×

Modular degrees of elliptic curves and some quotient of \(L\)-values. (English) Zbl 1469.11139

Summary: By the modular degree we mean the degree of a modular parametrization of an elliptic curve, namely the mapping degree of the surjection from a modular curve to an elliptic curve. Its arithmetic significance is discussed by D. Zagier [Can. Math. Bull. 28, 372–384 (1985; Zbl 0579.14027)] and A. Agashe et al. [in: Number theory, analysis and geometry. In memory of Serge Lang. Berlin: Springer. 19–49 (2012; Zbl 1276.11087)] in terms of the congruence of modular forms. Given an elliptic curve \(E_f\) attached to a rational newform \(f\), we explicitly relate its modular degree to a quotient of special values of some two \(L\)-functions attached to \(f\). We also provide several numerical examples of the formula.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields

Software:

ecdata
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] \BibAuthorsA. Abbes and E. Ullmo, À propos de la conjecture de Manin pour les courbes elliptiques modulaires, Compositio Math. 103 (1996), 269-286. · Zbl 0865.11049
[2] \BibAuthorsA. Agashe, K. A. Ribet and W. A. Stein, The modular degree, congruence primes, and multiplicity one, Number theory, analysis and geometry, Springer, New York (2012), 19-49. · Zbl 1276.11087
[3] \BibAuthorsC. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over \(\Q \): wild 3-adic exercises, J. Amer. Math. Soc. 14 (4) (2001), 843-939. · Zbl 0982.11033 · doi:10.1090/S0894-0347-01-00370-8
[4] \BibAuthorsF. Calegari and M. Emerton, Elliptic curves of odd modular degree, Israel J. Math. 169 (2009), 417-444. · Zbl 1275.11096 · doi:10.1007/s11856-009-0017-x
[5] \BibAuthorsA. C. Cojocaru and E. Kani, The modular degree and the congruence number of a weight 2 cusp form, Acta Arith. 114 no.2 (2004), 159-167. · Zbl 1108.11042
[6] \BibAuthorsJ. E. Cremona, Algorithms for modular elliptic curves, Second edition, Cambridge University Press, Cambridge, 1997. · Zbl 0872.14041
[7] \BibAuthorsC. Delaunay, Computing modular degrees using \(L\)-functions, J. Théor. Nombres Bordeaux 15 no.3 (2003), 673-682. · Zbl 1070.11021
[8] \BibAuthorsC. Delaunay, Critical and ramification points of the modular parametrization of an elliptic curve, J. Théor. Nombres Bordeaux 17 no.1 (2005), 109-124. · Zbl 1082.11033
[9] \BibAuthorsF. Diamond and J. Shurman, A first course in modular forms, Springer, New York, 2005. · Zbl 1062.11022
[10] \BibAuthorsM. Flach, On the degree of modular parametrizations, Séminaire de Théorrie des Nombres, Paris, 1991-92, Progr. Math., (116), Birkhäuser Boston, Boston, MA (1993), 23-36. · Zbl 0840.14019
[11] \BibAuthorsB. Gross, Heights and the special values of \(L\)-series, Number Theory, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, RI, 1987, 115-187. · Zbl 0623.10019
[12] \BibAuthorsH. Hida, Congruences of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), 225-261. · Zbl 0459.10018
[13] \BibAuthorsH. Hida, On congruence divisors of cusp forms as factors of the special values of their zeta functions, Invent. Math. 64 (1981), 221-262. · Zbl 0472.10028
[14] \BibAuthorsA. Knapp, Elliptic curves, Princeton University Press, Princeton, New Jersey, 1992. · Zbl 0804.14013
[15] \BibAuthorsY. Manin, Parabolic points and zeta-functions of modular curves, Math. USSR-Izv. 6 (1972), 19-64. · Zbl 0248.14010
[16] \BibAuthorsP. Michel and D. Ramakrishnan, Consequences of the Gross/Zagier formulae, Stability of average \(L\)-values, subconvexity, and non-vanishing mod \(p\), Number theory, analysis and geometry, Springer, New York (2012), 437-459. · Zbl 1276.11057
[17] \BibAuthorsT. Miyake, Modular forms, Springer-Verlag, 2006. · Zbl 1159.11014
[18] \BibAuthorsM. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular L-series, Ann. Math. 133 (3) (1991), 447-475. · Zbl 0745.11032 · doi:10.2307/2944316
[19] \BibAuthorsG. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 no. 6 (1976), 783-804. · Zbl 0348.10015
[20] \BibAuthorsM. Watkins, Computing the modular degree of an elliptic curve, Experiment. Math. 11 no. 4 (2002), 487-502. · Zbl 1162.11349
[21] \BibAuthorsA. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math. 141 (2) (1995), 443-551. · Zbl 0823.11029 · doi:10.2307/2118559
[22] \BibAuthorsD. · Zbl 0579.14027 · doi:10.4153/CMB-1985-044-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.