Existence of discrete cocompact subgroups of reductive groups over local fields.
Existence of rational points on smooth projective varieties
Explicit descriptions of quadratic maps on ℙ¹ defined over a field K
Explicit -descent for elliptic curves in characteristic
Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial
Given a multivariate polynomial with integral coefficients verifying an hypothesis of analytic regularity (and satisfying ), we determine the maximal domain of meromorphy of the Euler product and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.