2 research outputs found
Efficient Point-Counting Algorithms for Superelliptic Curves
In this paper, we present efficient algorithms for computing the number of
points and the order of the Jacobian group of a superelliptic curve over finite
fields of prime order p. Our method employs the Hasse-Weil bounds in
conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries
we express in terms of multinomial coefficients. We present a fast algorithm
for counting points on specific trinomial superelliptic curves and a slower,
more general method for all superelliptic curves. For the first case, we reduce
the problem of simplifying the entries of the Hasse-Witt matrix modulo p to a
problem of solving quadratic Diophantine equations. For the second case, we
extend Bostan et al.'s method for hyperelliptic curves to general superelliptic
curves. We believe the methods we describe are asymptotically the most
efficient known point-counting algorithms for certain families of trinomial
superelliptic curves
Hasse-Witt and Cartier-Manin matrices: A warning and a request
Let X be a curve in positive characteristic. A Hasse--Witt matrix for X is a
matrix that represents the action of the Frobenius operator on the cohomology
group H^1(X,O_X) with respect to some basis. A Cartier--Manin matrix for X is a
matrix that represents the action of the Cartier operator on the space of
holomorphic differentials of X with respect to some basis. The operators that
these matrices represent are adjoint to one another, so Hasse--Witt matrices
and the Cartier--Manin matrices are related to one another, but there seems to
be a fair amount of confusion in the literature about the exact nature of this
relationship. This confusion arises from differences in terminology, from
differing conventions about whether matrices act on the left or on the right,
and from misunderstandings about the proper formulae for iterating semilinear
operators. Unfortunately, this confusion has led to the publication of
incorrect results. In this paper we present the issues involved as clearly as
we can, and we look through the literature to see where there may be problems.
We encourage future authors to clearly distinguish between Hasse--Witt and
Cartier--Manin matrices, in the hope that further errors can be avoided.Comment: Corrected version; an error in Section 2.5 has been fixe