scholarly journals Generalizations of Dedekind sums and their reciprocity laws

2003 ◽  
Vol 106 (4) ◽  
pp. 355-378 ◽  
Author(s):  
Yumiko Nagasaka ◽  
Kaori Ota ◽  
Chizuru Sekine
Keyword(s):  
Dedekind Sums ◽  
Reciprocity Laws

scholarly journals p-Adic q-Twisted Dedekind-Type Sums

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1756
Author(s):  
Abdelmejid Bayad ◽  
Yilmaz Simsek
Keyword(s):  
Bernoulli Polynomials ◽  
Dedekind Sums ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Finite Sums ◽  
Symmetry Relations

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.

scholarly journals P-adic interpolation of dedekind sums

1988 ◽  
Vol 37 (2) ◽  
pp. 293-301 ◽  
Author(s):  
C. Snyder
Keyword(s):  
Positive Integer ◽  
Measure Theory ◽  
Explicit Representation ◽  
Dedekind Sums ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Dirichlet Characters

In this article we give an explicit representation of p-adic Dedekind sums and their reciprocity laws by using p-adic measure theory. We then study the consequences of the p-adic reciprocity law for particular positive integer values in which case we can recover a reciprocity law for Dedekind sums attached to particular Dirichlet characters. This gives a proof different from that of Nagasaka.

scholarly journals Identities on poly-Dedekind sums

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Lee-Chae Jang
Keyword(s):  
Modular Group ◽  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Transformation Behavior ◽  
Dedekind Eta Function ◽  
Bernoulli Function ◽  
Bernoulli Functions

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe 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.

Dedekind sums and the signature of f (x, y) + z N , II

1999 ◽  
Vol 5 (1) ◽  
pp. 161-179 ◽  
Author(s):  
A. Némethi
Keyword(s):  
Dedekind Sums
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