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<p>We consider elliptic curves defined by an equation of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(y^2=x^3+f(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f\in k[t]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> has coefficients in a perfect field <i>k</i> of characteristic not 2 or 3. By performing 2 and 3-descent, we obtain, under suitable assumptions on the factorization of <i>f</i>, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman.</p>

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Integral points on elliptic curves with j-invariant 0 over k(t)

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Jean Gillibert,
  • Emmanuel Hallouin,
  • Aaron Levin

摘要

We consider elliptic curves defined by an equation of the form \(y^2=x^3+f(t)\) y 2 = x 3 + f ( t ) , where \(f\in k[t]\) f ∈ k [ t ] has coefficients in a perfect field k of characteristic not 2 or 3. By performing 2 and 3-descent, we obtain, under suitable assumptions on the factorization of f, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman.