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Arithmetic functions on a Dedekind domain

Andrew Phillips

TL;DR

The work extends arithmetic-function theory to unique factorization monoids and connects it to formal power-series rings, establishing a functorial isomorphism A(M) ≅ K[[X_p]] and analyzing the ringed-space structure on totally multiplicative arithmetic functions over Dedekind-domain ideals. By introducing a Zariski-type topology and a refined quotient M_1(A), the authors show that the space of such functions determines the Dedekind domain up to isomorphism and that the correspondence is faithful (and fully faithful under refinement). They develop a scheme-like, ringed-space framework (affine arithmetic spaces) and prove an anti-equivalence between Dedekind domains/fields and affine arithmetic spaces, enabling a global-arithmetic-space perspective with gluing, dimension theory, and spectral properties. The results provide a novel, category-theoretic, geometric lens on algebraic number theory objects and pave the way for a global, scheme-like treatment of arithmetic functions through ringed spaces.

Abstract

We study functions from a unique factorization monoid to a field. The set of all such functions is a commutative ring isomorphic to a ring of formal power series over the field, with indeterminates indexed by the prime elements of the monoid. The set of all totally multiplicative functions on the monoid of integral ideals in a Dedekind domain has a ringed space structure, which, after identifying functions with the same prime ideal zeros, determines the Dedekind domain up to isomorphism.

Arithmetic functions on a Dedekind domain

TL;DR

The work extends arithmetic-function theory to unique factorization monoids and connects it to formal power-series rings, establishing a functorial isomorphism A(M) ≅ K[[X_p]] and analyzing the ringed-space structure on totally multiplicative arithmetic functions over Dedekind-domain ideals. By introducing a Zariski-type topology and a refined quotient M_1(A), the authors show that the space of such functions determines the Dedekind domain up to isomorphism and that the correspondence is faithful (and fully faithful under refinement). They develop a scheme-like, ringed-space framework (affine arithmetic spaces) and prove an anti-equivalence between Dedekind domains/fields and affine arithmetic spaces, enabling a global-arithmetic-space perspective with gluing, dimension theory, and spectral properties. The results provide a novel, category-theoretic, geometric lens on algebraic number theory objects and pave the way for a global, scheme-like treatment of arithmetic functions through ringed spaces.

Abstract

We study functions from a unique factorization monoid to a field. The set of all such functions is a commutative ring isomorphic to a ring of formal power series over the field, with indeterminates indexed by the prime elements of the monoid. The set of all totally multiplicative functions on the monoid of integral ideals in a Dedekind domain has a ringed space structure, which, after identifying functions with the same prime ideal zeros, determines the Dedekind domain up to isomorphism.
Paper Structure (7 sections, 35 theorems, 93 equations)

This paper contains 7 sections, 35 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Arithmetic functions
  3. Ring structure
  4. Topological ring structure
  5. Ringed space structure
  6. A refined space
  7. Global spaces

Key Result

Theorem 1.1

There is an isomorphism of $K$-algebras $\mathscr{A}(M) \to K\llbracket (X_p)_p\rrbracket$ which is functorial in $M$. In particular, $\mathscr{A}(M)$ is a UFD. Also, there is a discrete valuation on $\mathscr{A}(M)$ making it into a complete metric space.

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • ...and 75 more