On the computational complexity of modular symbols
Author:
Dorian Goldfeld
Journal:
Math. Comp. 58 (1992), 807814
MSC:
Primary 11F67; Secondary 11Y35
MathSciNet review:
1122069
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Efficient algorithms are obtained for integrating holomorphic differential oneforms along simple geodesic lines on those compact Riemann surfaces which are given as quotients of the upper halfplane by a congruence subgroup of . We may assume that every geodesic line passes through a cusp which is unique up to equivalence. The algorithms we construct run in polynomial time in the height of this cusp.
 [1]
A.
O. L. Atkin and J.
Lehner, Hecke operators on Γ₀(𝑚), Math.
Ann. 185 (1970), 134–160. MR 0268123
(42 #3022)
 [2]
J. E. Cremona, Computation of modular elliptic curves and the BirchSwinnerton Dyer conjecture, preprint.
 [3]
Dorian
Goldfeld, Modular elliptic curves and Diophantine problems,
Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990,
pp. 157–175. MR 1106659
(92e:11061)
 [4]
P. T. Lockhart, Diophantine equations and the arithmetic of hyperelliptic curves, Ph.D. Thesis, Columbia University, 1990.
 [5]
Ju.
I. Manin, Parabolic points and zeta functions of modular
curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972),
19–66 (Russian). MR 0314846
(47 #3396)
 [6]
Goro
Shimura, Introduction to the arithmetic theory of automorphic
functions, Publications of the Mathematical Society of Japan, No. 11.
Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton,
N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766
(47 #3318)
 [7]
Goro
Shimura, On the factors of the jacobian variety of a modular
function field, J. Math. Soc. Japan 25 (1973),
523–544. MR 0318162
(47 #6709)
 [1]
 A. O. L. Atkin and J. Lehner, Hecke operators on , Math. Ann. 185 (1970), 134160. MR 0268123 (42:3022)
 [2]
 J. E. Cremona, Computation of modular elliptic curves and the BirchSwinnerton Dyer conjecture, preprint.
 [3]
 D. Goldfeld, Modular elliptic curves and diophantine problems, Proc. First Canadian Number Theory Assoc. (Banff, Canada 1988), de Gruyter, New York, 1989. MR 1106659 (92e:11061)
 [4]
 P. T. Lockhart, Diophantine equations and the arithmetic of hyperelliptic curves, Ph.D. Thesis, Columbia University, 1990.
 [5]
 Ju. I. Manin, Parabolic points and zetafunctions of modular curves, Math. USSR Izvestija 6 (1972), 1964. MR 0314846 (47:3396)
 [6]
 G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, 1971. MR 0314766 (47:3318)
 [7]
 G. Shimura, On the factors of the Jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523544. MR 0318162 (47:6709)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
11F67,
11Y35
Retrieve articles in all journals
with MSC:
11F67,
11Y35
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199211220695
PII:
S 00255718(1992)11220695
Article copyright:
© Copyright 1992
American Mathematical Society
