We consider the parametric family of elliptic curves over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$E_{m}: y^{2} = x(x - n_{1})(x - n_{2}) + t^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>m</mi> </msub> <mo>:</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and t are particular polynomial expressions in an integral variable m. In this paper, we investigate the torsion group <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$E_{m}(\mathbb {Q})_{\textrm{tors}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>m</mi> </msub> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> <mtext>tors</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation>, a lower bound for the Mordell–Weil rank <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$r({E_{m}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>m</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the 2-Selmer group <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\textrm{Sel}}_{2}(E_{m})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Sel</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>m</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under certain conditions on m. This extends the previous works done in this direction, which are mostly concerned only with the Mordell–Weil ranks of various parametric families of elliptic curves.

On the Mordell–Weil rank and 2-Selmer group of a family of elliptic curves

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发表日期：

2026年1月30日

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Pankaj Patel,
Debopam Chakraborty,
Jaitra Chattopadhyay

摘要

We consider the parametric family of elliptic curves over $\mathbb {Q}$ $Q$ of the form $E_{m}: y^{2} = x(x - n_{1})(x - n_{2}) + t^{2}$ $E_{m} : y^{2} = x (x - n_{1}) (x - n_{2}) + t^{2}$ , where $n_{1}$ $n_{1}$ , $n_{2}$ $n_{2}$ and t are particular polynomial expressions in an integral variable m. In this paper, we investigate the torsion group $E_{m}(\mathbb {Q})_{\textrm{tors}}$ $E_{m} {(Q)}_{tors}$ , a lower bound for the Mordell–Weil rank $r({E_{m}})$ $r (E_{m})$ and the 2-Selmer group ${\textrm{Sel}}_{2}(E_{m})$ ${Sel}_{2} (E_{m})$ under certain conditions on m. This extends the previous works done in this direction, which are mostly concerned only with the Mordell–Weil ranks of various parametric families of elliptic curves.