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Zeta integrals on arithmetic surfaces


Author: T. Oliver
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 1003-1028
MSC (2010): Primary 11M99, 11R56
DOI: https://doi.org/10.1090/spmj/1432
Published electronically: September 30, 2016
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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 $ \mathbb{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.


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Additional Information

T. Oliver
Affiliation: Heilbronn Institute for Mathematical Research, University of Bristol, United Kingdom
Email: tdoliver163@gmail.com

DOI: https://doi.org/10.1090/spmj/1432
Keywords: Scheme of finite type, zeta function, local field, Hasse--Weil $L$-function, complete discrete valuation field, adeles
Received by editor(s): February 27, 2015
Published electronically: September 30, 2016
Dedicated: To Professor S. V. Vostokov on the occasion of his 70th birthday
Article copyright: © Copyright 2016 American Mathematical Society