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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
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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] 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
  • [GP] PARI-GP
  • [Ko] D. Kohel, Hecke module structure of quaternions, Class field theory--its centenary and prospect (Tokyo, 1998), 177-195, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001.MR 2002i:11059
  • [Pi] A. Pizer, Theta Series and Modular Forms of Level $p^2M$, Compositio Mathematica, Vol. 40, Fasc. 2, 1980, p. 177-241. MR 81k:10040
  • [Pi2] A. Pizer, An Algorithm for Computing Modular Forms on $\Gamma_0(N)^{*}$, Journal of Algebra 64, 1980, 340-390. MR 83g:10020
  • [PRV] A. Pacetti and F. Rodriguez-Villegas,
  • [Se] J.-P., Serre, Quelques applications du théoreème de Chebotarev, Publ. Math. IHES, 54 (1981), 123-201. 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

Fernando Rodriguez Villegas
Affiliation: Department of Mathematics, University of Texas at Austin, Texas 78712

B. Gross
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massacusetts 02138

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

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