Computing weight modular forms of level

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 | References | Similar Articles | Additional Information

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, 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**

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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:
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