Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Computing weight $2$ modular forms of level $p^2$


Authors: Ariel Pacetti and Fernando Rodriguez Villegas; with an appendix by B. Gross
Journal: Math. Comp. 74 (2005), 1545-1557
MSC (2000): Primary 11F11; Secondary 11E20, 11Y99
Published electronically: September 10, 2004
MathSciNet review: 2137017
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a prime $p$ we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space $V$ of modular forms of weight $2$ and level $p^2$. For $p \equiv 3 \bmod 4$ we define a special Hecke stable subspace $V_0$ of $V$ which contains the space of modular forms with CM by the ring of integers of $\mathbb Q(\sqrt{-p})$ and we describe the calculation of the corresponding Brandt matrices.


References [Enhancements On Off] (What's this?)

  • [Ei] M. Eichler, Lectures on modular correspondences, Bombay, Tata Institute of Fundamental Research, 1955-56.
  • [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
  • [Ma] Magma computational algebra system http://magma.maths.usyd.edu.au/magma/.
  • [GP] PARI-GP http://www.parigp-home.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
  • [Pi] Arnold Pizer, Theta series and modular forms of level 𝑝²𝑀, Compositio Math. 40 (1980), no. 2, 177–241. MR 563541
  • [Pi2] Arnold Pizer, An algorithm for computing modular forms on Γ₀(𝑁), J. Algebra 64 (1980), no. 2, 340–390. MR 579066, 10.1016/0021-8693(80)90151-9
  • [PRV] A. Pacetti and F. Rodriguez-Villegas, www.ma.utexas.edu/users/villegas/cnt/cnt.html.
  • [Se] Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
  • [Vi] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11F11, 11E20, 11Y99

Retrieve articles in all journals with MSC (2000): 11F11, 11E20, 11Y99


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: https://doi.org/10.1090/S0025-5718-04-01709-0
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 (DMS-99-70109); 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