We establish the existence of hyperelliptic curves of genus <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> defined over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> whose Jacobians possess rational torsion points of order N where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N=4g^2+2g-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> <msup> <mi>g</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>g</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$4\,g^2+ 2\,g -4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mspace width="0.166667em" /> <msup> <mi>g</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mspace width="0.166667em" /> <mi>g</mi> <mo>-</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$N = 2\,g^{2} + 7\,g + 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mspace width="0.166667em" /> <msup> <mi>g</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>7</mn> <mspace width="0.166667em" /> <mi>g</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we introduce a 1-parameter family of polynomials <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f_{t}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of degree <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2g+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>g</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. For all but finitely many rational values of t, if the discriminant of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f_{t}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is nonzero, then the hyperelliptic curve defined by <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$y^{2} = f_{t}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <msub> <mi>f</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has a rational point of order N on its Jacobian.

Quadratic torsion orders on Jacobian varieties

期刊论文
发表日期：

2026年7月20日

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Mohammad Sadek,
Hamide Suluyer

摘要

We establish the existence of hyperelliptic curves of genus $g\ge 2$ $g \geq 2$ defined over $\mathbb {Q}$ $Q$ whose Jacobians possess rational torsion points of order N where $N=4g^2+2g-2$ $N = 4 g^{2} + 2 g - 2$ or $4\,g^2+ 2\,g -4$ $4 g^{2} + 2 g - 4$ . For $N = 2\,g^{2} + 7\,g + 1$ $N = 2 g^{2} + 7 g + 1$ , we introduce a 1-parameter family of polynomials $f_{t}(x)$ $f_{t} (x)$ of degree $2g+1$ $2 g + 1$ . For all but finitely many rational values of t, if the discriminant of $f_{t}(x)$ $f_{t} (x)$ is nonzero, then the hyperelliptic curve defined by $y^{2} = f_{t}(x)$ $y^{2} = f_{t} (x)$ has a rational point of order N on its Jacobian.