Analysis on arithmetic schemes. II

Abstract

Kak qislo v ume, na peske ostavl sled, okean gromozdits vo t me, milliony let mertvo i zyb ba ka wepku. I esli rezko xagnut s debarkadera vbok, vovne, budex dolgo padat , ruki po xvam; no ne vosposleduet vspleska. J. Brodsky AbstractWe 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.Key Words: topological Milnor K-groups, explicit two-dimensional class field theory, adeles for arithmetic schemes, translation invariant integration and harmonic analysis, two-dimensional adelic analysis, zeta functions, zeta integral, analytic duality on arithmetic surfaces, boundary term, elliptic curves over global fields.

Geometric and analytic structures on the higher adèles

“…Also, no thoughts on infinite places will appear here. See [19] for adèles directed towards arithmetic considerations.…”

“…Analogously to the case of higher local fields, the adèles of a scheme can also be equipped with sequential topologies [19,20].…”

“…This yields further, more complicated, rings of integers, along with corresponding notions of adèles. See Fesenko [17,19]. Finally, instead of allowing just quasi-coherent sheaves as coefficients, one may also allow other sheaves as coefficients.…”

“…where L i+1 → L i+1 are Beilinson lattices for the flag 19) where K j are the n-local field factors. Our K is one of these factors.…”

Ramification of higher local fields approaches and questions

“…It would be interesting to understand what kind of ramification data are needed in adelic theory of arithmetic surfaces. For example, the non-wild part of the conductor of the curve appears in [Fe10, Subsection 3.4]; can we allow wild ramification here?…”