Computing modular degrees using L-functions
Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 673-682

We give an algorithm to compute the modular degree of an elliptic curve defined over ℚ. Our method is based on the computation of the special value at s=2 of the symmetric square of the L-function attached to the elliptic curve. This method is quite efficient and easy to implement.

Nous donnons un algorithme pour calculer le degré modulaire d’une courbe elliptique définie sur ℚ. Notre méthode est basée sur le calcul de la valeur spéciale en s=2 du carré symétrique de la fonction L attachée à la courbe elliptique. Cette méthode est assez efficace et facile à implémenter.

@article{JTNB_2003__15_3_673_0,
     author = {Christophe Delaunay},
     title = {Computing modular degrees using $L$-functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {673--682},
     year = {2003},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     doi = {10.5802/jtnb.420},
     zbl = {1070.11021},
     mrnumber = {2142230},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.420/}
}
TY  - JOUR
AU  - Christophe Delaunay
TI  - Computing modular degrees using $L$-functions
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2003
SP  - 673
EP  - 682
VL  - 15
IS  - 3
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.420/
DO  - 10.5802/jtnb.420
LA  - en
ID  - JTNB_2003__15_3_673_0
ER  - 
%0 Journal Article
%A Christophe Delaunay
%T Computing modular degrees using $L$-functions
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 673-682
%V 15
%N 3
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.420/
%R 10.5802/jtnb.420
%G en
%F JTNB_2003__15_3_673_0
Christophe Delaunay. Computing modular degrees using $L$-functions. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 673-682. doi: 10.5802/jtnb.420

[1] A. Atkin, J. Lehner, Hecke operators on Γ0(m). Math. Ann. 185 (1970), 134-160. | EuDML | MR | Zbl

[2] A. Atkin, W. Li, Twists of newforms and pseudo-eigenvalues of W-operators. Invent. Math. 48 (1978), 221-243. | EuDML | MR | Zbl

[3] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 no. 4 (2001), 843-939. (electronic). | MR | Zbl

[4] J. Coates, C.-G. Schmidt, Iwasawa theory for the symmetric square of an elliptic curve. J. reine angew. Math. 375 (1987), 104-156. | EuDML | MR | Zbl

[5] H. Cohen, Advanced topics in computational algebraic number theory. Graduate Texts in Mathematics, 193, Springer-Verlag, New-York, 2000. | MR | Zbl

[6] J. Cremona, Algorithms for modular elliptic curves. Second edition, Cambridge University Press, 1997. | MR | Zbl

[7] J. Cremona, Computing the degree of the modular parametrization of a modular elliptic curve. Math. Comp. 64 (1995), 1235-1250. | MR | Zbl

[8] D. Goldfeld, J. Hoffstein, D. Lieman, An effective zero-free region. Ann. of Math. (2) 140 no. 1 (1994), 177-181. | MR | Zbl

[9] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, pari-gp, available by anonymous ftp.

[10] G. Shimura, The special values of the zeta functions associated with cusp forms. Com. Pure Appl. Math. 29 (1976), 783-804. | MR | Zbl

[11] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 no. 3 (1995), 553-572. | MR | Zbl

[12] E. Tollis, Zeros of Dedekind zeta functions in the critical strip. Math. Comp. 66 (1997), 1295-1321. | MR | Zbl

[13] M. Watkins, Computing the modular degree of an elliptic curve. Experimental Maths 11 no. 4 (2003), 487-502. | MR | Zbl

[14] A. Wiles, Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 no. 3 (1995), 443-551. | MR | Zbl

[15] D. Zagier, Modular parametrizations of elliptic curves. Canad. Math. Bull. 28 no. 3 (1985), 372-384. | MR | Zbl

Cité par Sources :