Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces
This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodic...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2014
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| Online Access: | https://eprints.nottingham.ac.uk/14518/ |
| _version_ | 1848791980594692096 |
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| author | Oliver, Thomas David |
| author_facet | Oliver, Thomas David |
| author_sort | Oliver, Thomas David |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes.
1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series.
2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis.
The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity. |
| first_indexed | 2025-11-14T18:37:08Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-14518 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T18:37:08Z |
| publishDate | 2014 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-145182025-02-28T11:31:23Z https://eprints.nottingham.ac.uk/14518/ Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces Oliver, Thomas David This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes. 1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis. The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity. 2014-10-15 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/14518/1/ThomasOliverFinalThesis.pdf Oliver, Thomas David (2014) Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces. PhD thesis, University of Nottingham. Number Theory arithmetic geometry Zeta Functions L-Functions arithmetic surfaces adeles automorphic representations mean-periodicity. Hasse--Weil conjecture. |
| spellingShingle | Number Theory arithmetic geometry Zeta Functions L-Functions arithmetic surfaces adeles automorphic representations mean-periodicity. Hasse--Weil conjecture. Oliver, Thomas David Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| title | Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| title_full | Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| title_fullStr | Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| title_full_unstemmed | Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| title_short | Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| title_sort | higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces |
| topic | Number Theory arithmetic geometry Zeta Functions L-Functions arithmetic surfaces adeles automorphic representations mean-periodicity. Hasse--Weil conjecture. |
| url | https://eprints.nottingham.ac.uk/14518/ |