An arithmetic study of the formal Laplace transform in several variables

Nodal length fluctuations for arithmetic random waves

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

Arithmetic properties related to the shuffle-product

Properties of the shuffle product suggest the definition of a quadratic form with domain and values in formal power series over a field of characteristic 2. This quadratic form preserves rational (respectively algebraic) power series and its restriction to the affine subspace of series with constant term 1 is bijective. Conjecturally, this bijection restricts to a bijection of rational (respectively algebraic) formal power series.

Arithmetic and equidistribution of measures on the sphere

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moment…

Quasi-projective and formal-analytic arithmetic surfaces

This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surf…

Semi-galois Categories II: An arithmetic analogue of Christol's theorem

In connection with our previous work on semi-galois categories, this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series over finite field is algebraic over the polynomial ring. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F_1) variant of Christol's theorem in the sense that it replaces the polynomial ring with the ring of integers of a number field and the ring of formal power series with the r…

On the arithmetic transfer conjecture for exotic smooth formal moduli spaces

In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.

算术子群的形式西格尔模形式

Formal Siegel modular forms for arithmetic subgroups

本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。

On sections of arithmetic fundamental groups of open p-adic annuli

We show the non-existence of sections of arithmetic fundamental groups of open p-adic annuli of small radii, this implies the non-existence of sections of arithmetic fundamental groups of formal boundaries of formal germs of p-adic curves.

更正则的形式模空间与算术转移猜想：分歧二次情形

More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case

本文针对 p 进域二次分歧扩张上的酉群，构造了带仿射仿射层次的正则形式模空间及其间的对应，提出算术转移猜想，并在最低维情形下给出证明。

Regular formal moduli spaces and arithmetic transfer conjectures

We define various formal moduli spaces of p-divisible groups which are regular, and morphisms between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture of the third author in the presence of ramification. These conjectures include the AT conjecture of our previous joint paper. We prove these conjectures in low-dimensional cases.

形式-解析算术曲面上的正则函数

Regular Functions on Formal-Analytic Arithmetic Surfaces

本文研究形式-解析算术曲面上正则函数的性质，通过复分析猜想推导出连续统基数的结论，并提出均衡格林函数的推前公式。

算术jet空间的核与Frobenius映射

Kernels of Arithmetic Jet Spaces and Frobenius Morphism

本文研究了算术jet空间的核与Frobenius映射，揭示了其与Witt向量结构的联系。

Planck-scale distribution of nodal length of arithmetic random waves

We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use o…

Arithmetic theory of q-difference equations (G_q-functions and q-difference modules of type G, global q-Gevrey series)

In the first part of the paper we give a definition of G_q-function and we establish a regularity result, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem [And89, VI] and Katz Theorem [Kat70, §13]. In the second part of the paper, we combine it with some formal q-analogous Fourier transformations, obtaining a statement on the irrationality of special values of the formal $q$-Borel transformation of a G_q-function.

More Arithmetic Fundamental Lemma conjectures: the case of Bessel subgroups

We define some formal moduli space of quasi-isogenies of isoclinic $p$-divisible groups with a non-reductive group as the "structure group". We then formulate new Arithmetic Fundamental Lemma conjectures for Bessel subgroups in the context of the arithmetic Gan-Gross-Prasad conjectures. Some (very limited) evidence is presented.

Kashiwara's theorem for twisted arithmetic differential operators

We establish a version of Kashiwara's theorem for twisted sheaves of Berthelot's arithmetic differential operators for a closed immersion between smooth p-adic formal schemes. As an application, we construct simple modules for crystalline distribution algebras of reductive groups using the twisted version of the crystalline localisation theorem.

具有大中心的非交换空间的算术几何

Arithmetic Geometry of Non-Commutative Spaces with Large Centres

本文利用非交换形式形变理论引入多项式恒等代数的算术几何，获取交换角度无法观测的局部信息，为非交换框架下的算术几何深入研究奠基。

Composition inverses of the variations of the Baum-Sweet sequence

Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem. This problem was studied before for the Thue-Morse sequence. In this paper, we study arithmetic properties of the formal inverses of two sequences closely related to the well-known Baum-Sweet sequence. We give the recurrence relations for their formal inverses and we determine whether the sequences of indices at which these formal inverses take value $0$ and $…

D^\dagger-affinity of formal models of flag varieties

Let G be the group of L-rational points of a connected split reductive group over a finite extension L of Q_p. We show that formal models of the algebraic flag variety X of G are D-affine for certain sheaves of arithmetic differential operators. We then introduce the category of coadmissible G-equivariant arithmetic D-modules on the system of formal models of X and prove that it is anti-equivalent to the category of admissible locally L-analytic G-representations with trivial infinitesimal character. We compute the equivariant arithmetic D-modules of certain classes of representations.

对n个变量上不等价算术表达式的计数

Counting the number of inequivalent arithmetic expressions on $n$ variables

本文提出了一种时间复杂度为Θ(n²)的算法，用于计算n个不同变量上不等价的算术表达式的数量。