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

DOI:
https://doi.org/10.1090/S0025-5718-03-01567-9

Published electronically:
July 7, 2003

MathSciNet review:
2034126

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use an embedding of the symmetric th power of any algebraic curve of genus into a Grassmannian space to give algorithms for working with divisors on , using only linear algebra in vector spaces of dimension , and matrices of size . When the base field is finite, or if has a rational point over , these give algorithms for working on the Jacobian of that require 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 . 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 curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of .

**[ACGH85]**E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,*Geometry of algebraic curves. Vol. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**86h:14019****[ADH94]**Leonard M. Adleman, Jonathan DeMarrais, and Ming-Deh Huang,*A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields*, Algorithmic number theory ``ANTS-I'' (Ithaca, NY, 1994) (Leonard M. Adleman and Ming-Deh Huang, eds.), Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 28-40. MR**96b:11078****[BLR90]**Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud,*Néron models*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR**91i:14034****[Can87]**David G. Cantor,*Computing in the Jacobian of a hyperelliptic curve*, Math. Comp.**48**(1987), no. 177, 95-101. MR**88f:11118****[GH94]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley Classics Library (reprint of 1978 edition), John Wiley & Sons Inc., New York, 1994. MR**95d:14001****[GPS02]**S. D. Galbraith, S. M. Paulus, and N. P. Smart,*Arithmetic on superelliptic curves*, Math. Comp.**71**(2002), no. 237, 393-405 (electronic). MR**2002h:14102****[Har77]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. MR**57:3116****[Hes99]**Florian Hess,*Zur Divisorenklassengruppenberechnung in globalen Funktionenkörpern*, Ph.D. thesis, Technische Universität Berlin, 1999, may be downloaded from the web at`http://www.math.tu-berlin.de/~kant/publications/diss/diss_FH.ps.gz`

.**[Hes02]**Florian Hess,*Computing Riemann-Roch spaces in algebraic function fields and related topics*, J. Symbolic Comput.**33**(2002), no. 4, 425-445.**[HI94]**Ming-Deh Huang and Doug Ierardi,*Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve*, J. Symbolic Comput.**18**(1994), no. 6, 519-539. MR**96h:14077****[HS00]**Ryuichi Harasawa and Joe Suzuki,*Fast Jacobian group arithmetic on curves*, Algorithmic number theory ``ANTS-IV'' (Leiden, 2000) (Wieb Bosma, ed.), Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 359-376. MR**2002f:11073****[Laz89]**Robert Lazarsfeld,*A sampling of vector bundle techniques in the study of linear series*, Lectures on Riemann surfaces (Trieste, 1987) (M. Cornalba, X. Gomez-Mont, and A. Verjovsky, eds.), World Sci. Publishing, Teaneck, NJ, 1989, pp. 500-559. MR**92f:14006****[Mil86]**J. S. Milne,*Jacobian varieties*, Arithmetic geometry (Storrs, Conn., 1984) (Gary Cornell and Joseph H. Silverman, eds.), Springer, New York, 1986, pp. 167-212. MR**89b:14029****[Vol94]**Emil J. Volcheck,*Computing in the Jacobian of a plane algebraic curve*, Algorithmic number theory ``ANTS-I'' (Ithaca, NY, 1994) (Leonard M. Adleman and Ming-Deh Huang, eds.), Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 221-233. MR**96a:14033**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11Y16,
14Q05,
14H40,
11G20

Retrieve articles in all journals with MSC (2000): 11Y16, 14Q05, 14H40, 11G20

Additional Information

**Kamal Khuri-Makdisi**

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

Email:
kmakdisi@aub.edu.lb

DOI:
https://doi.org/10.1090/S0025-5718-03-01567-9

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