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Explicit rational group law on hyperelliptic Jacobians of any genus

Urbanik David

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 1

35--52

It is well-known that abelian varieties are projective, and so there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford’s classic “Lectures on Theta”, where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, we extend this work to give explicit equations for the group law on a dense open set. One can view these equations as generalizations of the usual chord-based group law on elliptic curves.

Jacobians of Hyperelliptic Curves over ℤn and Factorization of n

Dryło Robert, Pomykała Jacek

Fundamenta Informaticae

2019

Vol. 169, nr 4

275--283

E. Bach showed that factorization of an integer n can be reduced in probabilistic polynomial time to the problem of computing exponents of elements in ℤn* (in particular the group order of ℤ*n). It is also known that factorization of square-free integer n can be reduced to the problem of computing the group order of an elliptic curve E/ℤn. In this paper we describe the analogous reduction for computing the orders of Jacobians over ℤn of hyperelliptic curves C over ℤn using the Mumford representation of divisor classes and Cantor’s algorithm for addition. These reductions are based on the group structure of the Jacobian. We also propose other reduction of factorization to the problem of determining the number of points |C(ℤn)|, which makes use of elementary properties of twists of hyperelliptic curves.