Convergent Arithmetic Power Series

Abstract

This paper considers rings of power series on discs of radius r > 0 with coefficients lying in a subring A of the rational numbers. One such ring, denoted here by Ar [[t]], is an arithmetic version of the disc algebra studied in analysis. A slightly smaller ring, Ar+ [[t]], consists of the holomorphic functions on (a of) the closed disc of radius r with coefficients lying in A. The rings Z1 [[t]] and Z1 + [[t]] are in fact equal to Z[t], and so the families {Zr[[tI] 0 ? < r < I} and {Zr+ [[t]] I 0 < r < 1} can be regarded as interpolating between Z[[t]] and Z[t]. These convergent power series rings may thus be of use in a deformation-theoretic context. The maximal spectra of Ar [[t]] and Ar+ [[t]] are described in section 1, and the latter ring is shown to be a noetherian regular unique factorization domain. The ring Zr [[t]], meanwhile, is shown to be complete under the uniform norm. Section 2 shows that every finitely generated projective module over Ar+ [[t]] is free. The proof uses a patching lemma for free modules. This section also contains a more general module-patching result which can be viewed as an arithmetic analog of Grothendieck's Existence Theorem. From this point of view, Z is analogous to the polynomial ring k [xl over a field k, Z[[t]] and Z[t] correspond to k [x] [[t]] and k [x, t], and Zr+ [[t]] corresponds to k [[t]] [x], whose spectrum is a tubular neighborhood of the affine k-line. Section 3 considers the subring Ar+ [kItIh of Ar+ [[t]] consisting of algebraic power series. Every such power series is shown to lie in an etale neighborhood of the disc. It is also shown that the ring Z{ t)}h of algebraic power series over Z on the open unit disc consists entirely of rational functions. Given an algebraic structure over Z{t}h (obtained, for example, by