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

DOI:
https://doi.org/10.1090/S0025-5718-04-01709-0

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]**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]**PARI-GP`http://www.parigp-home.de/`.**[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*, Compositio Mathematica, Vol.**40**, Fasc. 2, 1980, p. 177-241. MR**81k:10040****[Pi2]**A. Pizer,*An Algorithm for Computing Modular Forms on*, Journal of Algebra**64**, 1980, 340-390. MR**83g:10020****[PRV]**A. Pacetti and F. Rodriguez-Villegas,`www.ma.utexas.edu/users/villegas/cnt/cnt.html`.**[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

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