Luochen Zhao
Given primes $\ell\ne p$, we record here a $p$-adic valued Fourier theory on a local field over $\mathbf{Q}_\ell$, which is developed under the perspective of group schemes. As an application, by substituting rigid analysis for complex analysis, it leads naturally to the $p$-adic local functional equation at $\ell$, which strongly resembles the complex one in Tate's thesis.
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We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
A Note on p-Adic Invariant Integral in the Rings of p-Adic Integers
在[2]中，我构造了Zp上的p-adic q-积分。本文研究了p-adic整数环中q=-1时p-adic不变p-adic q-积分的性质。最后给出了q=-1时p进q积分的一些应用。这些性质对欧拉数和多项式的研究是有用的和值得的。
We prove that the algebra of p-adic multi-zeta values are contained in another algebra which is defined explicitly in terms of series.
The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number. The proof is based on elementary facts from $p$-adic analysis.
A note on the p-adic log-gamma functions
本文证明了p-adic解析q-log Γ函数的某些Stirling型方程的系数中存在q-欧拉数
In this paper we study some properties of the fermionic p-adic integrals on Zp arising from the umbral calculus
These are notes on adic spaces. They are made available upon some requests in order to make quoting them easier.
A note on p-adic q-integrals associated with q-Euler numbers
本文给出了Z_p上费米子p-adic q-测度的详细证明，并讨论了与欧拉数和多项式的q-扩张有关的一些有趣的公式。
We extend previous work of the author using an idea of Buzzard and give an elementary construction of non-ordinary $p$-adic families of Hilbert Modular Eigenforms.
A note on p-adic q-Euler measure
本文主要研究修正的q-欧拉数和多项式的性质。本文的主要目的是构造p-adic q-Euler测度。
A note on $p$-adic higher Mahler measures
本文引入p进高维马勒测度，推广Kurokawa、Lalín和Ochiai的工作，并证明Akatsuka相关结果的p进版本。
We prove an interpolation formula for the values of certain $p$-adic Rankin--Selberg $L$-functions associated to non-ordinary modular forms.
A note on the Fourier transform of p-adic q-integral
作者在文献[1，3]中定义了$p$ -进$q$ -积分（= $I_q$ -积分）.本文考虑$I_q$ -傅里叶变换，并研究了与这种变换有关的一些性质。
In this short note we formalized the definition for the Ducci operator $D$ in the context of the $p$-adic field $\mathbb{Q}_p$ as a natural extension of the classical one. Moreover we will describe the behavior of the operator and will provide some simple results as a counterpart to the classical one.
Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers. The aim of this note is to give another proof of this classification.
In this paper, we study Catalan numbers which can be represented by the p-adic integral on Zp and we investigate some properties and formulae related to Catalan numbers and special numbers.
We study a class of semistable ordinary hyperelliptic curves over 2-adic fields and the special fibre of their minimal regular model. We show that these curves can be controlled using `cluster pictures', similarly to the case of odd residue characteristic.
We discuss the deep relationship between $\infty$-categorical functional analysis and the anticipated theory of $p$-adic motives. The motivation fundamentally comes from applications essentially in arithmetics from very broad perspectives. The $\infty$-categorical homotopical aspects we considered here come mainly from Bambozzi-Ben-Bassat-Kremnizer, Ben-Bassat-Mukherjee, Bambozzi-Kremnizer, Clausen-Scholze and Kelly-Kremnizer-Mukherjee, based on deep robust foundation of $\infty$-categorical solids and $\infty$-categorical ind-Banach or bornological modules over general Banach rings.
We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable $p$-adic $L$-functions around critical points via Emerton's representation theoretic approach.
This note explores the consequences of Koenigsmann's model theoretic argument from the proof of the birational p-adic section conjecture for curves in the context of higher dimensional varieties over p-adic local fields.
A note on the representation theory of central extensions of reductive $p$-adic groups
本文验证了约化p进群表示理论中的若干基本结果可推广到其有限中心扩张，为相关表示论研究提供了更一般的适用框架。
The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator.
A note on the $p$-adic inhomogeneous Littlewood conjecture
本文针对p进非齐次Littlewood猜想，在Bugeaud等人2011年解决p进Littlewood猜想度量版本的基础上，建立了完全非齐次推广，并在混合情形下得到Harrap等人2013年结果的部分非齐次推广。
Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the $p$-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.
In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.
These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic groups, and the modular Langlands correspondence. These notes give an overview of the subject with several examples.
For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork's conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya's decomposition theorem of $p$-adic differential equations defined over annulus.
We consider Bernoulli distributions and their regularizations, which are measures on the $p$-adic integers $\mathbb{Z}_p$. It is well known that their Mellin transform can be used to define $p$-adic $L$-functions. We show that for $p>2$ one of the regularized Bernoulli distributions is particularly simple and equal to a measure on $\mathbb{Z}_p$ that takes the values $\pm \frac{1}{2}$ on clopen balls. We apply this to $p$-adic $L$-functions for Dirichlet characters of $p$-power conductor and obtain Dirichlet series expansions similar to the complex case. Such expansions were studied by D. Delb…
In this note, presented as a ``community service", followed by the PhD research of the author, we draw the relation between Casselman's theorem regarding the asymptotic behavior of matrix coefficients of reductive algebraic groups over $p$-adic fields and its expression as a finite sum of finite functions. In addition, we write the expansion explicitly for general linear groups.
A note on the $q$-adic valuation of $σ_k(n)$
本文通过LTE引理和 cyclotomic多项式，推导出σ_k(n)的q进位的精确公式，并给出优于 Zhao 之前结果的渐进行为上界。
A note on some $p$-adic analytic Hecke actions
研究Hecke算子对p-进过度模形式空间的作用，证明其在某种意义上局部解析，并推导Galois表示的Hodge-Tate-Sen权重，验证Gouvêa猜想。
Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such that $G/H\cong \mathbb{Z}_p$. Under various assumptions, we establish asymptotic upper bounds for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that $H\cong \mathbb{Z}_p$.
By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, and allows us to establish an asymptotic formula for the number of simultaneous zeros of non-singular pairs of cubic forms in at least 131 variables. As a by-product, we obtain a version of Hensel's Lemma for linear spaces.
In this paper, we answer a question due to Y. André related to B. Dwork's conjecture on a specialization of the logarithmic growth of solutions of $p$-adic linear differential equations. Precisely speaking, we explicitly construct a $\nabla$-module $M$ over $\mathbb{Q}_p[[X]]_0$ of rank 2 such that the left endpoint of the special log-growth Newton polygon of $M$ is strictly above the left endpoint of the generic log-growth Newton polygon of $M$.
Invariant Functions on $p$-divisible Groups and the $p$-adic Corona Problem II
本注记消除了同名论文主要结果的一维限制，拓展了p-可除群与p进 corona 问题相关研究的适用范围。
Notes on the LVP and CVP in $p$-adic Fields
本文提出在$p$-adic域中通过多项式时间算法解决LVP和CVP问题，利用$p$-adic范数的非阿基米德性质，构造$p$-adic晶格的正交基。
For varieties over a perfect field of characteristic p, etale cohomology with Q_l-coefficients is a Weil cohomology theory only when l is not equal to p; the corresponding role for l = p is played by Berthelot's rigid cohomology. In that theory, the coefficient objects analogous to lisse l-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with l-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship bet…
An introduction to $p$-adic and motivic zeta functions and the monodromy conjecture
这些笔记给出了一个基本的介绍理论的$p$ -进和motivic zeta函数，motivic整合，和单值猜想。
A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$
本文研究 $ \mathrm{GL}_2(\mathbb{Q}_p) $ 的 $ p $-adic 表示的扩展，通过 $ p $-adic 局部朗兰兹对应中通用伽罗瓦表示得到的表示进行分类，并应用于证明有限层级德林费尔德空间上étale上同调中可约表示与超 cuspidal 异型分量之间扩展的消失。
In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on $\mathrm{WCart}_X$ in favorable situations.
Coleman calculated the absolute Frobenius on Fermat curves explicitly. In this paper we show that a kind of $p$-adic continuity implies a large part of his formula. To do this, we study a relation between functional equations of the ($p$-adic) gamma function and monomial relations on ($p$-adic) CM-periods.
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.
We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.
We prove a functional equation for the three-variable $p$-adic $L$-function attached to the Rankin-Selberg convolution of a Coleman family and a CM Hida family, which was studied by Loeffler and Zerbes. Consequentially, we deduce that an anticyclotomic $p$-adic $L$-function attached to a $p$-non-ordinary modular form vanishes identically in the indefinite setting.
Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even…
In this note, we give a criteria whether given two Eisenstein polynomials over a padic field define the same extension (Proposition 1.6). In particular, we completely identify Eisenstein polynomials of degree p (Theorem 1.16). This note is an English translation of a part of the original paper [12].
A Note on a result of Iwasawa
利用p-adic L-函数在1处的值恢复了Iwasawa关于主单位的p-adic对数的一个结果。我们推导出一个Iwasawa样的结果在主单位的奇数部分。
it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.
We prove that the first two coefficients in the series expansion around $s=1$ of the $p$-adic $L$-function of an elliptic curve over $\mathbb{Q}$ are related by a formula involving the conductor of the curve. This is analogous to a recent result of Wuthrich for the classical $L$-function, which makes use of the functional equation. We present a few other consequences for the $p$-adic $L$-function and a generalisation to the base-change to an abelian number field.
In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight alpha by using p-adic invariant q-integral on Zp.
Topics in Adic Spaces
本文为海德堡春季学校迷你课程的扩展讲义，系统介绍 adic 空间中的凝聚层、态射性质及其与形式几何和 Berkovich 空间的联系。
In this note we prove that for any given prime number $p$ the Morita $p$-adic gamma function $Γ_p$ is differentially transcendental over $\mathbb{C}_p(X)$.
We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer argument. In this paper, we present rather large class of such p-adic functional series with integer coefficients which contain factorials. By recurrence relations, we constructed sequence of polynomials A_k(n;x) which are a generator for a few other sequences also relevant to some problems in number theory and combinatorics.
Functional equation of the $p$-adic $L$-function of Bianchi modular forms
本文研究类数为 1 的虚二次域上 Bianchi 模形式的 $p$-进 $L$-函数，证明小斜率 $p$-稳定化情形的函数方程，并借助 $p$-进族推广至 $\Sigma$-光滑基变换形式。
Recently, Boole polynomials have been studied by Kim and Kim over the p-adic number field. In this paper, we consider a q-extension of Boole polynomials by using the fermionic p-adic integrals on Zp and give some new identities related to those polynomials.
A note on purity of crystalline local systems
本文利用棱镜描述证明了光滑p-adic仿射形式概形上晶体局部系统的纯性定理。
The Galois action on the pro-$\ell$ étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with $\ell$-adic coefficients, called the $\ell$-adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its $\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in…
A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture
本文研究p进Littlewood猜想的假定反例，证明若反例存在则存在基p展开为一致回复的反例，并给出形态词条件下猜想成立的充分条件，特别证明p=2时猜想对所有纯形态词成立。
On the 3-adic Valuation of a Cubic Binomial Sum
本文证明了关于立方二项式和的3-进制估值的猜想，提供了该领域的新的数学结果。
A note on the cohomology of $p$-adic analytic group actions
本文研究紧$p$进李群解析作用的连续上同调，证明可通过pro-$p$ pro循环子群的对应性质检测其上同调的同时消失或有限维性，并提出推广猜想及李代数上同调的相关猜想。
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