Counting points on superelliptic curves in average polynomial time
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866929722887241728 |
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| author | Sutherland, Andrew V. |
| author_facet | Sutherland, Andrew V. |
| contents | We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\ge 2$ and $f\in \mathbb Z[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X(\mathbb F_p)$ for all $p\le N$ not dividing $m\mathrm{lc}(f)\mathrm{disc}(f)$ in time $O(md^3 N\log^3 N\log\log N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_10189 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Counting points on superelliptic curves in average polynomial time Sutherland, Andrew V. Number Theory Algebraic Geometry 11G40 (Primary), 14G10, 14H25 11Y16 (Secondary) We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\ge 2$ and $f\in \mathbb Z[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X(\mathbb F_p)$ for all $p\le N$ not dividing $m\mathrm{lc}(f)\mathrm{disc}(f)$ in time $O(md^3 N\log^3 N\log\log N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$. |
| title | Counting points on superelliptic curves in average polynomial time |
| topic | Number Theory Algebraic Geometry 11G40 (Primary), 14G10, 14H25 11Y16 (Secondary) |
| url | https://arxiv.org/abs/2004.10189 |