Counting points on smooth plane quartics

Abstract

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.