A General Reciprocity Law on arbitrary Vector Spaces

Harbingers of Artin's Reciprocity Law. IV. Bernstein's Reciprocity Law

In the last article of this series we will first explain how Artin's reciprocity law for unramified abelian extensions can be formulated with the help of power residue symbols, and then show that, in this case, Artin's reciprocity law was already stated by Bernstein in the case where the base field contains the roots of unity necessary for realizing the Hilbert class field as a Kummer extension. Bernstein's article appeared in 1904, almost 20 years before Artin conjectured his version of the reciprocity law, and seems to have been overlooked completely.

热带 Weil 互反律与 Weil 配对

Tropical Weil's reciprocity law and Weil's pairing

本文证明 Weil 互反律的热带版本，并据此给出经典互反律的组合式证明，同时在零次除子上构造热带 Weil 配对。

Iwasawa理论中的显式互反律——以Lubin-Tate设为侧重点的综述

Explicit Reciprocity Laws in Iwasawa Theory -- A survey with some focus on the Lubin-Tate setting

本文从二次互反律出发，梳理高阶与广义互反律的发展，并聚焦Lubin-Tate设定下的Iwasawa理论显式互反公式。

Harbingers of Artin's Reciprocity Law. III. Gauss's Lemma and Artin's Transfer

We briefly review Artin's reciprocity law in the classical ideal theoretic language, and then study connections between Artin's reciprocity law and the proofs of the quadratic reciprocity law using Gauss's Lemma.

A Supplement to Scholz's Reciprocity Law

In this note we will present a supplement to Scholz's reciprocity law and discuss applications to the structure of 2-class groups of quadratic number fields.

Hermite's identity and the quadratic reciprocity law

We give a short proof of the quadratic reciprocity law using Gauss's Lemma and Hermite's identity.

Yet another proof of the quadratic reciprocity law

We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.

A Minimalist Proof of the Law of Quadratic Reciprocity

The shortest known proof of the law of quadratic reciprocity (without supplements) is presented.

Fermat curves and the reciprocity law on cyclotomic units

We define a "period ring-valued beta function" and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark's conjecture implies that the exponential of the derivatives at $s=0$ of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler's formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative (and partial) proof by using the reci…

Analytic Tate spaces and reciprocity laws

We consider a functional analytic variant of the notion of Tate space, namely the category of those topological vector spaces which have a direct sum decomposition where one summand is nuclear Frechet space and the other is the dual of a nuclear Frechet. We show that, both in the complex and in the p-adic setting, one can use this formalism to define symbols for analytic functions which satisfy Weil-type reciprocity laws.

Harbingers of Artin's Reciprocity Law. II. Irreducibility of Cyclotomic Polynomials

In the first article of this series we have presented the history of auxiliary primes from Legendre's proof of the quadratic reciprocity law up to Artin's reciprocity law. We have also seen that the proof of Artin's reciprocity law consists of several steps, the first of which is the verification of the reciprocity law for cyclotomic extensions. In this article we will show that this step can be identified with one of Dedekind's proofs of the irreducibility of the cyclotomic polynomial.

Noncommutative reciprocity laws on algebraic surfaces: a case of tame ramification

We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a finite last residue field the constructed local central extension is isomorphic to a central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov.

Reciprocity laws à la Iwasawa-Wiles

This paper is a brief survey on explicit reciprocity laws of Artin-Hasse-Iwasawa-Wiles type for the Kummer pairing on local fields.

Siegel模函数的Shimura互反律的一个显式版本

An explicit version of Shimura's reciprocity law for Siegel modular functions

本文给出了Siegel模函数奇异值的Shimura互反律的显式版本，并据此构造了比Igusa不变量更小的四次CM域类不变量，同时为Tonghai Yang的定理提供了新证明。

Harbingers of Artin's Reciprocity Law. I. The Continuing Story of Auxiliary Primes

In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's genus theory.

New arithmetical proof of the reciprocity law for Dedekind sums

In this paper, for coprime numbers p and q we consider the well known Dedekind sums S(p,q) First, we give an improvement of the proof given by H. Rademacher and A. Whiteman, and we construct a new arithmetical proof for the reciprocity law

New trigonometric identities and reciprocity laws of generalized Dedekind sums

We obtain new trigonometric identities, which are some product-to-sum type formulas for the higher derivative of the cotangent and cosecant functions. Further, from specializations of our formulas, we derive not only various known reciprocity laws of generalized Dedekind sums but also new reciprocity laws of generalized Dedekind sums.

A categorical proof of the Parshin reciprocity laws on algebraic surfaces

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.

The reciprocity law for the twisted second moment of Dirichlet L-functions

We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We show that these sums satisfy an elegant reciprocity formula. In the case that the modulus is prime, these sums are closely related to the twisted second moment, and the reciprocity formula for these sums implies Conrey's reciprocity formula. We also extend the range of uniformity of Conrey's formula.

关于 Lubin-Tate 正则映射与 Kato 显式互反律

On Lubin-Tate regulator maps and Kato's explicit reciprocity law

本文推广 Lubin-Tate 正则映射的插值性质至 Artin 特征，证明互反律，并给出 Kato 显式互反律的新证明。

Reciprocity Laws for the Higher Tame Symbol and the Witt Symbol on an Algebraic Surface

Parshin's higher Witt pairing on an arithmetic surface can be combined with the higher tame pairing to form a symbol taking values in the absolute abelian Galois group of the function field. We prove reciprocity laws for this symbol using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol.

The two-dimensional Contou-Carrère symbol and reciprocity laws

We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We relate also the two-dimensional Contou-Carrère symbol with the two-dimensional class field theory.

Reciprocity laws for Legendre symbols of the type $(a+b\sqrt{m}/p)$ - Long Version

We announce a very general statement involving the rational quartic residue symbol $(m/p)_4$ and, more generally, Legendre symbols of the type ${a+b\sqrt{m}/p$. We show how our main theorem can be used to produce many older results such as Scholz's, Lehmer's or Burde's reciprocity laws and many others. It is very likely that all existing reciprocity laws of this type can be obtained from our result. This is a corrected and improved version of version 2.

Theta Functions and Reciprocity Laws

In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using a Möbius transform, and show they have a functional equation, Euler product factorization, and roots only on the critical line. In the second part, we prove a general reciprocity law for sums of exponentials of rational quadratic forms in any number of variables, using the transformation formula of the Riemann theta function on the Siegel upper half-space.

Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring

The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed fiel…

Explicit reciprocity laws for higher local fields

Using the previously constructed explicit reciprocity laws for the generalized Kummer pairing of an arbitrary (one-dimensional) formal group, in this article a special consideration is given to Lubin-Tate formal groups. In particular, this allows for a completely explicit description of the Kummer pairing in terms of multidimensional p-adic differentiation. The results obtained here constitute a generalization, to higher local fields, of the formulas of Artin-Hasse, Iwasawa, Kolyvagin and Wiles.

Higher exponential maps and explicit reciprocity laws I

This paper has been withdrawn, as it is superseded by arXiv:0806.2122 (Bloch-Kato exponential maps for local fields with imperfect residue fields), which is a more recent version of the same paper.

A note on additive twists, reciprocity laws and quantum modular forms

We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.

A Hilbert reciprocity law on 3-manifolds

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.

Reciprocity laws for $(φ_L,Γ_L)$-modules over Lubin-Tate extensions

In the Lubin-Tate setting we study pairings for analytic $(φ_L,Γ_L)$-modules and prove an abstract reciprocity law which then implies a relation between the analogue of Perrin-Riou's Big Exponential map as developed by Berger and Fourquaux and a $p$-adic regulator map whose construction relies on the theory of Kisin-Ren modules generalising the concept of Wach modules to the Lubin-Tate situation.

Siegel 模形式权重 $(3,3)$ 的 Kato 显式互反律

Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$

本文将 Kato 显式互反律从模曲线推广到两条模曲线的乘积，并利用嵌入 Siegel 三维簇，得到权重 $(3,3)$ 的 Siegel 模形式所附 $p$-进表示的唯一次临界扭转的显式互反律。

The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain …

The Reciprocity Law for Dedekind Sums via the constant Ehrhart coefficient

We obtain a new motivated proof of the reciprocity law for Dedekind sums by computing the constant coefficient of the Ehrhart polynomial for a rectangular triangle in two ways. On the one hand, the constant term is the Euler characteristic, and on the other hand it is given by a cotangent Laurent expansion.

New trigonometric identities and reciprocity laws of generalized Dedekind sums II

We obtain new trigonometric identities, which are product-to-sum type formulas for derivative of the cosecant and cotangent functions. Further, from specializations of our formulas, we derive new reciprocity laws of generalized Dedekind sums.

Central extensions and reciprocity laws on algebraic surfaces

We consider an algebraic surface. For an irreducible curve on this surface and for a point on this curve one can associate an artinian ring, which is a sum of two-dimensional local fields. An example of two-dimensional local field is iterated Laurent series. We consider an integer value symbol on a two-dimensional local field. This symbol appears for the description of non-ramified coverings of two-dimensional local fields and for the description of intersection index of divisors on surfaces by adeles. We interpret this symbol as the commutator of lifting of elements in the central extension o…

关于 $\mathbb{F}_{q}[t]$ 中互易律的探讨

On the reciprocity law in $\mathbb{F}_{q}[t]$

本文在Rousseau方法基础上，为$\mathbb{F}_{q}[t]$中的$d$次剩余符号互易律提供新的初等证明，其中$d$为$q-1$的任意因数。

Non-abelian reciprocity laws and higher Brauer-Manin obstructions

We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…

On an explicit reciprocity law in local class field theory via $(φ, Γ)$-modules

Let $K$ be an unramified extension of $\mathbb{Q}_2$ and $μ_{2^n}$ the group of $2^n$-th root of unity for a fixed integer $n\geqslant 2$. In this paper, we give an explicit formula for the $μ_{2^n}$-valued Hilbert symbol over $K_n := K(μ_{2^n})$ using the theory of $(φ, Γ)$-modules.

On the reciprocity law for the twisted second moment of Dirichlet $L$-functions

We investigate the reciprocity law, studied by Conrey~\cite{Con07} and Young~\cite{You11a}, for the second moment of Dirichlet L-functions twisted by $χ(a)$ modulo a prime $q$. We show that the error term in this reciprocity law can be extended to a continuous function of $a/q$ with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula involving also shifted moments. We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of $a/q$, and, consequently, we improve upon a classical resu…

Rankin--Eisenstein类与显式互反律

Rankin--Eisenstein classes and explicit reciprocity laws

本文构造了模形式Rankin卷积的三变量$p$-adic伽罗瓦上同调类族，证明了其与L函数临界值相关的显式互反律，并由此得出L值非零时椭圆曲线Selmer群的有限性结果。

对角类的显式互反律：更高阶情形

Explicit reciprocity laws for diagonal classes: higher level cases

本文将平衡对角类的$p$-进显式互反律推广至模特征形式几何平衡三元组情形，允许其中两个形式在$p$处为超尖点型或分歧主级数。

Rational Quartic Reciprocity

We provide a simple proof of the general rational quartic reciprocity law due to Williams, Hardy and Friesen.

模形式的显式互反律与岩泽理论

Explicit reciprocity laws and Iwasawa theory for modular forms

本研究证明模形式本征形式的Mazur-Tate元素位于分圆Zp扩张对偶塞尔梅克群的菲廷理想中，通过p进局部朗兰兹对应构造局部上同调类与代数类似物，结合Kato欧拉系统和显式互反律完成证明。

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck and of Faisant. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p,$ extending and simplifying earlier work of Zhi-Hong Sun (2003).

Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $ε> 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(ε)$, the number of prime cubic residues $q < p^{1/2+ε}$ exceeds $p^{ε/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{1/9}$ pr…

Bettin–Conrey 余切和互反律的一个渐近性质

An asymptotic property on a reciprocity law for the Bettin--Conrey cotangent sum

本文针对 Bettin–Conrey 余切和互反律中隐式全纯函数的 Taylor 系数，利用第二类合流超几何函数将其渐近公式改进为渐近级数展开，从而解决了 Zagier 的猜想。

On $3$-dimensional foliated dynamical systems and Hilbert type reciprocity law

We show some fundamental results concerning $3$-dimensional foliated dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give a decomposition theorem for an FDS$^3$, which yields a classification of FDS$^3$'s. Secondly, for each type of the classification, we construct concrete examples of FDS$^3$'s. Finally, by using the integration theory for smooth Deligne cohomology, we introduce geometric analogues of local symbols and show a Hilbert type reciprocity law for an FDS$^3$. Our results answer the question posed by Deninger.

科尔梅兹猜想与模形式的互反定律

The conjecture of Colmez and reciprocity laws for modular forms

本文提出改进科尔梅兹猜想，明确CM阿贝尔簇的Faltings高度与Artin L函数的对数导数关系，并证明其对椭圆曲线成立，揭示模形式在CM点的精细互反律。

Quadratic reciprocity in a finite group

In this article we define a quadratic symbol for a finite group and prove a law of reciprocity for its value.

Explicit Reciprocity Laws for Formal Drinfeld Modules

In this paper, we prove explicit reciprocity laws for a class of formal Drinfeld modules having stable reduction of height one, in the spirit of those existing in characteristic zero (cf. the work of Wiles). We begin by defining the Kummer pairing in the language of formal Drinfeld modules defined over local fields of positive characteristic. We then prove explicit formulas for this pairing in terms of the logarithm of the formal Drinfeld module, a certain Coleman power series, torsion points and the trace. Our results extend the explicit formulas already proved by Anglès for Carlitz modules, …

Rational Quartic Reciprocity II

We continue investigating rational quartic reciprocity laws and, at the suggestion of the editor of AA, provide details of a proof of a remark in the first article with this title.

超有限域上多项式环的高次互反律及 Grunwald–Wang 定理的类比

Higher reciprocity law and An analogue of the Grunwald--Wang theorem for the ring of polynomials over an ultra-finite field

本文研究有限域非主超积上的多项式环，建立显式高次互反律，并据此证明特征 0 与特征 p 下 Grunwald–Wang 定理的类比。

Eight lectures on quadratic reciprocity

Rousseau's simple proof of the quadratic reciprocity law, followed by the proof of its equivalence with Hilbert's product formula. The Hilbert symbol is explained in terms of the reciprocity isomorphism, and the places of Q are determined.

关于单位群Friedberg-Jacquet周期的首个显式互反律

First explicit reciprocity law for unitary Friedberg--Jacquet periods

本文证明了当自动形式π由单位Friedberg-Jacquet周期区分时，相关动机的Bloch-Kato Selmer群消失，为单位群的互反律提供了首个显式结果。