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Enumerating hyperelliptic curves over finite fields in quasilinear time

Everett W. Howe
Feb 2024
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摘要原文
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 $\widetilde{O}(q^{2g-1})$, where $q=\#k$. Such an algorithm already exists in the case $g=2$, thanks to the work of Mestre and Cardona and Quer on reconstructing curves from their Igusa invariants. Experimentally, it appears that our new algorithm is much faster in practice than one based on their work.
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