A Note on Critical p-adic L-functions

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.     

p-进单项式动力系统严格遍历分解的一个新证明

本文研究了p-进单项式动力系统的严格遍历分解.利用代数数论的理论和Halmos与von Neumann定理,给出了p-进单项式动力系统严格遍历分解的一个新证明.     

Iwasawa Theory for Elliptic Curves at Unstable Primes

In this paper we examine the Iwasawa theory of modular elliptic curves E defined over Q without semi-stable reduction at p. By constructing p-adic L-functions at primes of additive reduction, we formulate a "Main Conjecture" linking this L-function with a certain Selmer group for E over the Zp-extension. Thus the leading term is expressible in terms of III_E, E(Q)_tors and a p-adic regulator term.     

Variation of Heegner points in Hida families

Given a weight two modular form f with associated p-adic Galois representation V _f , for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V _f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida’s universal ordinary deformation of V _f . We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.     

Big Heegner point Kolyvagin system for a family of modular forms

The principal goal of this paper is to develop Kolyvagin’s descent to apply with the big Heegner point Euler system constructed by Howard  for the big Galois representation \(\mathbb T \) attached to a Hida family \(\mathbb F \) of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family \(\mathbb F \) at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.     

A Regularization of Quantum Field Hamiltonians with the Aid of p–adic Numbers

Gaussian distributions on infinite-dimensional p-adic spaces are introduced and the corresponding L₂-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L₂-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L₂-spaces with respect to a p-adic Gaussian distribution.     

单个生成元Walsh p-进制平移不变空间伸缩的交与并

p-进制MRA与GMRA是构造L~2(R_+)中小波框架的重要工具. L~2(R+)中嵌套子空间序列交集为{0},并集为L~2(R_+)是其构成p-进制MRA与GMRA的基本要求.本文研究单个生成元Walsh p-进制平移不变子空间伸缩的交与并,证明了:对任意单个生成元Walsh p-进制平移不变子空间,其p-进制伸缩的交是{0};若生成元分为Walsh p-细分函数,则其p-进制伸缩的并是L~2(R_+)中一个Walshp-进制约化子空间.特别地,其伸缩构成L~2(R_+)中p-进制GMRA当且仅当∪_(j∈z)p~j supp(■φ)=R+,其中■为定义在L~2(R_+)上的Walsh p-进制傅里叶变换.值得注意的是:形式上,我们的结果类似于通常L~2(R)的情形,然而其证明不是平凡的.这是因为定义在R_+上的p-进制加法"⊕"不同于定义在R上的通常加法"+".     

若干新的p进制Hardy-Littlewood-Pólya型不等式

本文建立了若干新的具最佳常数因子的p进制Hardy-Littlewood-Pólya型不等式，同时也给出了它们的等价形式以及一些特殊结果.     

p-Adic interpolation of square roots of central L-values of modular forms

We construct a meromorphic function on part of the eigencurve that interpolates a square root of a ratio of quadratic twists of the central modular $L$ -value.     

p-adic Modular Symbols and A-adic Modular Forms

A construction of A-adic modular forms from p-adic modular symbols is described. It shows that each A linear map satisfying some certain conditions from the module of p-adic modular symbols to A corresponds to a A-adic modular form.