<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> be a finite field of even characteristic and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in \mathbb {F}_{q}^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mmultiscripts> <mi mathvariant="double-struck">F</mi> <mrow> <mi>q</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>. For any non-constant monic square-free polynomial <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$A \in \mathbb {F}_{q}[t]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the polynomial <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">$X^2+AX+\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>A</mi> <mi>X</mi> <mo>+</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> is irreducible over <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and it defines a real quadratic extension <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$K_A$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which we call <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Chowla type. In this paper we investigate the distribution of the class numbers <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$h_{A}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> over real quadratic function fields <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{A}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> of Chowla type by comparing the distribution of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$L(1,\chi _{A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>χ</mi> <mi>A</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to that of a random Euler product <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1156_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$L(1,\mathbb {X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="double-struck">X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

The distribution of class numbers of real quadratic function fields of Chowla type in even characteristic

期刊论文
发表日期：

2025年7月16日

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Hwanyup Jung

摘要

Let $\mathbb {F}_{q}$ $F_{q}$ be a finite field of even characteristic and $\alpha \in \mathbb {F}_{q}^{*}$ $α \in F_{q}^{*}$ . For any non-constant monic square-free polynomial $A \in \mathbb {F}_{q}[t]$ $A \in F_{q} [t]$ , the polynomial $X^2+AX+\alpha $ $X^{2} + A X + α$ is irreducible over $\mathbb {F}_{q}(t)$ $F_{q} (t)$ and it defines a real quadratic extension $K_A$ $K_{A}$ of $\mathbb {F}_{q}(t)$ $F_{q} (t)$ , which we call $\alpha $ $α$ -Chowla type. In this paper we investigate the distribution of the class numbers $h_{A}$ $h_{A}$ over real quadratic function fields $K_{A}$ $K_{A}$ of Chowla type by comparing the distribution of $L(1,\chi _{A})$ $L (1, χ_{A})$ to that of a random Euler product $L(1,\mathbb {X})$ $L (1, X)$ .