Analysis on Arithmetic Schemes . I

A shift invariant measure on a two dimensional local field, taking values in formal power series over reals, is introduced and discussed. Relevant elements of analysis, including analytic duality, are developed. As a two dimensional local generalization of the works of Tate and Iwasawa a local zeta integral on the topological Milnor K 2-group of the field is introduced and its properties are studied. 2000 Mathematics Subject Classification: 28B99, 28C99, 28E05, 2899, 42-99, 11S99, 11M99, 11F99, 11-99, 19-99. The functional equation of the twisted zeta function of algebraic number fields was first proved by E. Hecke (for a recent exposition see [N, Ch. VII]). J. Tate [T] and, for unramified characters without local theory, K. Iwasawa [I1–I2] lifted the zeta function to a zeta integral defined on an adelic space. Their method of proving the functional equation and deriving finiteness of several number theoretical objects is a generalization of one of the proofs of the functional equation by B. Riemann. The latter in the case of rational numbers uses an appropriate theta formula derived from a summation formula which itself follows from properties of Fourier transform. Hence the functional equation of the zeta integral reflects symmetries of the Fourier transform on the adelic object and its quotients, and the right mixture of their multiplicative and additive structures. With slight modification the approach of Tate and Iwasawa for characteristic zero can be extended to a uniform treatment of any characteristic, e.g. [W2]. Earlier, the functional field case was treated similarly to Hecke’s method by E. Witt in 1936 (cf. [Rq, sect. 7.4]), and it was also established using essentially harmonic analysis on finite rings by H.L. Schmid and O. Teichmüller [ST] in 1943 (for a modern exposition see e.g. [M,3.5]); those works were not widely known. Documenta Mathematica · Extra Volume Kato (2003) 261–284