Zeta integrals on arithmetic surfaces : WestminsterResearch

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Title	Zeta integrals on arithmetic surfaces
Type	Journal article
Authors	Oliver, T.
Abstract	Given a (smooth, projective, geometrically connected) curve over a number field, one expects its Hasse–Weil L-function, a priori defined only on a right half-plane, to admit meromorphic continuation to C and satisfy a simple functional equation. Aside from exceptional circumstances, these analytic properties remain largely conjectural. One may formulate these conjectures in terms of zeta functions of two-dimensional arithmetic schemes, on which one has non-locally compact “analytic” adelic structures admitting a form of “lifted” harmonic analysis first defined by Fesenko for elliptic curves. In this paper we generalize his global results to certain curves of arbitrary genus by invoking a renormalizing factor which may be interpreted as the zeta function of a relative projective line. We are lead to a new interpretation of the gamma factor (defined in terms of the Hodge structures at archimedean places) and a (two-dimensional) adelic interpretation of the “mean-periodicity correspondence”, which is comparable to the conjectural automorphicity of Hasse–Weil L-functions.
Journal	St Petersburg Math. J.
Journal citation	27, pp. 1003-1028
Year	2016
Publisher	American Mathematical Society
Digital Object Identifier (DOI)	https://doi.org/10.1090/spmj/1432
Web address (URL)	https://www.ams.org/journals/spmj/2016-27-06/S1061-0022-2016-01432-0/
Published online	30 Sep 2016

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Zeta integrals on arithmetic surfaces

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