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Abstract 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 $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ 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 $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ 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 $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest.
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Reduction of dynatomic curves
In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families $${\mathcal{F}}$$ of polynomial maps, such as the family $$f_{c}(x)=x^{m}+c$$ , where $$m\geq 2$$ . We do this by making use of the dynatomic modular curves $$Y_{1}(n)$$ (respectively $$Y_{0}(n)$$ ) which parametrize maps $$f$$ in $${\mathcal{F}}$$ together with a point (respectively orbit) of period $$n$$ for $$f$$ . The key point in our strategy is to study the set of primes $$p$$ for which the reduction of $$Y_{1}(n)$$ modulo $$p$$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $$n$$ , a discriminant $$D_{n}$$ whose list of prime factors contains all the primes of bad reduction for $$Y_{1}(n)$$ . In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $$p$$ dividing $$D_{n}$$ : one guarantees that $$p$$ is in fact a prime of bad reduction for $$Y_{1}(n)$$ , yet this same criterion implies that $$Y_{0}(n)$$ is geometrically irreducible. The other guarantees that the reduction of $$Y_{1}(n)$$ modulo $$p$$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $$Y_{1}(n)$$ for several primes dividing $$D_{n}$$ when $n=7,8,11$ , and $$f_{c}(x)=x^{2}+c$$ . The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.
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- Award ID(s):
- 1901819
- PAR ID:
- 10274388
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 39
- Issue:
- 10
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 2717 to 2768
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2017.140
