On the Hasse norm principle.
A power digraph, denoted by , is a directed graph with as the set of vertices and as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of for and are determined....
Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally -adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper bounds from their paper and, under the assumption of integrality, to improve slightly upon their bounds.
Let be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function as x → ∞, where q grows with x in a definite way and j = 2,3,4.
Let be a normalized primitive holomorphic cusp form of even integral weight for the full modular group . Denote by the th normalized Fourier coefficient of . We are interested in the average behaviour of the sum for , where and is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power -functions and Rankin-Selberg -functions.
It is well known by results of Golod and Shafarevich that the Hilbert -class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian -extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian -extension over in which eight primes...
Let be an imaginary bicyclic biquadratic number field, where is an odd negative square-free integer and its second Hilbert -class field. Denote by the Galois group of . The purpose of this note is to investigate the Hilbert -class field tower of and then deduce the structure of .