Related papers: Adelic Rogers integral formula

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.

We establish unconditional $\Omega$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an…

We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on $\GL(n)\times\GL(n-1)$ for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation…

This paper contains two remarks on Beilinson's adeles with values in the De Rham complex of a scheme. The first is an interpretation, in terms of adeles, of the decomposition of the De Rham complex on a scheme defined modulo $p^{2}$ (the…

In this article we prove a reciprocity law in number fields with odd class number that specializes to Scholz's reciprocity law over the rationals.

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version…

We develop an explicit algebriac de Rham theory for relative completion of $\mathrm{SL}_2(\mathbb{Z})$. This allows the construction of iterated integrals involving modular forms of the second kind, generalizing iterated integrals of…

As a first application of a very old theorem, known as Herschel's theorem, we provide direct elementary proofs of several explicit expressions for some numbers and polynomials that are known in combinatorics. The second application deals…

The main result of this paper is that if E is a field extension of finite odd degree over a real field Q, and if E is a repeated radical extension of Q, then every intermediate field is also a repeated radical extension of Q. This paper…

One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real…

We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…

Legendre's relation for the complete elliptic integrals of the first and second kinds is generalized. The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott's identity.

A simple formal procedure makes the main properties of the lagrangian binomial extendable to functions depending to any kind of order of the time--derivatives of the lagrangian coordinates. Such a broadly formulated binomial can provide the…

Some aspects of analysis involving fields with absolute value functions are discussed, which includes the real or complex numbers with their standard absolute values, as well as ultrametric situations like the p-adic numbers.

We generalize Rado's extension theorem to complex spaces.

In two previous papers the author introduced a multiplication of distributions in one dimension and he proved that two one-dimensional Dirac delta functions and their derivatives can be multiplied, at least under certain conditions. Here,…

We provide a generalization of an algebraic linear combination for the trace of certain elliptic modular forms, and through specializing the expression at a suitable pair consisting of an elliptic curve over algebraic number fields and its…

In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.

In this note we show that any proof of Wallis's formula or of the probability integral formula proves both assertions.

We survey the classical results of the Dirichlet Approximation Theorem.