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<p>We use machine learning to study the moduli space of genus two curves, specifically focusing on detecting whether a genus two curve has (<i>n</i>,&#xa0;<i>n</i>)-split Jacobian. Based on such techniques, we observe that there are very few rational moduli points with small weighted moduli height and (<i>n</i>,&#xa0;<i>n</i>)-split Jacobian for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1393_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=2, 3, 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. We computational prove that there are only 34 genus two curves (resp. 44 curves) with (2,2)-split Jacobians (resp. (3,3)-split Jacobians) and weighted moduli height <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1393_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\le 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We discuss different machine learning models for such applications and demonstrate the ability to detect splitting with high accuracy using only the Igusa invariants of the curve. This shows that artificial neural networks and machine learning techniques can be highly reliable for arithmetic questions in the moduli space of genus two curves and may have potential applications in isogeny-based cryptography.</p>

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Machine learning for moduli space of genus two curves and an application to isogeny-based cryptography

  • 期刊论文

  • 发表日期：

    2025年2月25日

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  • Elira Shaska,
  • Tanush Shaska

摘要

We use machine learning to study the moduli space of genus two curves, specifically focusing on detecting whether a genus two curve has (n, n)-split Jacobian. Based on such techniques, we observe that there are very few rational moduli points with small weighted moduli height and (n, n)-split Jacobian for \(n=2, 3, 5\) n = 2 , 3 , 5 . We computational prove that there are only 34 genus two curves (resp. 44 curves) with (2,2)-split Jacobians (resp. (3,3)-split Jacobians) and weighted moduli height \(\le 3\) ≤ 3 . We discuss different machine learning models for such applications and demonstrate the ability to detect splitting with high accuracy using only the Igusa invariants of the curve. This shows that artificial neural networks and machine learning techniques can be highly reliable for arithmetic questions in the moduli space of genus two curves and may have potential applications in isogeny-based cryptography.