Let K be an imaginary quadratic field, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}_{K,f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be an order in K of conductor <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Let E be an elliptic curve with CM by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {O}_{K,f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, such that E is defined by a model over <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Q}(j_{K,f})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$j_{K,f}=j(E)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>j</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It has been shown by the author and Lozano-Robledo that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\operatorname {Gal}(\mathbb Q(j_{K,f},E[N])/\mathbb Q(j_{K,f}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Gal</mo> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>,</mo> <mi>E</mi> <mrow> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is abelian only for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$N=2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and 4. Let p be a prime and let <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be an integer. In this article, we bound the cardinality of the commutator subgroups of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\operatorname {Gal}(\mathbb Q(E[p^n])/\mathbb Q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Gal</mo> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>p</mi> <mi>n</mi> </msup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and classify the maximal abelian extensions contained in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb Q(E[p^n])/\mathbb Q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>p</mi> <mi>n</mi> </msup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> </mrow> </math></EquationSource> </InlineEquation>.

The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb {Q}$ with complex multiplication

期刊论文
发表日期：

2025年11月20日

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Asimina S. Hamakiotes

摘要

Let K be an imaginary quadratic field, and let $\mathcal {O}_{K,f}$ $O_{K, f}$ be an order in K of conductor $f\ge 1$ $f \geq 1$ . Let E be an elliptic curve with CM by $\mathcal {O}_{K,f}$ $O_{K, f}$ , such that E is defined by a model over $\mathbb {Q}(j_{K,f})$ $Q (j_{K, f})$ , where $j_{K,f}=j(E)$ $j_{K, f} = j (E)$ . It has been shown by the author and Lozano-Robledo that $\operatorname {Gal}(\mathbb Q(j_{K,f},E[N])/\mathbb Q(j_{K,f}))$ $Gal (Q (j_{K, f}, E [N]) / Q (j_{K, f}))$ is abelian only for $N=2,3$ $N = 2, 3$ , and 4. Let p be a prime and let $n\ge 1$ $n \geq 1$ be an integer. In this article, we bound the cardinality of the commutator subgroups of $\operatorname {Gal}(\mathbb Q(E[p^n])/\mathbb Q)$ $Gal (Q (E [p^{n}]) / Q)$ and classify the maximal abelian extensions contained in $\mathbb Q(E[p^n])/\mathbb Q$ $Q (E [p^{n}]) / Q$ .

The maximal abelian extension contained in a division field of an elliptic curve over \(\mathbb {Q}\) with complex multiplication

摘要