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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. 

Abstract Building on work of Boneh, Durfee and Howgrave-Graham, we present a deterministic algorithm that provably finds all integers p such that $$p^r \mathrel {|}N$$ p r | N in time $$O(N^{1/4r+\epsilon })$$ O ( N 1 / 4 r + ϵ ) for any $$\epsilon > 0$$ ϵ > 0 . For example, the algorithm can be used to test squarefreeness of N in time $$O(N^{1/8+\epsilon })$$ O ( N 1 / 8 + ϵ ) ; previously, the best rigorous bound for this problem was $$O(N^{1/6+\epsilon })$$ O ( N 1 / 6 + ϵ ) , achieved via the Pollard–Strassen method.