Rigid cohomology and p -adic point counting

Alan G.B. Lauder[1]

  • [1] Mathematical Institute Oxford University 24-29 St Giles Oxford OX1 3LB

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 169-180
  • ISSN: 1246-7405

Abstract

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I discuss some algorithms for computing the zeta function of an algebraic variety over a finite field which are based upon rigid cohomology. Two distinct approaches are illustrated with a worked example.

How to cite

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Lauder, Alan G.B.. "Rigid cohomology and $p$-adic point counting." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 169-180. <http://eudml.org/doc/249426>.

@article{Lauder2005,
abstract = {I discuss some algorithms for computing the zeta function of an algebraic variety over a finite field which are based upon rigid cohomology. Two distinct approaches are illustrated with a worked example.},
affiliation = {Mathematical Institute Oxford University 24-29 St Giles Oxford OX1 3LB},
author = {Lauder, Alan G.B.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {varieties over finite fields; zeta function; point counting; -adic cohomology},
language = {eng},
number = {1},
pages = {169-180},
publisher = {Université Bordeaux 1},
title = {Rigid cohomology and $p$-adic point counting},
url = {http://eudml.org/doc/249426},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Lauder, Alan G.B.
TI - Rigid cohomology and $p$-adic point counting
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 169
EP - 180
AB - I discuss some algorithms for computing the zeta function of an algebraic variety over a finite field which are based upon rigid cohomology. Two distinct approaches are illustrated with a worked example.
LA - eng
KW - varieties over finite fields; zeta function; point counting; -adic cohomology
UR - http://eudml.org/doc/249426
ER -

References

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  3. J-P. Dedieu, Newton’s method and some complexity aspects of the zero-finding problem. In “Foundations of Computational Mathematics”, (R.A. DeVore, A. Iserles, E. Suli), LMS Lecture Note Series 284, Cambridge University Press, 2001, 45–67. Zbl0978.65048MR1836614
  4. B. Dwork, On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648. Zbl0173.48501MR140494
  5. B. Dwork, On the zeta function of a hypersurface II. Ann. Math. (2) 80 (1964), 227–299. Zbl0173.48601MR188215
  6. N. Elkies, Elliptic and modular curves over finite fields and related computational issues. In “Computational perspectives in number theory: Proceedings of a conference in honour of A.O.L. Atkin” , (D.A. Buell and J.T. Teitelbaum), American Mathematical Society International Press 7 (1998), 21–76. Zbl0915.11036MR1486831
  7. M.D. Huang, D. Ierardi, Counting points on curves over finite fields. J. Symbolic Comput. 25 (1998), 1–21. Zbl0919.11046MR1600606
  8. K. Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. Journal of the Ramanujan Mathematical Society 16 (2001), 323–338. Zbl1066.14024MR1877805
  9. K. Kedlaya, Finiteness of rigid cohomology with coefficients, preprint 2002. Zbl1133.14019MR2239343
  10. A.G.B Lauder, Deformation theory and the computation of zeta functions, Proceedings of the London Mathematical Society 88 (3) (2004), 565-602. Zbl1119.11053MR2044050
  11. A.G.B. Lauder, Counting solutions to equations in many variables over finite fields, Foundations of Computational Mathematics 4 (3) (2004), 221-267. Zbl1076.11040MR2078663
  12. A.G.B. Lauder, D. Wan, Counting points on varieties over finite fields of small characteristic. To appear in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography (Mathematical Sciences Research Institute Publications), J.P. Buhler and P. Stevenhagen (eds), Cambridge University Press. Available at: http://www.maths.ox.ac.uk/lauder/ Zbl1188.11069
  13. J. Pila, Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp. 55 No. 192 (1990), 745–763. Zbl0724.11070MR1035941
  14. R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p . Math. Comp. 44 no. 170 (1985), 483–494. Zbl0579.14025MR777280
  15. N. Tsuzuki, Bessel F-isocrystals and an algorithm for computing Kloosterman sums, preprint 2003. 
  16. D. Wan, Algorithmic theory of zeta functions. To appear in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography (Mathematical Sciences Research Institute Publications), J.P. Buhler and P. Stevenhagen (eds), Cambridge University Press. Available at: http://www.math.uci.edu/~dwan/preprint.html 

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