An Introduction To Tate’s Thesis

Tate’s thesis may be viewed as giving the theory of automorphic representations and L-functions of the simplest connected reductive group G=GL(1), and so it remains a fundamental reference and starting point for anyone interested in mordern theory of automorphic representations. In this note, we will introduce the functional equation and meromorphic continuation of the Zeta function both in the local case and the global case. In the global case, we will use the adelic language. The knowledge of local field and algebraic number theory are assumed known.