The Hilbert class field of the quaternion algebra B is an algebra <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_595_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr {H}}(B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that every two-sided ideal of B is principal in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_595_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr {H}}(B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We study the avatars of B and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_595_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr {H}}(B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, i.e. algebraic surfaces attached to the quaternion algebras. It is proved that the avatar of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_595_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathscr {H}}(B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is obtained from the avatar of B by a birational map. We apply this result to the analogy between number fields and function fields.

Birational geometry of quaternions

期刊论文
发表日期：

2025年5月25日

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Igor V. Nikolaev

摘要

The Hilbert class field of the quaternion algebra B is an algebra ${\mathscr {H}}(B)$ $H (B)$ such that every two-sided ideal of B is principal in ${\mathscr {H}}(B)$ $H (B)$ . We study the avatars of B and ${\mathscr {H}}(B)$ $H (B)$ , i.e. algebraic surfaces attached to the quaternion algebras. It is proved that the avatar of ${\mathscr {H}}(B)$ $H (B)$ is obtained from the avatar of B by a birational map. We apply this result to the analogy between number fields and function fields.