<p>In this paper, we study the Iwasawa <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-invariant of the cyclotomic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb Z _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-extension of a real quadratic field <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb Q (\sqrt{p_1p_2p_3p_4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mn>2</mn> </msub> <msub> <mi>p</mi> <mn>3</mn> </msub> <msub> <mi>p</mi> <mn>4</mn> </msub> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p_1, p_2, p_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> are distinct odd prime numbers satisfying certain arithmetic conditions. We give a new infinite family of real quadratic fields for which Greenberg’s conjecture holds.</p>

On $\mathbb {Z}_2$-extensions of real quadratic fields with four odd ramified prime numbers

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2026年5月5日

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Yasushi Mizusawa,
Yuito Saito

摘要

In this paper, we study the Iwasawa $\lambda $ $λ$ -invariant of the cyclotomic $\mathbb Z _2$ $Z_{2}$ -extension of a real quadratic field $\mathbb Q (\sqrt{p_1p_2p_3p_4})$ $Q (\sqrt{p_{1} p_{2} p_{3} p_{4}})$ , where $p_1, p_2, p_3$ $p_{1}, p_{2}, p_{3}$ and $p_4$ $p_{4}$ are distinct odd prime numbers satisfying certain arithmetic conditions. We give a new infinite family of real quadratic fields for which Greenberg’s conjecture holds.

On \(\mathbb {Z}_2\)-extensions of real quadratic fields with four odd ramified prime numbers

摘要