L-functions Exponential sums Newton polygon .

In this paper, we will calculate the number of Galois extensions of local fields with Galois group $A_{n}$ or $S_{n}$ .

In this paper, we study the algebraic structure of principal units in the tower of the maximal abelian extensions of local fields of characteristic zero and the corresponding Galois groups at each level. As an application, we show the finiteness result for the number of coverings with a given degree of the maximal abelian extension of a local field in characteristic zero. The number of p-coverings for Qp is computed explicitly.

We construct a finite subgroup of Brauer–Manin obstruction for detecting the existence of integral points on integral models of principle homogeneous spaces of multi-norm tori. Several explicit examples are provided.

Integral point Linear algebraic group of multiplicative type Galois cohomology Brauer–Manin obstruction Strong approximation Sum of two squares

We introduce descent methods to the study of strong approximation on algebraic varieties. We apply them to two classes of varieties defined by P(t)=N_{K/k}(z): firstly for quartic extensions of number fields K/k and quadratic polynomials P(t) in one variable, and secondly for k=Q, an arbitrary number field K and P(t) a product of linear polynomials over Q in at least two variables. Finally, we illustrate that a certain unboundedness condition at archimedean places is necessary for strong approximation.