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Cuspidal Modular Symbols are Transportable

Published online by Cambridge University Press:  01 February 2010

William A. Stein  and
Helena A. Verrill
William A. Stein
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA, [email protected], http://modular.fas.harvard.edu/
Helena A. Verrill
Affiliation:
Institute for Mathematics, University of Hannover, Welfengarten 1, 30167 Hannover, Germany, [email protected], http://hverrill.net

Abstract

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Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} for all α,β in the extended upper half plane and γ ∊ Γ. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 4 , 2001 , pp. 170 - 181
Copyright
Copyright © London Mathematical Society 2001

References

1.Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
2.Cremona, J. E., Algorithms for modular elliptic curves, 2nd edn (Cambridge University Press, Cambridge, 1997).Google Scholar
3.Cremona, J. E., ‘Computing periods of cusp forms and modular elliptic curves’, Experiment. Math. 6 (1997) 97–107.CrossRefGoogle Scholar
4.Manin, J. I., Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 19–66.Google Scholar
5.Merel, L., Universal Fourier expansions of modular forms. On Artin's conjecture for odd 2-dimensional representations (Springer, 1994) 59–94.Google Scholar
6.Stein, W. A., ‘Explicit approaches to modular abelian varieties’, Ph.D. thesis, University of California, Berkeley (2000).Google Scholar