Duc Hiep Pham
In this paper, we formulate and prove the so-called $p$-adic non-commutative analytic subgroup theorem. This result is seen as the $p$-adic analogue of a recent theorem given by Yafaev.
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On the leading terms of Zeta isomorphisms and p-adic L-functions in non-commutative Iwasawa theory
本文讨论了岩泽理论在K_1-群和Kato的局部化K_1-群下的下降形式。然后，我们证明插值公式的“首项”的全球Zeta同构，与某些泰特动机和p-进L函数，与某些关键动机。
We prove a unicity result for the $L$-functions appearing in the non-commutative Iwasawa main conjecture over totally real fields. We then consider continuous representations $ρ$ of the absolute Galois group of a totally real field $F$ on adic rings in the sense of Fukaya and Kato. Using our unicity result, we show that there exists a unique sensible definition of a non-commutative $L$-function for any such $ρ$ that factors through the Galois group of a possibly infinite totally real extension. We also consider the case of CM-extensions and discuss the relation with the equivariant main conjec…
We prove a p-adic analogue of Wüstholz's analytic subgroup theorem. We apply this result to show that a curve embedded in its Jacobian intersects the p-adic closure of the Mordell-Weil group transversely whenever the latter has rank equal to 1. This allows us to give some theoretical justification to Chabauty techniques applied to finding rational points on curves whose Jacobian has Mordel-Weil group of rank 1.
Comparing direct limit and inverse limit of even $K$-groups in non-commutative $p$-adic Lie extensions
本文研究非交换 $p$-adic 丽扩展中偶 $K$-群的直接极限与逆极限的对偶性，扩展了之前在 $\mathbb{Z}_p^d$-扩展上的结果。
In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bellaïche asked the same question for the special case of Abelian varieties. These problems are $p$--adic analogues of a question raised earlier by B. Mazur on the density of rational points for the real topology. For a simple Abelian variety over the field of rational numbers, we show that the actual $p$--adic rank is at least the third of the expected value.
Let E/Q be an elliptic curve with good supersingular reduction at p with a_p(E)=0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the Zp-cyclotomic extension of a finite Galois extension of Q where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.
We use a $p$-adic analogue of the analytic subgroup theorem of Wüstholz to deduce the transcendence and linear independence of some new classes of $p$-adic numbers. In particular we give $p$-adic analogues of results of Wüstholz contained in [G. Wüstholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkhäuser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes à un paramètre $p$-adique de variétés de groupe, Invent. Math. 40 (1977), no. 2, 171-1…
It is well-known that the Wüstholz' analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual way the transcendence of a large class of complex numbers, e.g. the transcendence of $π$ which is originally due to Lindemann. In this paper we revisit the $p$-adic analogue of the analytic subgroup theorem and present a proof based on the method described and developed by the authors in a recent related paper.
On some l-adic representations of Gal(Qbar/Q) attached to noncongruence subgroups
SL_2（Z）的非同余子群上的l-adic抛物上同调群是Gal（Qbar/F）对数域F的有限维表示.我们展示的例子（F=Q）产生的伽罗瓦表示的图像是开子群的完全群的辛相似（任意维）。伽罗瓦群的象的确定依赖于Katz关于包含主幂零元元的gl（N）的约化子代数的分类定理;本文还包含了I.格罗诺夫斯基
Rigidity of p-adic cohomology classes of congruence subgroups of GL(n, Z)
我们通过引入和研究算术Hecke本征类的p-adic刚性的概念，扩展了Ash和Stevens [Ash-Stevens 97]关于GL（N，Q）的p-普通算术上同调类的p-adic解析族的工作。一个算术本征类被称为“刚性的”，如果（模扭曲）它不允许包含算术专门化的Zebriki稠密集的非平凡p-adic变形。本文开发了工具，显式研究GL（N）的特征簇的结构。我们利用这些工具证明了GL（3）的非自同调尖形的已知例子是刚性的。此外，我们猜想，对于GL（3），刚性等价于非自对偶。
In this paper we prove that the motivic Eisenstein classes associated to polylogarithms of commutative group schemes can be $p$-adically interpolated in étale cohomology. This generalizes results for elliptic curves obtained in our former work. Already for one dimensional commutative groups the results proved here are much more flexible as they allow to treat degeneration questions easily.
Chow group of 0-cycles on surface over a p-adic field with infinite torsion subgroup
给出了p进域上具有无穷挠子群的曲面上0-圈的Chow群的一个例子。
The goal of this paper is to illustrate different approaches to understand Euler characteristics in the setting of totally real commutative and non-commutative Iwasawa theory. In addition to this, and in the spirit of Hesselholt and Mitchell, we show how these objects interact with homotopy theoretic invariants such as ($K(1)$-local) $K$-theory and Topological Cyclic homology.
The p-adic closure of a subgroup of rational points on a commutative algebraic group
设G是Q上的交换代数群.设Gamma是G（Q）的一个子群，包含在G（Q_p）的紧子群的并中.给出了G（Q_p）中Gamma闭包维数的一个猜想，并证明了它对某些环面的正确性等价于Leopoldt猜想.
We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…
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 $p$.
In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.
How to find "best rational approximations" of maximal commutative subgroups of GL(n,R)? In this paper we pose and make first steps in the study of this problem. It contains both classical problems of Diophantine and simultaneous approximations as a particular subcases but in general is much wider. We prove estimates for n=2 for both totaly real and complex cases and write the algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sail…
We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of $ p $-adic fields or number fields which is $ H $-Galois for a commutative Hopf algebra $ H $. Firstly, we show that if $ L/K $ is a tame Galois extension of $ p $-adic fields then each fractional ideal of $ L $ is free over its associated order in $ H $. We also show that this conclusion remains valid if $ L/K $ is merely almost classically Galois. Finally, we show that if $ L/K $ is an abelian extension of number fields then every ambiguous fractional ideal of $ L…
We establish the Iwasawa main conjecture for semi-stable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field which contain the constant $\mathbb Z_p$-extension and are everywhere unramified. Under the classical $μ=0$ hypothesis we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [TV] together with the trace formulas of [EL1].
Polarizations on abelian varieties and self-dual ell-adic representations of inertia groups
GL_d（Q_{\ell}）的每个有限子群都共轭于GL_d（Z_{\ell}）的一个子群.然而，如果我们用辛群代替一般线性群，这就不成立了。我们称G是惯性型群，如果G是有限群，且G的正规Sylow-p-子群具有循环商.证明了若\ell >d+1，且G是Sp_{2d}（Q_{\ell}）的惯性型子群，则G在GL_{2d}（Q_{\ell}）中共轭到\Sp_{2d}（Z_{\ell}）的子群.尽管G不能在\GL_{2d}（Q_ \ell）中共轭到\Sp_{2d}（Z_ \ell）的一个子群，但我们证明了只要\ell >3，它仍然可以嵌入到\Sp_{2d}（F_ \ell）中，使得特征多项式保持不变（mod \ell）.后一个结果适用于任意有限群，不一定是惯性型的，也适用于对称形式，而不仅仅是交替形式。我们给出的例子表明，边界是尖锐的。我们应用这些结果来构建，为每一个奇素数\ell，issues类的阿贝尔簇所有的极化度可分的\ell。本文是ANT-0151的修订版，标题为“有限群的自对偶ell-adic表示”。
In arXiv:math/0404297 a non-commutative Iwasawa Main Conjecture for elliptic curves over $\mathbb{Q}$ has been formulated. In this note we show that it holds for all CM-elliptic curves $E$ defined over $\mathbb{Q}$. This was claimed in (loc.\ cit.) without proof, which we want to provide now assuming that the torsion conjecture holds in this case. Based on this we show firstly the existence of the (non-commutative) $p$-adic $L$-function of $E$ and secondly that the (non-commutative) Main Conjecture follows from the existence of the Katz-measure, the work of Yager and Rubin's proof of the 2-var…
It is well known that, for any finitely generated torsion module M over the Iwasawa algebra Z_p [[Γ ]], where Γ is isomorphic to Z_p, there exists a continuous p-adic character ρ of Γ such that, for every open subgroup U of Γ, the group of U-coinvariants M(ρ)_U is finite; here M( ρ) denotes the twist of M by ρ. This twisting lemma was already applied to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a non commutative generalization of this twisting lemma replacing torsion modules over Z_p [[ Γ ]] by ce…
For a curve $X$ over a $p$-adic field $k$, using the class field theory of $X$ due to S. Bloch and S. Saito we study the abelian geometric fundamental group $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ of $X$. In particular, it is investigated a subgroup of $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ which classifies the geometric and abelian coverings of $X$ which allow possible ramification over the special fiber of the model of $X$. Under the assumptions that $X$ has a $k$-rational point, $X$ has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of …
Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and $\mathrm{GL}_2(\mathbf{Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q…
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.
We study the behaviour of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups $Γ_1(N2^r)$ for $r \geq 2$, and prove a control theorem for these homology modules.
On non-commutative Iwasawa theory and derivatives of Euler systems
本文利用约化行列式函子描述非交换代数的相对 $K_0$-群，结合 zeta element 推广，提出了非交换 $p$-进岩泽理论的主猜想及 Rubin-Stark 欧拉系统的高阶导数公式，为等变 Tamagawa 数猜想提供了广泛推广的证据。
In this article I generalise previous computations (by K. Kato, T. Hara and myself) of K_1 (only up to p-power torsion) of p-adic group rings of finite non-abelian p-groups in terms of p-adic group rings of abelian subquotients of the group. Such computation have applications in non-commutative Iwasawa theory due to a strategy proposed by D. Burns, K. Kato (and a modification by T. Hara) for deducing non-commutative main conjectures from commutative main conjectures and certain congruences between special L-values. However, I do not say anything about Iwasawa theory in this article. The detail…
We develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.
It is well known that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb Z_p[[Γ]]$, where $Γ\cong \mathbb Z_p$, there exists a continuous $p$-adic character $ρ$ of $Γ$ such that, for the twist $M(ρ)$ of $M$, the $Γ_n := Γ^{p^n}$ Euler characteristic, i.e. $χ(Γ_n, M(ρ))$, is finite for every $n$. We prove a generalization of this result by considering modules over the Iwasawa algebra of a general $p$-adic Lie group $G$, instead of $Γ$. We relate this twisted Euler characteristic to the evaluation of the {\it Akashi series} at the twist and in turn use it to indicate…
The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kat…
Let F be a p-adic field and let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n is odd then all the genuine irreducible representations of G(n) are induced from a normal subgroup of finite index closely related to G`(n). Thus, we reduce, in this case, the theory of genuine admissible representations of G(n) to the better understood corresponding theory of G`(n). For odd n we also prove the uniqueness of certain Whittaker functionals along with Rodier type of Heredit…
Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry
在阿拉克洛夫理论中，算术曲面的完备化是通过“无穷远处的闭纤维”的正式线性组合扩大因子群来实现的。马宁描述了任何这样的闭纤维的对偶图在一个无限纠缠的有界测地线在一个双曲双曲体赋予一个肖特基均匀化。本文研究数域中整数环上的算术曲面，其纤维亏格为$g\geq 2$。我们利用Connes的谱三重数理论将双曲几何的双曲体与Deninger的阿基米德上同调和本文第一作者所介绍的局部单值$N$在算术无穷远处的锥的上同调联系起来。
Approximate lattices and S-adic linear groups
本文为S-adic线性群中的近似格建立自然研究框架，定义算术性，推广Schreiber和Meyer定理至幂零S-adic群，并研究其与特定子群交集，提供更精确结论。
We prove the transfer congruence between $p$-adic Hecke $L$-functions for CM fields over cyclotomic extensions, which is a non-abelian generalization of the Kummer's congruence. The ingredients of the proof include the comparison between Hilbert modular varieties, the $q$-expansion principle, and some modification of Hsieh's Whittaker model for Katz' Eisenstein series. As a first application, we prove explicit congruence between special values of Hasse-Weil $L$-function of a CM elliptic curve twisted by Artin representations. As a second application, we prove the existence of a non-commutative…
A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety
研究p进对数下阿贝尔簇非 torsion 点非消失问题，主要针对GL₂型阿贝尔簇及其与希格纳点相关的情形，采用p进解析子群定理作为主要工具。
Intrinsic Subgroups and the $\ell$-adic Galois image
本文研究域k上光滑射影曲线Picard群扭子群上的双加性对称配对，定义内在子群，并对椭圆曲线给出其扭子群内在子群的纯代数分类，推广Yamazaki等人的解析方法至任意域k⊂ℂ，并给出k=ℚ时的显式计算算法。
On non-commutative Euler systems, I: preliminaries on `det' and `Fit'
本文将交换代数的经典构造扩展至非交换半单代数序上的模，提出了高阶Fitting不变量等概念的简化版本，为发展$p$-adic表示的非交换Euler系统理论奠定基础。
Difference subgroups of commutative algebraic groups over finite fields
工作的Chatzidakis和Hrushovski模型论的差异领域的特征零表明，集团定义的差分方程有一个非常有限的结构。最近的工作Chatzidakis，Hrushovski和Peterzil [CHP]扩展了这类结果是已知的正特征的差域类。本文分析了有限域上可由这种差分方程构造的单交换代数群的扭点的子群。我们的结果是相当完整的模数论中的一些著名的代数。在一种情况下，我们需要四个指数猜想的$p$ -adic版本，在另一种情况下，我们需要关于原根的Artin猜想的版本。我们恢复部分定理Boxall交叉品种与组的$m$ -权力扭转点，但在一般情况下，这一定理并不遵循从模型理论分析，因为可能没有领域自同构$σ$，使$m$ -权力扭转群包含在一个模群定义与$σ$。另一方面，某些由模差分方程定义的群比$m$ -幂扭点群大得多，因此我们的结果在另一个方向上更强。在某些方面，模型论方法扩展了Bogomolov的方法和Lang的原始方法。
Gelfand's trick shows that the spherical Hecke algebra of a $p$-adic split reductive group is commutative. We adapt this strategy in order to show that the spherical derived Hecke algebra is graded-commutative under mild assumptions on the coefficient ring.
In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular…
Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions
本文通过发展非尖点Siegel模形式的Hida理论，利用Greenberg-Stevens方法构造改进的p进L-函数，研究了P-普通Siegel模形式的标准p进L-函数在半稳定平凡零点处的导数，从而部分证明了Greenberg关于p进L-函数平凡零点的阶数与首项系数的猜想。
We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$ and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of loc…
Iwasawa theory of totally real fields for certain non-commutative $p$-extensions
本文利用Oliver和Taylor引入的积分定理，证明了关于某些特定的非交换$p$ -adic李扩张的全实域的Iwasawa主猜想.我们的结果给出了Kazuya Kato关于Heisenberg型伽罗瓦扩张的主要猜想的证明的某些推广。
We investigate generalizations along the lines of the Mordell--Lang conjecture of the author's $p$-adic formal Manin--Mumford results for $n$-dimensional $p$-divisible formal groups $\mathcal{F}$. In particular, given a finitely generated subgroup $Γ$ of $\mathcal{F}(\overline{\mathbb{Q}}_p)$ and a closed subscheme $X\hookrightarrow \mathcal{F}$, we show under suitable assumptions that for any points $P\in X(\mathbb{C}_p)$ satisfying $nP\inΓ$ for some $n\in\mathbb{N}$, the minimal such orders $n$ are uniformly bounded whenever $X$ does not contain a formal subgroup translate of positive dimens…
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。
We propose a definition of characters in the context of Schneider-Teitelbaum's theory of locally analytic representations of p-adic reductive groups. This character will be a function on a compact subgroup of a maximal torus of the reductive group in question. As an example we treat the locally analytic principal series of SL(2,Q_p).
Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of $SL_1(D)$ admits self-similar actions on regular rooted trees. Moreover, we prove results on $\mathbb{Z}_p$-Lie lattices that allow to deal with the case where the center of $D$ is bigger than $\mathbb{Q}_p$, and lead to the classification of the torsion-free $p$-adic analytic pro-$p$ groups $G$ of dimension less than $p$ with the property that all the nontrivial c…
Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $τ$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G τ$ and we show that $\mathrm{Ind…
A Nomizu-van Est theorem in Ekedahl's derived $\ell$-adic setting
本文在Ekedahl导出的ℓ-adic设定中，证明了关于unipotent线性代数群的紧开子群的上同调定理，结合了Pink公式和自动形式 perverse sheaves的应用。
Let $G$ be a commutative algebraic group embedded in projective space and $Γ$ a finitely generated subgroup of $G$. From these data we construct a chain of algebraic subgroups of $G$ which is intimately related to obstructions to multiplicity or interpolation estimates. Let $γ_1,...,γ_l$ denote a family of generators of $Γ$ and, for any $S>1$, let $Γ(S)$ be the set of elements $n_1γ_1+..+n_lγ_l$ with integers $n_j$ such that $|n_j| < S$. Then this chain of subgroups controls, for large values of $S$, the distribution of $Γ(S)$ with respect to algebraic subgroups of $G$. As an application we es…
Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks
本文研究了 $p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系，通过构建增强的 $L$-参数集并应用到各种群上，证明了某些猜想并简化了深度零情况。
Let $k$ be a number field, $Ω$ be a finite symmetric subset of $\mathbb{GL}_{n_0}(k)$, and $Γ=\langle Ω\rangle$. Let \[ C(Γ):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} Γ\text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\}, \] and $Γ_{\mathfrak{p}}$ be the closure of $Γ$ in $\mathbb{GL}_{n_0}(k_{\mathfrak{p}})$. Assuming that the Zariski-closure of $Γ$ is semisimple, we prove that the family of left translation actions $\{Γ\curvearrowright Γ_{\mathfrak{p}}\}_{\mathfrak{p}\in C(Γ)}$ has {\em uniform spectral gap}. As a corollary we get that the left translation action $Γ\curvearrowr…
We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of $\mathbf{SL}_2$ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the $\mathbf{SL}_2$-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in $\mathbf{SL}_2$ is restricted from one in $\mathbf{GL}_2$. From this it follows that any archetype must be induced from a Bushnell--Kutzko type. Given a supercuspidal representation $π$, we give an additional explicit description…
A wavelet theory for local fields and related groups
设G是局部紧阿贝尔群，其紧开子群H.这样一个群最著名的例子是G=Q_p，p进有理数域（作为加法下的群），它有紧开子群H=Z_p，p进整数环。经典的小波理论，这需要一个非平凡的离散子群的翻译，不适用于G，它可能没有这样的子群。利用G的对偶群的商的陪集表示，发展了G上的小波理论。小波基的构造是通过一种迭代法，在对偶群中产生所谓的小波集。
In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups $\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural $p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. We work with a general connected reductive split group over some $p$-adic field (with simply connected derived group) and study the $\Bbb{I}$-analytic vectors in principal series representations. Our main result is an irreducibility criterion which generalizes results of Clozel and Ray in the $\text{GL}_n$-case.
An irreducible decomposition of the Weil representation restricted to open compact subgroups
本文利用 Schrödinger 模型，对任意剩余特征的 p-进域，给出了 Weil 表示限制到极大开紧子群和 Iwahori 子群的不可约分解，并证明其无重数性。
Let $F$ be a $p$-adic field of characteristic 0, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $Π_{i=1}^{r} SL_{n_i} \subseteq M \subseteq Π_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under \textit{the local Jacquet-Langlands type correspondence} between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Mui{ć} and Savin (2000)…
Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}…
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