<p>In this article, we investigate holomorphy and non vanishing of Artin <i>L</i>-functions for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Re (s) &gt; 1/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℜ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>&gt;</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. First, we show that Artin <i>L</i>-functions of a solvable Galois extension <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{K}/\textrm{F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>K</mtext> <mo stretchy="false">/</mo> <mtext>F</mtext> </mrow> </math></EquationSource> </InlineEquation> with Galois group <i>G</i> are holomorphic at a point <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(s_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>s</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> by comparing the order of vanishing of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\zeta _\textrm{K}(s)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mtext>K</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq5.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">\(\zeta _{\textrm{K}^{G^{(2)}}}(s)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <msup> <mtext>K</mtext> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(G^{(2)}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> is the second commutator subgroup of <i>G</i>. This extends a result of Foote and Kumar Murty. Finally, we derive a criterion which is equivalent to the assertion that all the poles (except for a possible pole at <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(s=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) and non-trivial zeros of Artin <i>L</i>-functions necessarily lie on the line <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2060_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Re (s) = 1/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℜ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

错误:搜索内容不能为空,请输入英文关键词
错误:关键词超出字数限制,请精简
高级检索

On holomorphy and non-vanishing of Artin L-functions

  • Sanoli Gun,
  • Suhita Hazra,
  • Dhananjaya Sahu

摘要

In this article, we investigate holomorphy and non vanishing of Artin L-functions for \(\Re (s) > 1/2\) ( s ) > 1 / 2 . First, we show that Artin L-functions of a solvable Galois extension \(\textrm{K}/\textrm{F}\) K / F with Galois group G are holomorphic at a point \(s_0\) s 0 by comparing the order of vanishing of \(\zeta _\textrm{K}(s)\) ζ K ( s ) with \(\zeta _{\textrm{K}^{G^{(2)}}}(s)\) ζ K G ( 2 ) ( s ) , where \(G^{(2)}\) G ( 2 ) is the second commutator subgroup of G. This extends a result of Foote and Kumar Murty. Finally, we derive a criterion which is equivalent to the assertion that all the poles (except for a possible pole at \(s=1\) s = 1 ) and non-trivial zeros of Artin L-functions necessarily lie on the line \(\Re (s) = 1/2\) ( s ) = 1 / 2 .