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

解析曲线的$p$-进平展上同调的算术对偶性

Arithmetic duality for $p$-adic pro-étale cohomology of analytic curves

该研究证明了$\textbf{Q}_p$有限扩张上光滑dagger曲线的算术$p$-进pro-étale上同调的庞加莱对偶性，推导结合了谱序列、几何比较定理与经典对偶，还验证了乘积相容性并得到相关上同调的泛函分析性质。

Arithmetic of $p$-adic curves and sections of geometrically abelian fundamental groups

Let $X$ be a proper, smooth, and geometrically connected curve of genus $g(X)\ge 1$ over a $p$-adic local field. We prove that there exists an effectively computable open affine subscheme $U\subset X$ with the property that $period (X)=1$, and $index (X)$ equals $1$ or $2$ (resp. $period(X)=index (X)=1$, assuming $period (X)=index (X)$), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of $U$ splits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to $X$, and give a new characteri…

A p-adic regulator map and finiteness results for arithmetic schemes

A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension …

p-进曲线算术基本群的局部截面

Local sections of arithmetic fundamental groups of p-adic curves

本文研究p-进曲线算术基本群的截面，证明其可无条件提升为尖点阿贝尔伽罗瓦群的截面，并由此导出紧化指数为1的结论及截面来自有理点的充要条件。

On arithmetic families of filtered phi-modules and crystalline representations

We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrizing integral data and determine the image of this morphism.

Structure of ordinary $Λ$-adic arithmetic cohomology groups

We study the $Λ$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $Λ$-adic Selmer groups coming from two different towers have "almost same" $Λ$-module structures. We also prove the cotorsionness of $Λ$-adic Tate-Shafarevich group under mild assumptions.

The cuspidalisation of sections of arithmetic fundamental groups II

In this paper we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally i-th and 2/p-th step prosolvable arithmetic fundamental groups. As a consequence we exhibit two, necessary and sufficient, conditions for sections of arithmetic fundamental groups of hyperbolic curves over p-adic local fields to arise from rational points. We also exhibit a class of sections of arithmetic fundamental groups of p-adic curves which are orthogonal to Pic, and which satisfy (unconditionally) one of the above conditions.

On some p-adic power series attached to the arithmetic of $\mathbb Q(ζ\_p).$

In this paper, we prove that the derivative of the Iwasawa power series associated to p-adic L-functions of $\mathbb Q(ζ\_p)$ are not divisible by p. This extends previous results obtained by Ferrero and Washington in 1979.

刚性解析空间的ℓ进平展上同调与邻近环的算术性质

Arithmetic Properties Of $\ell$-adic Étale Cohomology and Nearby Cycles of Rigid-Analytic Spaces

本文研究p进非阿基米德局部域上刚性解析簇的ℓ进平展上同调，建立Frobenius特征值界、强局部单值化定理、邻近环混合性，并用perfectoid分析证明Deligne权-单值化猜想的局部版本，同时发展Deligne拓扑斯上的可构造ℓ进复形理论。

算术导数到p进域与数域的推广

A Generalization of Arithmetic Derivative to $p$-adic Fields and Number Fields

本文将算术偏导数推广到p进域、算术导数推广到数域，研究其迭代的p进赋值动力系统，证明任意非负整数n都存在无穷多个恰有n个反偏导数的元素，并考察算术导数的p进连续性。

Product formula for p-adic epsilon factors

Let X be a smooth proper curve over a finite field of characteristic p. We prove a product formula for p-adic epsilon factors of arithmetic D-modules on X. In particular we deduce the analogous formula for overconvergent F-isocrystals, which was conjectured previously. The p-adic product formula is the equivalent in rigid cohomology of the Deligne-Laumon formula for epsilon factors in l-adic étale cohomology (for a prime l different from p). One of the main tools in the proof of this p-adic formula is a theorem of regular stationary phase for arithmetic D-modules that we prove by microlocal te…

Derived Hecke action at $p$ and the ordinary $p$-adic cohomology of arithmetic manifolds

We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of semisimple groups $\mathrm G(\mathbb Q)$, i.e., we study the derived version of Hida's theory for ordinary Hecke algebras. This is the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the tame case. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.

p进数域中的连分数算术

The arithmetic of continued fractions in the field of $p$-adic numbers

本文研究了p进数域中连分数的算术，提出了一种计算莫比乌斯变换和双线性分数变换的完整方法，并证明了某些情况下部分商无法通过初始连分数恢复，但此类元素集合在p进数域上具有零测度。

Obstructions to weak approximation for reductive groups over $p$-adic function fields

We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theore…

On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions

Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) $p$-adic heights on $B$ along anticyclotomic $\Z_{p}$-extensions of $E$. This provides evidence towards Schneider's conjecture on the non-vanishing of $p$-adic heights. For CM elliptic curves over $\Q$, the result was previously known as a consequence of work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based o…

Rigid cohomology over Laurent series fields III: Absolute coefficients and arithmetic applications

In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and Frobenius structures, and therefore are naturally $(φ,\nabla)$-modules over $\mathcal{E}_K^\dagger$ whenever they are finite dimensional. We also introduce a category of `absolute' coefficients for the theory; the same results are true for cohomology groups with coefficients. We moreover prove a $p$-adic version of the weight monodromy conjecture for smooth (not …

Fast arithmetic in unramified p-adic fields

Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmuller lifting and computing minimal polynomials.

Rigidity of p-adic cohomology classes of congruence subgroups of GL(n, Z)

We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cusp…

一个具有晶体局部系系数的算术平展-晶体比较

An arithmetic étale-crystalline comparison with coefficients in crystalline local systems

本文利用栈化方法将 Colmez--Niziol 的算术平展-晶体比较定理推广到任意晶体局部系系数，并建立带系数的 Beilinson 纤维方块及 syntomic 上同调与 p 进平展上同调的比较，同时描述 Z_p 上完美 F-规范的等ogeny范畴。

Arithmetic $\D$-modules and Representations

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic analogue of Kashiwara's theory of $D$-modules for real Lie groups i.e. it should give a realization of the $p$-adic representations of a $p$-adic Lie group as spaces of overconvergent solutions of arithmetic $D$-modules which will come equipped with an action of the Galois group. We shall discuss the ca…

Bounding Picard numbers of surfaces using p-adic cohomology

Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F_2 and F_3 with arithmetic Picard number 1, and a smooth quintic over F_2 with geometric …

Arithmetic properties of Fredholm series for p-adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

On the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field

We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$ (arXiv:1909.02697), but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out the strategy, we also generalize results of Howard (arXiv:1303.0545) on CM-cycle intersection and of Ehlen--Sankaran (arXiv…

On p-adic comparison theorems for rigid analytic varieties, I

We compute, in a stable range, the arithmetic p-adic etale cohomology of smooth rigid analytic and dagger varieties (without any assumption on the existence of a nice integral model) in terms of differential forms using syntomic methods. The main technical input is a construction of a Hyodo-Kato cohomology and a Hyodo-Kato isomorphism with de Rham cohomology.

一个p进算术内积公式

A $p$-adic arithmetic inner product formula

本文构造了偶秩拟分裂酉群上自守表示的分圆p进L-函数，证明其在平凡特征处消没阶为1时Selmer群秩至少为1，并利用酉Shimura簇上的特殊圈构造Selmer theta提升，建立了p进高度与p进L-函数导数之间的精确公式，即算术内积公式的p进类比。

$p$-进函数域上齐次空间的算术

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。

Arithmetic {B}reuil-{K}isin-{F}argues modules and comparison of integral {p}-adic Hodge theories

Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(φ,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered $(φ,N,G_K)$-modules to $G_K$-equivariant modifications of vector bundles over the Fargues-Fontaine curve $X_{FF}$, with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over $X_{FF}$ and the isogeny category of Breuil-Kisin-Fargue…

算术动力学中的非常稀疏返回集

Very sparse return sets in arithmetic dynamics

本文研究动力莫德尔-朗猜想中返回集的稀疏性，探讨p进版本、复解析版本及后向轨道版本的类似现象。

Arithmetic of p-irregular modular forms: families and p-adic L-functions

Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three…

On the existence of non-geometric sections of arithmetic fundamental groups

We show the existence of group-theoretic sections of the "etale-by-geometrically abelian" quotient of the arithmetic fundamental group of hyperbolic curves over $p$-adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points.

$S$-算术（上）同调与 $p$-进自守形式

$S$-arithmetic (co)homology and $p$-adic automorphic forms

本文研究数域上约化群的 $S$-算术（上）同调，基于局部代数与局部解析表示构造了抛物子群的特征簇，并证明其与过收敛同调及 Breuil-Ding 构造的 Bernstein 特征簇一致或密切相关。

An arithmetic property of intertwining operators for p-adic groups

If one proposes to use the theory of Eisenstein cohomology to prove algebraicity results for the special values of automorphic L-functions as in my work with Harder for Rankin-Selberg L-functions, or its generalizations as in my work with Bhagwat for L-functions for orthogonal groups and independently with Krishnamurthy on Asai L-functions, then in a key step, one needs to prove that the normalised standard intertwining operator between induced representations for p-adic groups has a certain arithmetic property. The principal aim of this article is to address this particular local problem in t…

Further examples of non-geometric sections of arithmetic fundamental groups

We show the existence of group-theoretic sections of certain geometrically pro-nilpotent by abelian arithmetic fundamental groups of hyperbolic curves over p-adic local fields which are non-geometric, i.e., which do not arise from rational points. Among these quotients is the geometrically metabelian arithmetic fundamental group.

Remarks on cycle classes of sections of the arithmetic fundamental group

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stix's theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a sectio…

Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry

In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus $g\geq 2$. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology…

A note on arithmetic congruences

By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.

Birational smooth minimal models have equal Hodge numbers in all dimensions

This is a resume of the author's talk at the Worhshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry (July 23-29, 2001), the Fields Institute. The aim of this note is to prove that birational smooth minimal models over C have equal Hodge numbers in all dimensions by an arithmetic method. Our method is a refinement of the method of V. Batyrev and C.-L. Wang on Betti numbers who used p-adic integration and the Weil conjecture. Our ingredient is to use further arithmetic results such as the Chebotarev density theorem and p-adic Hodge theory.

临界p-adicL函数的算术

Arithmetic of critical $p$-adic $L$-functions

本文研究临界p-adicL函数的算术理论，提出新的Iwasawa理论框架和厚Selmer复形，推导出主猜想的增强版本。

Some arithmetic and geometric aspects of algebraic cycles and motives

This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on cohomologies of algebraic varieties, periods, algebraic cycles and motives. I might one day find the energy to expand these notes and maybe translate them in English. In the meantime, please feel free to ask questions. The first sections of this text propose an introduction to the theory, I tried to present points of view and examples which are not always stressed in t…

$p$-进流形上 Weil 丛的算术方面

Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds

本文建立 $p$-进解析流形上 Weil 丛的系统理论，证明其继承典范解析结构，并给出提升定理、Galois 等变结构及与晶体上同调的比较同构，应用于 $p$-进模形式和椭圆曲线的无穷小形变研究。

Local and global methods in arithmetic (in Russian)

Let $p$ be a prime. We discuss methods of solution of congruences modulo $p^n$ using $p$-adic numbers; these methods are similar to computations with real numbers (local methods). Examples of relations between local and global methods are given producing a passage from congruences to solutions in integers (global methods). The use of a computer is illistrated for the study of $p$-adic numbers and algebraic curves.

Geometry, dynamics, and arithmetic of $S$-adic shifts

This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux-Rauzy as we…

Arithmetic of weighted Catalan numbers

In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to $2$-adic integers. These methods are also extended to $q$-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to t…

On the arithmetic of the BC-system

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of an algebraic closure of F_p. The obtained representations are the p-adic analogues of the complex, extremal KMS states at zero temperature of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over complex numbers is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose values…

Galois covers of the open p-adic disc

This paper investigates Galois branched covers of the open $p$-adic disc and their reductions to characteristic $p$. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.

Simultaneous Diophantine Approximation in Non-degenerate p-adic Manifolds

S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.

Path Integrals and p-adic L-functions

We prove an arithmetic path integral formula for the inverse p-adic absolute values of the Kubota-Leopoldt p-adic L-functions at roots of unity.

Arithmetic properties of Apéry-like numbers

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers' conjectures on the p-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p.

Poitou--Tate sequence for complex of tori over $p$-adic function fields

We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.

Existence of compatible systems of lisse sheaves on arithmetic schemes

Deligne conjectured that a single l-adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of l'-adic lisse sheaves with various l'. Drinfeld used Lafforgue's result as an input and proved this conjecture when the variety is smooth. We consider an analogous existence problem for a regular flat scheme over Z and prove some cases using Lafforgue's result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.

On ramification filtrations and $p$-adic differential modules, I: equal characteristic case

Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors a…

法格斯-福恩塔因曲线：对 $p$-adic 伽罗华表示理论研究的激励

La curva de Fargues--Fontaine: Una motivación al estudio de la teoría de representaciones de Galois $p$-ádicas

本文综述了法格斯-福恩塔因曲线在 $p$-adic Hodge 理论中的核心地位及其在分类 $p$-adic 伽罗华表示中的作用，分析了 Fontaine 时期环的性质及其对曲线构造的贡献。

Carlitz动机的$v$-进周期与Chowla-Selberg公式再探

$v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited

该研究在函数域数论框架下，利用带复乘的Carlitz动机的$v$-进晶体-德拉姆周期阐释$v$-进算术伽马值，建立Ogus型Chowla-Selberg公式，证明周期的代数无关性，并推导伽马值代数关系的来源。