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 Algebra i Analiz, 2015, Volume 27, Issue 6, Pages 199–233 (Mi aa1473)

Research Papers

Zeta integrals on arithmetic surfaces

T. Oliver

Heilbronn Institute for Mathematical Research, University of Bristol, UK

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 an (two-dimensional) adelic interpretation of the “mean-periodicity correspondence”, which is comparable to the conjectural automorphicity of Hasse–Weil $L$-functions.

Keywords: scheme of finite type, zeta function, local field, Hasse–Weil $L$-function, complete discrete valuation field, adeles.

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English version:
St. Petersburg Mathematical Journal, 2016, 27:6, 1003–1028

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Citation: T. Oliver, “Zeta integrals on arithmetic surfaces”, Algebra i Analiz, 27:6 (2015), 199–233; St. Petersburg Math. J., 27:6 (2016), 1003–1028

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/aa1473
• http://mi.mathnet.ru/eng/aa/v27/i6/p199

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. T. Oliver, “Automorphicity and mean-periodicity”, J. Math. Soc. Jpn., 69:1 (2017), 25–51
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