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Linear algebra algorithms for divisors on an algebraic curve

Author: Kamal Khuri-Makdisi
Journal: Math. Comp. 73 (2004), 333-357
MSC (2000): Primary 11Y16, 14Q05, 14H40, 11G20
Published electronically: July 7, 2003
MathSciNet review: 2034126
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Abstract: We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and $C_{ab}$curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of $O(g^2)$.

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Additional Information

Kamal Khuri-Makdisi
Affiliation: Mathematics Department and Center for Advanced Mathematical Sciences, American University of Beirut, Bliss Street, Beirut, Lebanon

Received by editor(s): November 7, 2001
Received by editor(s) in revised form: March 29, 2002
Published electronically: July 7, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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