Reciprocity Laws Research Articles

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

Ramification theory of reciprocity sheaves, II Higher local symbols

We construct a theory of higher local symbols along Paršin chains for reciprocity sheaves. Applying this formalism to differential forms, gives a new construction of the Paršin–Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato’s higher local class field theory in the geometric case. The higher local symbols satisfy various reciprocity laws. The main result of the paper is a characterization of the modulus attached to a section of a reciprocity sheaf in terms of the higher local symbols.

The Residue Theorem

Cauchy's Residue Theorem, commonly known as the residue Theorem, is a key theorem in complex variables because it enables people to determine the enclosed curve line of integrals for functionalities. Real integrals and infinite series may also be calculated using it. Our ability to handle complicated analysis is improved by the residue theorem. By connecting the internal Ehrhart polynomials to the closures, we illustrate these tetrahedron Ehrhart-Macdonald reciprocity laws. We calculate the Earhart coefficient of the codimensions to demonstrate our methodology. We conclude by demonstrating how to use our ideas to locate any paste applied with a convex lattice.

Obstructions to reciprocity laws related to division fields of Drinfeld modules

Let q q be an odd prime power, denote by F q \mathbb {F}_q the finite field with q q elements, and set A ≔ F q [ T ] A ≔\mathbb {F}_q[T] , F ≔ F q ( T ) F ≔\mathbb {F}_q(T) . Let ψ : A → F { τ } \psi : A \to F\{\tau \} be a Drinfeld A A -module over F F , of rank r ≥ 2 r \geq 2 , with E n d F ¯ ( ψ ) = A End_{\overline {F}}(\psi ) = A . For a non-zero ideal n \mathfrak {n} of A A , denote its unique monic generator by n n , denote the degree of n n as a polynomial in T T by deg ⁡ n \deg n , and denote the n \mathfrak {n} -division field of ψ \psi by F ( ψ [ n ] ) F(\psi [\mathfrak {n}]) . A reciprocity law for ψ \psi asserts that, if gcd ( c h a r F , r ) = 1 \gcd (charF, r) = 1 or if n \mathfrak {n} is prime, then a non-zero prime ideal p ∤ n \mathfrak {p} \nmid \mathfrak {n} of A A splits completely in F ( ψ [ n ] ) F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a 1 , p ( ψ ) a_{1, \mathfrak {p}}(\psi ) of ψ \psi at p \mathfrak {p} and the first component b 1 , p ( ψ ) b_{1, \mathfrak {p}}(\psi ) of the Frobenius index of ψ \psi at p \mathfrak {p} satisfy the congruences a 1 , p ( ψ ) ≡ − r ( mod n ) a_{1, \mathfrak {p}}(\psi ) \equiv -r \pmod n and b 1 , p ( ψ ) ≡ 0 ( mod n ) b_{1, \mathfrak {p}}(\psi ) \equiv 0 \pmod n . We find the Dirichlet density of the set of non-zero prime ideals p \mathfrak {p} for which the latter congruence never holds, that is, for which b 1 , p ( ψ ) = 1 b_{1, \mathfrak {p}}(\psi ) = 1 . Using similar methods, we prove an asymptotic formula for the function of x x defined by the average 1 # { p : deg ⁡ p = x } ∑ p : deg ⁡ p = x τ A ( b 1 , p ( ψ ) ) \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi )) , where p = A p \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A A whose monic generator p ∈ A p \in A has degree x x and where τ A ( b 1 , p ( ψ ) ) \tau _A(b_{1, \mathfrak {p}}(\psi )) denotes the number of monic divisors of b 1 , p ( ψ ) b_{1, \mathfrak {p}}(\psi ) .

On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward the Iwasawa main conjecture in this setting. The proof relies on the first and second reciprocity laws relating theta elements to Heegner point Euler systems on Shimura curves. As a by-product we also prove a result towards the rank 0 case of certain Bloch–Kato conjecture and a parity conjecture.

Remarks on classical number-theoretic aspects of Milnor–Witt K-theory

We record a few observations on number theoretic aspects of Milnor-Witt K-theory, focusing on generalizing classical results on reciprocity laws, Hasse's norm theorem and K_2 of number fields and rings of integers.

P-Adic q-Twisted Dedekind-Type Sums

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p-adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a fine study of the existing symmetry relations between the finite sums, considered in our study, and their symmetries through permutations of initial parameters.

A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions

We generalize Contou-Carrère symbols to higher dimensions. To an ( n + 1 ) (n+1) -tuple f 0 , … , f n ∈ A ( ( t 1 ) ) ⋯ ( ( t n ) ) × f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times } , where A A denotes an algebra over a field k k , we associate an element ( f 0 , … , f n ) ∈ A × (f_0,\dots ,f_n) \in A^{\times } , extending the higher tame symbol for k = A k = A , and earlier constructions for n = 1 n = 1 by Contou-Carrère, and n = 2 n = 2 by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic K K -theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.

Multiple Dedekind Type Sums and Their Related Zeta Functions

The main purpose of this paper is to use the multiple twisted Bernoulli polynomials and their interpolation functions to construct multiple twisted Dedekind type sums. We investigate some properties of these sums. By use of the properties of multiple twisted zeta functions and the Bernoulli functions involving the Bernoulli polynomials, we derive reciprocity laws of these sums. Further developments and observations on these new Dedekind type sums are given.

The norm residue symbol for higher local fields

In this paper we investigate the Kummer pairing associated to an arbitrary (one-dimensional) formal group. In particular, we obtain formulae describing the values of the pairing in terms of multidimensional p-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. The results are a generalization to higher-dimensional local fields of Kolyvagin's explicit reciprocity laws. In particular, they constitute a generalization of the reciprocity laws of Artin-Hasse, Iwasawa and Wiles.

Automorphic Forms and Arithmetic

The workshop was at the interface of automorphic forms and analytic number theory. The aim was to disseminate, discuss and develop important new methods and results in the analytic theory of automorphic forms, in particular on higher rank groups, as well as their arithmetic applications. This includes, for instance, the study of various aspects of L -functions such as moments, reciprocity laws, and probabilistic aspects, counting problems with automorphic forms, applications of results of algebraic geometry to automorphic forms, as well as analytic aspects of automorphic forms over function fields.

On the Bloch\u2013Kato conjecture for Hilbert modular forms

The aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

Using decomposition groups to prove theorems about quadratic residues

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.

Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions

Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions

The exceptional zero phenomenon for elliptic units

The exceptional zero phenomenon has been widely studied in the realm of p -adic L -functions, where the starting point lies in the foundational works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among others. This phenomenon also appears in the study of Euler systems, and in this case one is led to study higher order derivatives of cohomology classes in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws which make the connection with p -adic L -funtions. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a derived elliptic unit either to special values of Katz’s two variable p -adic L -function or to its derivatives. Further, we interpret this fact in terms of an \mathcal L -invariant and relate this result to other approaches to the exceptional zero phenomenon, most notably to the work of Bley, and we also compare this setting with other scenarios concerning Heegner points and Beilinson–Flach elements.

The Chevalley–Gras formula over global fields

Dans cet article, nous donnons une preuve adélique de la formule de Chevalley–Gras pour les corps de nombres qui, elle-même, est une généralisation de la formule du nombre de classes ambiges. L’idée est de réduire cette formule au théorème de la norme de Hasse et à des lois de réciprocité globaux. Nous donnons également une preuve adélique de la formule de Chevalley–Gras pour les groupes des classes des diviseurs de degré 0 dans le cas des corps de fonctions, qui étend un résultat de Rosen.

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.

Nonabelian 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 curve, these take values in Galois cohomology of modular forms, and give obstructions to an adelic elliptic curve with global Tate module underlying a global elliptic curve.

Explicit reciprocity laws for higher local fields

Using previously constructed 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.

Review of Number Theory

There are probably at least as many different approaches to number theory as there are books written about it. Some broad distinctions include those taking an historical versus (say) a purely modern approach, with many gradations in between, or those that are algebraically oriented (e.g., with an emphasis on reciprocity laws, or questions that relate to algebraic geometry), or still others that are more analytic. The book under review is definitely in the latter category. The "message" of the book is in the title, as primes and their density are the principle concern. In accordance with that theme, a highlight of the book is a complete proof of the prime number theorem. However, the theme and its variations are taken as springboards to other important fields, including aspects of algebraic number theory, as well as applications, such as primality testing and cryptography.