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This chapter contains descriptions of functions designed to perform
calculations with hyperelliptic curves and their Jacobians.
A hyperelliptic curve, which is taken to include the genus one case,
is given by a nonsingular generalized Weierstrass equation
y2 + h(x) y = f(x),
where h(x) and f(x) are polynomials over a field K. The curve
is viewed as embedded in a weighted projective space, with weights
1, g + 1, and 1, in which the points at infinity are nonsingular.
Functionality for hyperelliptic curves includes optimized algorithms
for working on genus two curves over Q, including heights on the
Jacobian, and a datatype for the Kummer surface of the Jacobian.
For Jacobians of curves over finite fields, there exist specialized
algorithms for computing the group structure of the set of rational
points.
The category of hyperelliptic curves is CrvHyp
and points on curves are of type PtHyp.
Jacobians of hyperelliptic curves
are of type JacHyp and
points JacHypPt. Similarly, the
Kummer surface of a genus two curve is of type
SrfKum
with points of type SrfKumPt.
The initial development of machinery for hyperelliptic curves was
undertaken by Michael Stoll, supported by members of the Magma group.
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