Zeta integrals on arithmetic surfaces

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.