Algebraic solution of the Jacobi inverse problem and explicit addition laws
Yaacov Kopeliovich
TL;DR
The paper addresses the problem of producing explicit, coordinate-free addition laws on the Jacobians of algebraic curves by solving the algebraic Jacobi inverse problem for non-special divisors. It develops a Vandermonde-type determinant construction at a Weierstrass point to recover complementary divisors, enabling explicit addition on Sym^g(X) and extending to multiplication by n and division polynomials. These algebraic methods yield a practical framework for divisor arithmetic that applies to general (n,s) curves and clarifies the role of Mumford divisors in this context. The work has potential cryptographic and computational applications, including explicit isogenies and coordinate-free formulations, with the (3,4) example illustrating the approach and suggesting avenues for a fully abstract theory.
Abstract
We formulate a solution to the Algebraic version of the Inverse Jacobi problem. Using this solution we produce explicit addition laws on any algebraic curve generalizing the law suggested by Leykin [2] in the case of (n, s) curves. This gives a positive answer to a question asked by T. Shaska whether addition laws appearing in [2] can be produced in a coordinate free manner.