2010/05/20 by Ivan Fesenko · 44 citations
Mathematics · #advanced mathematical theories #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals #Mathematics #Meromorphic function #Boundary (topology) #Analytic continuation #L-function #Mathematical analysis #Pure mathematics #Riemann zeta function #Elliptic curve #Algebraic number field #Surface integral #Arithmetic #Integral equation
paper · doi:10.1017/is010004028jkt103
published in Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology 5(3), 437-557 (Cambridge University Press)
openalex publication_date 2010/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/08
Abstract We construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K 2 -delic and K 1 × K 1 -delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.