Let L/K be a Galois extension of number fields and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G=\textrm{Gal}(L/K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mtext>Gal</mtext> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt’s conjecture at p for certain proper intermediate fields of L/K implies Leopoldt’s conjecture at p for L. We also obtain relations between the Leopoldt defects of intermediate fields of L/K. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation>, there exists an infinite family <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> of totally real <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-extensions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> such that Leopoldt’s conjecture for F at p holds for every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F \in \mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>∈</mo> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p \in \mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="script">P</mi> </mrow> </math></EquationSource> </InlineEquation>.

Applications of representation theory and of explicit units to Leopoldt’s conjecture

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2026年3月17日

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Fabio Ferri,
Henri Johnston

摘要

Let L/K be a Galois extension of number fields and let $G=\textrm{Gal}(L/K)$ $G = Gal (L / K)$ . We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt’s conjecture at p for certain proper intermediate fields of L/K implies Leopoldt’s conjecture at p for L. We also obtain relations between the Leopoldt defects of intermediate fields of L/K. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers $\mathcal {P}$ $P$ , there exists an infinite family $\mathcal {F}$ $F$ of totally real $S_{3}$ $S_{3}$ -extensions of $\mathbb {Q}$ $Q$ such that Leopoldt’s conjecture for F at p holds for every $F \in \mathcal {F}$ $F \in F$ and $p \in \mathcal {P}$ $p \in P$ .