Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_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 the finite field with q elements, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$F:=\mathbb {F}_q(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>:</mo> <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> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$F^{\operatorname {sep}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mo>sep</mo> </msup> </math></EquationSource> </InlineEquation> a separable closure of F. Set A to denote the polynomial ring <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </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>. Let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {p}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">p</mi> </math></EquationSource> </InlineEquation> be a non-zero prime ideal of A, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation> be the completion of A at <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {p}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">p</mi> </math></EquationSource> </InlineEquation>. Given any integer <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq8.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$r\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, I construct a Galois representation <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="196" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho :\operatorname {Gal}(F^{\operatorname {sep}}/F)\rightarrow \operatorname {GL}_r(\mathcal {O})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>:</mo> <mo>Gal</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mo>sep</mo> </msup> <mo stretchy="false">/</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mo>GL</mo> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which is unramified at all non-zero primes <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq10.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {l}\ne \mathfrak {p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">l</mi> <mo>≠</mo> <mi mathvariant="fraktur">p</mi> </mrow> </math></EquationSource> </InlineEquation> of A, and whose image is a finite index subgroup of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\operatorname {GL}_r(\mathcal {O})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>GL</mo> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, if the degree of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {p}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">p</mi> </math></EquationSource> </InlineEquation> is 1, then <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq13.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> is also unramified at <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1012_Article_IEq14.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>.

Galois representations over function fields that are ramified at one prime

期刊论文
发表日期：

2025年1月28日

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Anwesh Ray

摘要

Let $\mathbb {F}_q$ $F_{q}$ be the finite field with q elements, $F:=\mathbb {F}_q(T)$ $F : = F_{q} (T)$ and $F^{\operatorname {sep}}$ $F^{sep}$ a separable closure of F. Set A to denote the polynomial ring $\mathbb {F}_q[T]$ $F_{q} [T]$ . Let $\mathfrak {p}$ $p$ be a non-zero prime ideal of A, and $\mathcal {O}$ $O$ be the completion of A at $\mathfrak {p}$ $p$ . Given any integer $r\ge 2$ $r \geq 2$ , I construct a Galois representation $\rho :\operatorname {Gal}(F^{\operatorname {sep}}/F)\rightarrow \operatorname {GL}_r(\mathcal {O})$ $ρ : Gal (F^{sep} / F) \to {GL}_{r} (O)$ which is unramified at all non-zero primes $\mathfrak {l}\ne \mathfrak {p}$ $l \neq p$ of A, and whose image is a finite index subgroup of $\operatorname {GL}_r(\mathcal {O})$ ${GL}_{r} (O)$ . Moreover, if the degree of $\mathfrak {p}$ $p$ is 1, then $\rho $ $ρ$ is also unramified at $\infty $ $\infty$ .