Computing weight modular forms of level
Authors:
Ariel Pacetti and Fernando Rodriguez Villegas; with an appendix by B. Gross
Journal:
Math. Comp. 74 (2005), 15451557
MSC (2000):
Primary 11F11; Secondary 11E20, 11Y99
Published electronically:
September 10, 2004
MathSciNet review:
2137017
Fulltext PDF Free Access
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Abstract: For a prime we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space of modular forms of weight and level . For we define a special Hecke stable subspace of which contains the space of modular forms with CM by the ring of integers of and we describe the calculation of the corresponding Brandt matrices.
 [Ei]
M. Eichler, Lectures on modular correspondences, Bombay, Tata Institute of Fundamental Research, 195556.
 [Gr]
Benedict
H. Gross, Arithmetic on elliptic curves with complex
multiplication, Lecture Notes in Mathematics, vol. 776, Springer,
Berlin, 1980. With an appendix by B. Mazur. MR 563921
(81f:10041)
 [Ma]
Magma computational algebra system http://magma.maths.usyd.edu.au/magma/.
 [GP]
PARIGP http://www.parigphome.de/.
 [Ko]
David
R. Kohel, Hecke module structure of quaternions, Class field
theory—its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure
Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 177–195.
MR
1846458 (2002i:11059)
 [Pi]
Arnold
Pizer, Theta series and modular forms of level
𝑝²𝑀, Compositio Math. 40 (1980),
no. 2, 177–241. MR 563541
(81k:10040)
 [Pi2]
Arnold
Pizer, An algorithm for computing modular forms on
Γ₀(𝑁), J. Algebra 64 (1980),
no. 2, 340–390. MR 579066
(83g:10020), http://dx.doi.org/10.1016/00218693(80)901519
 [PRV]
A. Pacetti and F. RodriguezVillegas, www.ma.utexas.edu/users/villegas/cnt/cnt.html.
 [Se]
JeanPierre
Serre, Quelques applications du théorème de
densité de Chebotarev, Inst. Hautes Études Sci. Publ.
Math. 54 (1981), 323–401 (French). MR 644559
(83k:12011)
 [Vi]
MarieFrance
Vignéras, Arithmétique des algèbres de
quaternions, Lecture Notes in Mathematics, vol. 800, Springer,
Berlin, 1980 (French). MR 580949
(82i:12016)
 [Ei]
 M. Eichler, Lectures on modular correspondences, Bombay, Tata Institute of Fundamental Research, 195556.
 [Gr]
 B. Gross, Arithmetic on elliptic curves with complex multiplication, with an appendix by B. Mazur, Lecture Notes in Mathematics, 776, Springer, Berlin, 1980. MR 81f:10041
 [Ma]
 Magma computational algebra system http://magma.maths.usyd.edu.au/magma/.
 [GP]
 PARIGP http://www.parigphome.de/.
 [Ko]
 D. Kohel, Hecke module structure of quaternions, Class field theoryits centenary and prospect (Tokyo, 1998), 177195, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001.MR 2002i:11059
 [Pi]
 A. Pizer, Theta Series and Modular Forms of Level , Compositio Mathematica, Vol. 40, Fasc. 2, 1980, p. 177241. MR 81k:10040
 [Pi2]
 A. Pizer, An Algorithm for Computing Modular Forms on , Journal of Algebra 64, 1980, 340390. MR 83g:10020
 [PRV]
 A. Pacetti and F. RodriguezVillegas, www.ma.utexas.edu/users/villegas/cnt/cnt.html.
 [Se]
 J.P., Serre, Quelques applications du théoreème de Chebotarev, Publ. Math. IHES, 54 (1981), 123201. MR 83k:12011
 [Vi]
 M. F. Vigneras, Arithmetique des algebres de quaternions, Lecture Notes in Mathematics, 800.MR 82i:12016
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Additional Information
Ariel Pacetti
Affiliation:
Department of Mathematics, University of Texas at Austin, Texas 78712
Email:
apacetti@math.utexas.edu
Fernando Rodriguez Villegas
Affiliation:
Department of Mathematics, University of Texas at Austin, Texas 78712
Email:
villegas@math.utexas.edu
B. Gross
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massacusetts 02138
Email:
gross@math.harvard.edu
DOI:
http://dx.doi.org/10.1090/S0025571804017090
PII:
S 00255718(04)017090
Received by editor(s):
February 18, 2003
Received by editor(s) in revised form:
December 16, 2003
Published electronically:
September 10, 2004
Additional Notes:
The first and second authors were supported in part by grants from TARP and NSF (DMS9970109); they would like to thank the Department of Mathematics at Harvard University, where part of this work was done, for its hospitality
Article copyright:
© Copyright 2004
American Mathematical Society
