Anatoly N. Kochubei
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.
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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.
This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The coordinate representation can be useful, e.g., for applications to cryptography. Moreover, by using this representation we can consider non-smooth $p$-adic transformations. The basic technical toolsare van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theore…
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…
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。
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。
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。
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…
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…
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…
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。
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…
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.
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。
We study how the category of $q$-connections depends on the choice of coordinates. We exploit Bhatt's and Scholze's $q$-crystalline site, which is based on a coordinate free formulation of $q$-PD structures, in order to relate $q$-crystals and $q$-connections in the $p$-adic setting. This yields a natural equivalence between the the categories of $q$-connections for different choices of coordinates in the $p$-adic setting. The equivalence can be described explicitly in terms of differential operators. In order to obtain a global equivalence, we patch the $p$-adic differential operators to crea…
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 …
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。
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。
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.
Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute…
Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$. When $K$ is different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, we consider a certain specific model for the elliptic curve $E_K$ with $j(E_K)=j(\mathcal{O}_K)$ which is defined over $\mathbb{Q}(j(E_K))$. In this paper, for each positive integer $N$ we compare the extension field of $\mathbb{Q}$ generated by the coordinates of $N$-torsion points on $E_K$ with the ray class field $K_{(N)}$ of $K$ modulo $N\mathcal{O}_K$. By using this result we investigate the image of a $p$-adic…
Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(α_{1}, α_{2}, ... , α_{2g + 1})$, where the $α_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \prod_{i = 1}^{2g + 1} (x - α_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $Γ(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generator…
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