $k$. We study the deformation of $\ell$-adic Galois representations of the function field of $X$ while keeping the local Galois representations at all places undeformed.

On the rational approximation to $p$-adic Thue--Morse numbers

Let $p$ be a prime number and $ξ$ an irrational $p$-adic number. Its multiplicative irrationality exponent ${μ^{\times}} (ξ)$ is the supremum of the real numbers ${μ^{\times}}$ for which the inequality $$ |b ξ- a|_{p} \leq | a b |^{- {μ^{\times}} / 2} $$ has infinitely many solutions in nonzero integers $a, b$. We show that ${μ^{\times}} (ξ)$ can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of $ξ$. We establish that ${μ^{\times}} ({{ξ_{{\bf t}, p}}}) = 3$, where ${{ξ_{{\bf t}, p}}}$ is the $p$-adic number $1 - p - p^2 + p^…

Higher dimensional 3-adic CM construction

We find equations for the higher dimensional analogue of the modular curve X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime 3. We prove the existence of a quasi-quadratic time algorithm for computing a canonical lift in characteristic 3 based on these equations, with a detailed description of our method in genus 1 and 2.

Carlitz动机的$v$-进周期与Chowla-Selberg公式再探

$v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited

该研究在函数域数论框架下，利用带复乘的Carlitz动机的$v$-进晶体-德拉姆周期阐释$v$-进算术伽马值，建立Ogus型Chowla-Selberg公式，证明周期的代数无关性，并推导伽马值代数关系的来源。

非普通素点处模形式的p进L函数的同余

Congruences of $p$-adic $L$-functions of modular forms at non-ordinary primes

针对p非普通且权重相等的同余模形式，研究其p进L函数的Iwasawa不变量关联及带符号Iwasawa主猜想的等价性。

张量幂上卡尔蒂兹模上代数点之间的线性关系

Linear relations among algebraic points on tensor powers of the Carlitz module

本文研究了卡尔蒂兹模张量幂上代数点的线性关系，利用对偶t-动机和弗罗贝尼乌斯差分方程，推导出在∞-adic和v-adic设置中卡尔蒂兹多项式对数在代数点上的线性无关性条件。

Mahler's classification and a certain class of $p$-adic numbers

In this paper, we study a relation between digits of $p$-adic numbers and Mahler's classification. We show that an irrational $p$-adic number whose digits are automatic, primitive morphic, or Sturmian is an $S$-, $T$-, or $U_1$-number in the sense of Mahler's classification. Furthermore, we give an algebraic independence criterion for $p$-adic numbers whose digits are Sturmian.

Simultaneous Approximation to Real and p-adic Numbers

We study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. Davenport and W.M. Schmidt in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality requires to study the dual problem of approximation to successive powers of these numbers by rational numbers with the same denominators. Dirichlet's Box Principle provides estimates for such approximations but one can do better. In this thesis we establis…

p进数列的超越性与一个量化p进Roth定理

Transcendence of $p$-adic continued fractions and a quantitative $p$-adic Roth theorem

本文研究了p进数列的超越性，证明回文和拟周期性p进数列收敛于超越数或二次无理数，并提供了一个量化版的p进Roth定理。

Local theta-regulators of an algebraic number -- p-adic Conjectures

Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible representation. A linear representation L^theta=delta.V\_theta is associated withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) := Reg\_p(eta)/p^[K : Q] (norm…

Katz型p-adic L函数在素数p于CM域中不分裂时的情形

Katz type p-adic L-functions for primes p non-split in the CM field

本文为F为椭圆特征形式、K为二次虚域且p>3在K中不分裂的三元组构造了p-adic L函数，扩展了Katz及Bertolini-Darmon-Prasana的经典构造，并证明了Kronecker极限公式与p-adic Gross-Zagier公式。

Multiplicative $p$-adic metric Diophantine approximation on manifolds and dichotomy of exponents

In this paper we study $p$-adic Diophantine approximation on manifolds, specifically multiplicative Diophantine approximation on affine subspaces and a Diophantine dichotomy for analytic $p$-adic manifolds.