Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $\mathbb{Q}$. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.

In this paper, we find the number of representations of certain quadratic forms in 20 and 24 variables. We get this as an application of the evaluation of certain triple convolution sums of the divisor functions. Further, by comparing our formulas with that of Lomadze, we get expressions of certain cusp forms in terms of some finite sums involving the solution set of the quadratic form representation.

We extend a result of Cavachi on sums of relatively prime polynomials by proving that a polynomial of the form $f(X)+p^{k}g(X)$, with $f$ and $g$ relatively prime polynomials with integer coefficients, $\deg f<\deg g$, and $k$ a positive integer prime to $\deg g$ is irreducible over $\mathbb{Q}$ for all but finitely many prime numbers $p$.

Let $\mathbf{a}=(a_1,\cdots,a_k)$ denote a $k$-tuple of positive integers such that $a_1 \leq a_2 \leq\break \cdots \leq a_k$. We put $d(\mathbf{a};n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and let $\Delta(\mathbf{a};x)$ be the error term of the corresponding asymptotic formula for the summatory function of $d(\mathbf{a};n)$. In this paper we show an asymptotic formula of the mean square of $\Delta(\mathbf{a};x)$ under a certain condition. Moreover, when $k$ equals $2$ or $3$, we give unconditional asymptotic formulas for these mean squares.

In [SV] and [BMb], the authors proved the existence of a so-called higher rank amplifier and in [HRRa], the authors described an explicit version of a $GL(3)$ amplifier. This article provides, for $n\geq 4$, a totally explicit $GL(n)$ amplifier and gives all the results required to use it effectively.

We study a logarithmic version of the classical result of Artin-Furwängler on principalization of ideal classes in the Hilbert class-field by applying the group theoretic description of the transfert map to logarithmic class-groups of degree 0.

We establish, for a substantial class of elliptic curves, that the arithmetic local constants introduced by Mazur and Rubin agree with quotients of analytic root numbers.

In this paper, we present a method by which one can strip primes from the level of a vector-valued genus 2 Siegel modular form while preserving a congruence modulo this prime. An application of this result to four-dimensional Galois representations will also be presented.