Computing class groups by induction with generalised norm relations

TL;DR

The paper generalizes norm relations to generalized norm relations (GNRel) in the group algebra setting to enable inductive class-group computations beyond Galois extensions. It introduces compositum-based actions, provides equivalent field-theoretic and representation-theoretic characterizations, and proves a polynomial-time algorithm that computes S-unit bases (and thus class groups) from lower-degree subfields when a GNRel exists. The approach leverages Hecke algebras and Mackey functors to connect group-theoretic relations with arithmetic invariants, yielding rigorous complexity bounds and practical speedups. Empirical results on large-degree fields (including a degree-105 example) demonstrate substantial performance gains over classical methods, highlighting GNRel as a powerful tool for explicit class-field computations.

Abstract

We introduce a generalisation of norm relations in the group algebra Q[G], where G is a finite group. We give some properties of these relations, and use them to obtain relations between the S-unit groups of different subfields of the same Galois extension of Q, of Galois group G. Then we deduce an algorithm to compute the class groups of some number fields by reducing the problem to fields of lower degree. We compute the class groups of some large number fields.

Theorems & Definitions (91)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1.1
• Lemma 1.2
• Proposition 1.3
• proof
• Remark 1.5
• Remark 1.6
• Proposition 1.7