Computing $p$-Class Group Structure in Real Quadratic Fields: A New Approach

TL;DR

The paper develops a unified, deterministic framework for determining the $p^n$-torsion structure of class groups of real quadratic fields by linking ideal classes to degree $p^n$ polynomials through generalized higher composition laws and unit-norm indices. It blends relative Polya groups, transfer maps, and Galois cohomology to produce a simplified class-number formula and capitulation criteria, enabling unconditional computation of $p^n$-parts via ramified extensions $M/N$. A central result is a bijection between order $p^n$ classes and cyclic extensions, with a practical criterion for capitulation that can be checked using unit norms, illustrated by explicit PARI/GP calculations. This approach extends classical Gauss and Bhargava constructions to arbitrary primes, offering a computable, theory-driven route to decode the p-part structure of real quadratic class groups.

Abstract

This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for real quadratic fields and establishes a bridge between class field theory, composition laws of binary forms of degree $p^n$, and ideal classes of order $p^n$, where p is prime and n is an arbitrary positive integer.

Theorems & Definitions (56)

• Theorem 1.1: Cohen2013, Theorems 5.2.8 and 5.2.9
• Theorem 1.2: Wood Wood2011
• Theorem 1.3: Bhargava2004, Theorem 13
• Theorem 1.4: BhargavaVarma2016, Theorem 2
• Theorem 1.5: Main technical theorem
• Definition 1.6
• Theorem 1.7
• Remark 1.8
• Corollary 1.9
• Example 1.10