We present an algorithm that, for every fixed genus g, will enumerate all hyperelliptic curves of genus g over a finite field k of odd characteristic in quasilinear time; that is, the time required for the algorithm is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_594_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$\widetilde{O}(q^{2g-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>O</mi> <mo stretchy="true">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow> <mn>2</mn> <mi>g</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_594_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$q=\# k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mo>#</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>. Such an algorithm already exists in the case <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_594_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$g=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, thanks to work of Mestre and Cardona and Quer, and in the case <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2024_594_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$g=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.

Enumerating hyperelliptic curves over finite fields in quasilinear time

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发表日期：

2025年1月17日

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Everett W. Howe

摘要

We present an algorithm that, for every fixed genus g, will enumerate all hyperelliptic curves of genus g over a finite field k of odd characteristic in quasilinear time; that is, the time required for the algorithm is $\widetilde{O}(q^{2g-1})$ $\tilde{O} (q^{2 g - 1})$ , where $q=\# k$ $q = # k$ . Such an algorithm already exists in the case $g=2$ $g = 2$ , thanks to work of Mestre and Cardona and Quer, and in the case $g=3$ $g = 3$ , thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.