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LLL reduction and a conjecture of Gunnells
Author(s):
Darrin
Doud;
Russell
Ricks
Journal:
Proc. Amer. Math. Soc.
138
(2010),
409-415.
MSC (2000):
Primary 11H55;
Secondary 11F75
Posted:
September 17, 2009
MathSciNet review:
2557158
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Additional information
Abstract:
Paul Gunnells has developed an algorithm for computing actions of Hecke operators on arithmetic cohomology below the cohomological dimension. One version of his algorithm uses a conjecture concerning LLL-reduced matrices. We prove this conjecture for dimensions 2 through 5 and disprove it for all higher dimensions.
References:
-
- 1.
- Avner Ash, Darrin Doud, and David Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), no. 3, 521-579. MR 1896473 (2003g:11055)
- 2.
- Avner Ash, Paul E. Gunnells, and Mark McConnell, Cohomology of congruence subgroups of
 , J. Number Theory 94 (2002), no. 1, 181-212. MR 1904968 (2003f:11072) - 3.
- -, Cohomology of congruence subgroups of
 . II, J. Number Theory 128 (2008), no. 8, 2263-2274. MR 2394820 - 4.
- Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206 (94i:11105)
- 5.
- Paul E. Gunnells, Computing Hecke eigenvalues below the cohomological dimension, Experiment. Math. 9 (2000), no. 3, 351-367. MR 1795307 (2001k:11092)
- 6.
- A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515-534. MR 682664 (84a:12002)
- 7.
- The PARI-Group, Bordeaux, PARI/GP, Version 2.3.3, 2007, available from http:// pari.math.u-bordeaux.fr/.
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Additional Information:
Darrin
Doud
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
doud@math.byu.edu
Russell
Ricks
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
russellricks@byu.edu
DOI:
10.1090/S0002-9939-09-10131-4
PII:
S 0002-9939(09)10131-4
Keywords:
LLL-reduced lattices
Received by editor(s):
December 31, 2008
Posted:
September 17, 2009
Communicated by:
Ken Ono
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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