THESIS
2018
Abstract
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]=(x
^{n}) 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]=(x
^{n}) 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...[
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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]=(x
^{n}) 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]=(x
^{n}) 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.
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