Counting points on smooth plane quartics
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909581900251136 |
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| author | Costa, Edgar Harvey, David Sutherland, Andrew V. |
| author_facet | Costa, Edgar Harvey, David Sutherland, Andrew V. |
| contents | We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a prime of good reduction. We consider two approaches for computing $\#X(\mathbb F_p)$, one which runs in $O(p\log p\log\log p)$ time using $O(\log p)$ space and one which runs in $O(p^{1/2}\log^2\!p)$ time using $O(p^{1/2}\log p)$ space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $X/\mathbb Q$ that compute $\#X(\mathbb F_p)$ for good primes $p\le N$ in $O(N\log^3\! N)$ time using $O(N)$ space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $\mathbb P^1$, which in combination with previous results addresses all curves of genus $g\le 3$. Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_09890 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Counting points on smooth plane quartics Costa, Edgar Harvey, David Sutherland, Andrew V. Number Theory 11G40 (Primary), 14G10, 14H25 11Y16 (Secondary) We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a prime of good reduction. We consider two approaches for computing $\#X(\mathbb F_p)$, one which runs in $O(p\log p\log\log p)$ time using $O(\log p)$ space and one which runs in $O(p^{1/2}\log^2\!p)$ time using $O(p^{1/2}\log p)$ space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $X/\mathbb Q$ that compute $\#X(\mathbb F_p)$ for good primes $p\le N$ in $O(N\log^3\! N)$ time using $O(N)$ space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $\mathbb P^1$, which in combination with previous results addresses all curves of genus $g\le 3$. Our algorithms also compute Cartier-Manin/Hasse-Witt matrices that may be of independent interest. |
| title | Counting points on smooth plane quartics |
| topic | Number Theory 11G40 (Primary), 14G10, 14H25 11Y16 (Secondary) |
| url | https://arxiv.org/abs/2208.09890 |