The analytic class number formula for $1$-dimensional affine schemes

TL;DR

This paper derives an analytic class number formula applicable to orders in products of S-integers in global fields, extending classical results to a broader algebraic and geometric context.

Contribution

It introduces a new analytic class number formula for 1-dimensional affine schemes over integers, generalizing existing formulas for number fields and orders.

Findings

Established a class number formula for affine schemes of dimension 1 over Z

Unified algebraic and geometric perspectives on class numbers

Extended classical class number formulas to new algebraic structures

Abstract

We derive an analytic class number formula valid for an order in a product of $S$-integers in global fields, or equivalently for reduced finite-type affine schemes of pure dimension $1$ over $\mathbb{Z}$.