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On the existence of Minkowski units

David Burns, Donghyeok Lim, Christian Maire

Abstract

We investigate the Galois structure of algebraic units in cyclic extensions of number fields and thereby obtain strong new results on the existence of independent Minkowski $S$-units.

On the existence of Minkowski units

Abstract

We investigate the Galois structure of algebraic units in cyclic extensions of number fields and thereby obtain strong new results on the existence of independent Minkowski -units.
Paper Structure (8 sections, 9 theorems, 49 equations)

This paper contains 8 sections, 9 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Hypotheses and examples
  3. Galois cohomology
  4. Module structures
  5. Yakovlev diagrams
  6. The proof of Theorem \ref{['intro thm']}
  7. Some special cases
  8. Minkowski units

Key Result

Theorem 1.1

Fix a natural number $n$. Then, as $(L/K,S)$ ranges over $\mathcal{C}_n$, only finitely many isomorphism classes of indecomposable $\mathbb{Z}_p[(\mathbb{Z}/p^n)]$-lattices arise as direct summands of any $U_{L,S}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 2.2
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 11 more