Zeta Functions in Arithmetic and Geometry

In this talk we briefly introduce several definitions of Zeta functions that are used in the context of arithmetic varieties. These functions are in a vague sense related to each other and, at least the version defined by Hasse and Weil, arise from the Riemann zeta function. We will focus on the motivic zeta function attached to a Calabi-Yau variety X defined over a field K endowed with an ultrametric absolute value. We will explain what it means for a formal series with coefficients in the Grothendieck ring of varieties to be rational and how poles are defined. We will finally discuss the monodromy conjecture that relates those poles with the action of the absolute Galois group of K on the (étale) cohomology of X. We discuss how the Hilbert schemes of points on a surface behave with respect to the monodromy conjecture.