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<p>In his earlier work, the second author introduced a method of the Shimura lift that can lift a cusp form of weight 3/2 to a cusp form of weight 2. This method has been successfully applied to the congruent number elliptic curve, leading to new criteria for congruent numbers and new results in the congruent number problem. The purpose of this paper is to apply the same method to a new family of elliptic curves <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1626_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">\(E_n:y^2=x^3+2n^3. \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo>:</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>n</mi> <mn>3</mn> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Specifically, we establish formulas for the values of the <i>L</i>-series at <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1626_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(s=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1626_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(E_n \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and provide the criteria for the non-vanishing of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1626_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(L(E_n,1).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>n</mi> </msub> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Our proof relies on the main equality <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1626_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="750" /> </InlineMediaObject> <EquationSource Format="TEX">\(\#\{(x,y,z)\in \mathbb {Z}^3 \mid p^2=3x^2+4y^2+7z^2+2yz\}=\#\{(x,y,z,w)\in \mathbb {Z}^4 \mid p=3x^2+9y^2+4z^2+7w^2+2zw\} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> <mo>∣</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>3</mn> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mi>z</mi> <mo stretchy="false">}</mo> </mrow> <mo>=</mo> <mo>#</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>4</mn> </msup> <mo>∣</mo> <mi>p</mi> <mo>=</mo> <mn>3</mn> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>w</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mi>w</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any prime <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1626_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\equiv 1\pmod 6.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>6</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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L-values of elliptic curves \(E_n:y^2=x^3+2n^3\)

  • 期刊论文

  • 发表日期：

    2025年6月19日

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  • Jun Lu,
  • Hourong Qin

摘要

In his earlier work, the second author introduced a method of the Shimura lift that can lift a cusp form of weight 3/2 to a cusp form of weight 2. This method has been successfully applied to the congruent number elliptic curve, leading to new criteria for congruent numbers and new results in the congruent number problem. The purpose of this paper is to apply the same method to a new family of elliptic curves \(E_n:y^2=x^3+2n^3. \) E n : y 2 = x 3 + 2 n 3 . Specifically, we establish formulas for the values of the L-series at \(s=1\) s = 1 of \(E_n \) E n and provide the criteria for the non-vanishing of \(L(E_n,1).\) L ( E n , 1 ) . Our proof relies on the main equality \(\#\{(x,y,z)\in \mathbb {Z}^3 \mid p^2=3x^2+4y^2+7z^2+2yz\}=\#\{(x,y,z,w)\in \mathbb {Z}^4 \mid p=3x^2+9y^2+4z^2+7w^2+2zw\} \) # { ( x , y , z ) ∈ Z 3 ∣ p 2 = 3 x 2 + 4 y 2 + 7 z 2 + 2 y z } = # { ( x , y , z , w ) ∈ Z 4 ∣ p = 3 x 2 + 9 y 2 + 4 z 2 + 7 w 2 + 2 z w } for any prime \(p\equiv 1\pmod 6.\) p ≡ 1 ( mod 6 ) .