Abstract
We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus g defined over \({\mathbb {F}}_q\). It is based on the approaches by Schoof and Pila combined with a modelling of the \(\ell \)-torsion by structured polynomial systems. Our main result improves on previously known complexity bounds by showing that there exists a constant \(c>0\) such that, for any fixed g, this algorithm has expected time and space complexity \(O((\log q)^{c g})\) as q grows and the characteristic is large enough.
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Acknowledgements
We are grateful to Éric Schost and Guillermo Matera for fruitful discussions and for pointing out important references. We also wish to thank anonymous referees for their comments which helped improve the paper.
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Communicated by John Cremona.
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Abelard, S., Gaudry, P. & Spaenlehauer, PJ. Improved Complexity Bounds for Counting Points on Hyperelliptic Curves. Found Comput Math 19, 591–621 (2019). https://doi.org/10.1007/s10208-018-9392-1
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DOI: https://doi.org/10.1007/s10208-018-9392-1
Keywords
- Hyperelliptic curves
- Local zeta function
- Schoof–Pila’s algorithm
- Multi-homogeneous polynomial systems
- Geometric resolution