Yumiko Hironaka
We give a general expression of spherical functions on $p$-adic homogeneous spaces of $G$, based on data of $G$ and functional equations of spherical functions. Then, we show a unified method to obtain functional equations of spherical functions and relate them those of $p$-adic local zeta functions of small prehomogeneous vector spaces.
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We investigate the space $X$ of unitary hermitian matrices over $\frp$-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic, i.e., $2\notin \frp$. In the latter half we assume odd residual characteristic, and give explicit formulas of typical spherical functions on $X$, where Hall-Littlewood symmetric polynomials of type $C_n$ appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show the Schwartz space $\SKX$ is a free Hecke algebr…
Spherical functions and local densities on the space of $p$-adic quaternion hermitian matrices
本文在p-进域上的四元数埃尔米特形式空间中定义球面函数，利用局部密度建立归纳公式与函数方程，并研究相关施瓦茨空间的赫克模结构。
We are interested in the harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of odd size over a ${\mathfrak p}$-adic field of odd residual characteristic, which is a continuation of our previous paper where we have studied for even size matrices. First we give the explicit representatives of the Cartan decomposition of $X$ and introduce a typical spherical function $ω(x;z)$ on $X$. After studying the functional equations, we give an explicit formula for $ω(x;z)$, where Hall-Littlewood po…
On the space $\mathbb Q_p^n$, where $p\ne 2$ and $p$ does not divide $n$, we construct a p-adic counterpart of spherical coordinates. As applications, a description of homogeneous distributions on $\mathbb Q_p^n$ and a skew product decomposition of p-adic Lévy processes are given.
Spherical varieties and p-adic families of cohomology classes
本文证明了与对称空间相关的上同调类p-进族的扭转相容性，表明单一族可插值不同权重的经典上同调类，推广了Euler系统与p-进L-函数的先前结果，并给出了GSp(4)及相关群的Euler系统新应用。
This is the final version, to appear in Commentarii Mathematici Helvetici.
We study spherical functions on the space isomorphic to $U(2n)/(U(n)\times U(n))$ over a $p$-adic field; those functional equations with respect to the action of the Weyl group, the location of possible poles and zeros, explicit formulas, and spherical Fourier transforms. Then, as an application, we give a functional equation of $p$-adic local hermitian Siegel series.
Spherical varieties and non-ordinary families of cohomology classes
该研究构建了与约化群球对相关的局部对称空间上同调的非寻常p进族类，给出平展情形下到伽罗瓦上同调的映射，方法可用于新的p进欧拉系和p进L函数构造，并以非寻常西格尔模形式的范数相容类为例推广了已有欧拉系的p部分。
We are interested in the harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of size $m$ over a ${\mathfrak p}$-adic field $k$ and give unified description including dyadic case, which is a continuation of our previous papers on non-dyadic case. The space becomes complicated when $e = v_π(2) > 0$. First we introduce a typical spherical function $ω(x;z)$ on $X$, and study their functional equations, which depend on $m$ and $e$, we give an explicit formula for $ω(x;z)$, where Hall-Littlewo…
We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.
The Newform $K$-Type and $p$-adic Spherical Harmonics
本文研究非阿基米德局部域上单位球面局部常值函数的 $K$-模分解，定义 $p$-进球谐函数，并利用 $K$-型刻画新形式与导子指数，最后与阿基米德情形对比。
This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition fu…
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.
The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ``Langlands dual'' group. We generalize this description to an arbitrary spherical variety X of G as follows: Irreducible unramified quotients of the space $C_c^\infty(X)$ are in natur…
Parahoric level $p$-adic $L$-functions for automorphic representations of $\operatorname{GL}_{2n}$ with Shalika models
本文为全实域上 $\operatorname{GL}_{2n}$ 的 parahoric 球形正则细化尖点自守表示构造了 $p$-进 $L$-函数，证明了抛物特征簇的平坦性并在族中构造 $L$-函数，核心是通过改进 Shalika 泛函与测试向量并推广局部系数理论实现。
The Fourier coefficients of the Siegel-Eisenstein series are p-adically continued for all primes p, as meromorphic functions, using the reciprocal of a product of L-functions. A construction of p-adic meromorphic families of such series is given in relation to the geometry of homogeneous spaces. Applications are given to p-adic L-functions, to Siegel's Mass Formula, to p-adic analytic families of automorphic representations. Based on author's talk for the Conference Automorphic Forms and Related Geometry, Assessing the Legacy of I.I. Piatetski-Shapiro
p-adic Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound
本文引入p-进球面码与p-进接吻数问题，并将Pfender关于球面码Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein上界的一行证明推广到p-进Hilbert空间。
We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we d…
Anticyclotomic $p$-adic $L$-functions for Coleman families of $U_{n+1} \times U_{n}$
本文利用定酉群球对分支律的 $p$-进插值，构造并在重数一假设下推广至 Coleman 族的反循环 $p$-进 $L$-函数，用以插值中心临界 $L$-值的平方根。
Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local global principle for rational points with respect to discrete valuations of F.
This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dime…
We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. As an application, we show that the failure of the local-global principle for rational points on …
Arithmetics of homogeneous spaces over $p$-adic function fields
本文研究 $\mathbb{Q}_p$ 有限扩域上曲线函数域 $K$ 中 $\textrm{SL}_{n,K}$ 齐次空间的局部-全局与弱逼近问题，利用 Galois 上同调对偶定理证明任何有限阿贝尔群均为 $K$ 上的 Galois 群。
Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $χ:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\to B$ of the spherical Hecke algebra ${\mathfrak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]}{\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the …
Dual $p$-adic Diophantine approximation on manifolds
本文研究了双p-进逼近在流形上的收敛性问题，针对维度至少为三的超曲面及部分满足条件的流形，证明了同质p-进收敛结果，并在非同质情况下提供了较弱的结论。
The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${\rm PGL}_2(F)$, where $F$ is a $p$-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for ${\rm PGL}_2(F)$.
Spherical functions of symmetric forms and a conjecture of Hironaka
本文验证了Hironaka关于对称形式球函数空间在海克代数上自由秩为4^r的猜想。
We show that Artin's conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r>1.
We classify the closed orbits under the action of maximal tori on the S-adic homogeneous spaces. As an application, we prove that if the set of values at the integer points of any homogeneous non-degenerate split form is discrete, then the form is multiple of a form with rational coefficients. Our result is related to the notable Littlewood conjecture.
Spherical functions on symmetric spaces of Friedberg-Jacquet type
本文构造了特定$p$-adic群对的对称空间上球函数的显式模型，并描述了其Hecke模结构。
On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
本文推广p进McKay对应，将Denef的Igusa局部zeta函数公式拓展至Q-嵌入奇点消解情形，证明了相关强单值群猜想与指数和猜想对特定加权齐次多项式成立。
On $p$-adic $L$-functions for $GL_{2n}$ in finite slope Shalika families
本文提出了一种新的方法，通过构造p-adic L-函数来研究GL_{2n}在有限斜率Shalika家族中的特征空间几何性质。
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of geometrically irreducible ell-adic middle-extension sheaves on a curve over a finite field which are pointwise pure of weight 0 and have bounded ramification and rank. As an application, we show that "random" functions defined on a finite field can not usually be approximated by short linear combinations of trace functions of sheaves with small complexity.
Reciprocity obstruction to strong approximation over p-adic function fields
本文在p进曲线函数域上构造SL_n齐性空间反例，证明强逼近失效，并提出且刻画了作为唯一障碍的互反障碍。
$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-orbit closures on a $p$-adic homogeneous space of infinite volume
本文研究了无限体积 $p$-进齐性空间上的轨道闭包问题，通过引入高分支性条件，推广了 Ratner 定理并对 $\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ 的 $Γ$-轨道进行了分类。
We present a conceptual and uniform interpretation of the methods of integral representations of L-functions (period integrals, Rankin-Selberg integrals). This leads to: (i) a way to classify of such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture which would imply a vast generalization of the method, and (iii) an explanation of the phenomenon of "weight factors" in a relative trace formula. We also prove results of independent interest, such as the generalized Cartan decomposition for spherical varieties…
Single radius spherical cap discrepancy on compact two-point homogeneous spaces
本文在紧致两点齐性空间上研究单一半径球冠差异的平方积分下界，给出依赖于流形维数 d 的半径条件并证明阶为 N^{-1-1/d} 的估计，同时对 d≡1(mod 4) 的维数建立了较弱的下界。
Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicit…
Let $P\in \Bbb Q_p[x,y]$, $s\in \Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=\frac{1}{1-p^{-1}}\int_{\Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(\Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $χ$ of $\Bbb Z_p^\times$ the characteristic function $\det(1-uA_{P,s,χ})$ of the restriction $A_{P,s,χ}$ of $A_{P,s}$ to the eigenspace $L^2(\Bbb Z_p)_χ$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{P,s,χ}$ are the reciprocals of the zeros …
Formalization of non-Archimedean functional analysis 1: spherically complete spaces
本文提出非阿基米德泛函分析中球面完备空间的正式化，涵盖定义、性质及应用，如Birkhoff-James正交性和Hahn-Banach定理。
In this note, we give a formula for the Whittaker-Shintani functions for the p-adic symplectic groups, which is a generalization of the Zonal spherical functions and Whittaker functions. We then use the formula to give an alternative proof of a conjecture given by T.Shintani on the unramified calculation of L-function for $Sp_{2n}\times GL_{1}$.
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