In this thesis, we study the local and global zeta integral of a specific algebra associated with a base field k. In the local case, let k be a local field and char(k) = 0, and B = k[x]=(xⁿ) for a fixed integer n 1. In this case, we show that the zeta integrals on B have meromorphic continuation and they satisfy certain functional equations. In the global case, let k be an algebraic number field and B(A) = A[x]=(xⁿ) where A is the adele ring of k. In this case, we use Poisson summation formula to study the zeta integrals on B(A), and also prove the meromorphic continuation and certain functional equations of zeta integrals. The corresponding case "n = 1" is studied in detail by John Tate in his thesis[1] where he proved the functional equation and meromorphic continuation of t...[ Read more ]

In this thesis, we study the local and global zeta integral of a specific algebra associated with a base field k. In the local case, let k be a local field and char(k) = 0, and B = k[x]=(xⁿ) for a fixed integer n > 1. In this case, we show that the zeta integrals on B have meromorphic continuation and they satisfy certain functional equations. In the global case, let k be an algebraic number field and B(A) = A[x]=(xⁿ) where A is the adele ring of k. In this case, we use Poisson summation formula to study the zeta integrals on B(A), and also prove the meromorphic continuation and certain functional equations of zeta integrals. The corresponding case "n = 1" is studied in detail by John Tate in his thesis[1] where he proved the functional equation and meromorphic continuation of the zeta integrals and the zeta functions for an algebraic number field k. Godement and Jacquet[2] generalized the study of zeta integrals on any simple algebras in 1972. The work in this thesis can be regarded as the study of zeta integrals on certain non-simple algebras B and B(A) and this provides a new example where the theory of zeta integrals in Tate's thesis can be generalized to non-simple algebras.