Counting points on smooth plane quartics
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40993_2022_Article_397.pdf
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Author(s)
Costa, Edgar
Harvey, David
Sutherland, Andrew V.
Date Issued
November 21, 2022
Publisher
Springer International Publishing
Citation
Research in Number Theory. 2022 Nov 21;9(1):1
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s40993-022-00397-8
