We investigate the Iwasawa <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-invariant <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda _p(\chi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the p-adic L-function associated with a Dirichlet character <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> for odd primes p. Using special values of the p-adic L-function and its derivative, we derive novel and computationally efficient criteria to distinguish between the cases <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda \geqslant 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>⩾</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, we study the case when the p-adic L-function vanishes at <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$s=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Building on the results of Ferrero-Greenberg and of Gross-Koblitz, we establish explicit conditions for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\lambda _p(\chi ) >1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\lambda _p(\chi ) > 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we extend the methods of Ernvall-Metsänkylä and of Dummit et al. to calculate the <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-invariant. This is achieved by twisting <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> by characters <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> of p-power order and using the values of the p-adic L-function at <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$s=2-p, \dots , 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>2</mn> <mo>-</mo> <mi>p</mi> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we utilize the value at <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$s=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> to compute <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\lambda _p(\chi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, these results are applied to generate numerical data on the distribution of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-invariants, when either the prime p or the Dirichlet character <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> is fixed.

Special values of p-adic L-functions and Iwasawa $\lambda $-invariants of Dirichlet characters

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2026年7月23日

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Heiko Knospe

摘要

We investigate the Iwasawa $\lambda $ $λ$ -invariant $\lambda _p(\chi )$ $λ_{p} (χ)$ of the p-adic L-function associated with a Dirichlet character $\chi $ $χ$ for odd primes p. Using special values of the p-adic L-function and its derivative, we derive novel and computationally efficient criteria to distinguish between the cases $\lambda = 0$ $λ = 0$ , $\lambda = 1$ $λ = 1$ , $\lambda = 2$ $λ = 2$ , and $\lambda \geqslant 3$ $λ ⩾ 3$ . In particular, we study the case when the p-adic L-function vanishes at $s=0$ $s = 0$ . Building on the results of Ferrero-Greenberg and of Gross-Koblitz, we establish explicit conditions for $\lambda _p(\chi ) >1 $ $λ_{p} (χ) > 1$ and $\lambda _p(\chi ) > 2$ $λ_{p} (χ) > 2$ . Furthermore, we extend the methods of Ernvall-Metsänkylä and of Dummit et al. to calculate the $\lambda $ $λ$ -invariant. This is achieved by twisting $\chi $ $χ$ by characters $\psi $ $ψ$ of p-power order and using the values of the p-adic L-function at $s=2-p, \dots , 0$ $s = 2 - p, \dots, 0$ . Additionally, we utilize the value at $s=1$ $s = 1$ to compute $\lambda _p(\chi )$ $λ_{p} (χ)$ . Finally, these results are applied to generate numerical data on the distribution of $\lambda $ $λ$ -invariants, when either the prime p or the Dirichlet character $\chi $ $χ$ is fixed.

Special values of p-adic L-functions and Iwasawa \(\lambda \)-invariants of Dirichlet characters

摘要