In this article, we study the 2-Selmer rank of a family of elliptic curves of the Legendre type, given by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ y^2 = x(x + p)(x - q) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where p and q are primes. For the special case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ p \equiv 7 \pmod {8} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>7</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ p = q $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>, we rigorously prove that the 2-Selmer rank of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ y^2 = x(x + p)(x - p) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is exactly 1. Consequently, under the 2-Selmer rank one conjecture or the finiteness of the Shafarevich–Tate group, we deduce that such primes <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p \equiv 7 \pmod {8}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>7</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are congruent numbers. Furthermore, we extend our analysis to the more general scenario involving distinct primes p and q, and show that the 2-Selmer rank of these curves lies between 0 and 2.

On 2-Selmer Rank of the Legendre Curves and Congruent Numbers

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发表日期：

2025年11月11日

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Priyanka Gairola,
Abhishek Juyal

摘要

In this article, we study the 2-Selmer rank of a family of elliptic curves of the Legendre type, given by $ y^2 = x(x + p)(x - q) $ $y^{2} = x (x + p) (x - q)$ , where p and q are primes. For the special case $ p \equiv 7 \pmod {8} $ $p \equiv 7 (mod 8)$ with $ p = q $ $p = q$ , we rigorously prove that the 2-Selmer rank of $ y^2 = x(x + p)(x - p) $ $y^{2} = x (x + p) (x - p)$ is exactly 1. Consequently, under the 2-Selmer rank one conjecture or the finiteness of the Shafarevich–Tate group, we deduce that such primes $p \equiv 7 \pmod {8}$ $p \equiv 7 (mod 8)$ are congruent numbers. Furthermore, we extend our analysis to the more general scenario involving distinct primes p and q, and show that the 2-Selmer rank of these curves lies between 0 and 2.