Let K be an imaginary quadratic field. For an order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_801_Article_IEq1.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> in K and a positive integer N, let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_801_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{\mathcal {O},\,N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mi mathvariant="script">O</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be the ray class field of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_801_Article_IEq1.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> modulo <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_801_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$N\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mi mathvariant="script">O</mi> </mrow> </math></EquationSource> </InlineEquation>. We deal with various subjects related to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_801_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{\mathcal {O},\,N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mi mathvariant="script">O</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, mainly about Galois representations attached to elliptic curves with complex multiplication, form class groups and L-functions for orders.

Arithmetic properties of orders in imaginary quadratic fields

期刊论文
发表日期：

2025年6月30日

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Ho Yun Jung,
Ja Kyung Koo,
Dong Hwa Shin,
Dong Sung Yoon

摘要

Let K be an imaginary quadratic field. For an order $\mathcal {O}$ $O$ in K and a positive integer N, let $K_{\mathcal {O},\,N}$ $K_{O, N}$ be the ray class field of $\mathcal {O}$ $O$ modulo $N\mathcal {O}$ $N O$ . We deal with various subjects related to $K_{\mathcal {O},\,N}$ $K_{O, N}$ , mainly about Galois representations attached to elliptic curves with complex multiplication, form class groups and L-functions for orders.