Abstract
Given a (smooth, projective, geometrically connected) curve over a number field, one expects its HasseâWeil $L$-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 âanalyticâ adelic structures admitting a form of âliftedâ 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 âmean-periodicity correspondenceâ, which is comparable to the conjectural automorphicity of HasseâWeil $L$-functions.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.