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Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves

Author: Hao Chen
Journal: Math. Comp. 85 (2016), 2499-2514
MSC (2010): Primary 11G05
Published electronically: December 9, 2015
MathSciNet review: 3511290
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Abstract: Let $ E$ be an optimal elliptic curve defined over $ \mathbb{Q}$. The critical subgroup of $ E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $ E(\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $ E$. We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to $ E$ and develop two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical polynomials. Finally, a table of critical polynomials is obtained for all elliptic curves of rank two and conductor smaller than 1000, from which we deduce our result.

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  • [AO03] Scott Ahlgren and Ken Ono, Weierstrass points on $ X_0(p)$ and supersingular $ j$-invariants, Math. Ann. 325 (2003), no. 2, 355-368. MR 1962053 (2004b:11086),
  • [BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $ \mathbf {Q}$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843-939 (electronic). MR 1839918 (2002d:11058),
  • [BFH90] Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Nonvanishing theorems for $ L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543-618. MR 1074487 (92a:11058),
  • [Che] Hao Chen, Computing Fourier expansion of $ \Gamma _0(N)$ newforms at non-unitary cusps, in preparation.
  • [Cre] J. E. Cremona, Elliptic curve data,
  • [Del02] Christophe Delaunay, Formes modulaires et invariants de courbes elliptiques définies sur $ \mathbb{Q}$, Thèse de doctorat, Université Bordeaux 1 (décembre 2002).
  • [Del05] Christophe Delaunay, Critical and ramification points of the modular parametrization of an elliptic curve, J. Théor. Nombres Bordeaux 17 (2005), no. 1, 109-124 (English, with English and French summaries). MR 2152214 (2006c:11075)
  • [GZ86] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $ L$-series, Invent. Math. 84 (1986), no. 2, 225-320. MR 833192 (87j:11057),
  • [Lig75] Gérard Ligozat, Courbes modulaires de genre $ 1$, 1975 (French). Bull. Soc. Math. France, Mém. 43; Supplément au Bull. Soc. Math. France Tome 103, no. 3, Société Mathématique de France, Paris. MR 0417060 (54 #5121)
  • [Mah74] Kurt Mahler, On the coefficients of transformation polynomials for the modular function, Bull. Austral. Math. Soc. 10 (1974), 197-218. MR 0354556 (50 #7034)
  • [MS04] T. Mulders and A. Storjohann, Certified dense linear system solving, J. Symbolic Comput. 37 (2004), no. 4, 485-510. MR 2093448 (2006c:11151),
  • [MSD74] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1-61. MR 0354674 (50 #7152)
  • [S$^+$14] W.A. Stein et al., Sage Mathematics Software (Version 6.4), The Sage Development Team, 2014,
  • [Ste] William Stein, Algebraic number theory, a computational approach,
  • [Yan06] Yifan Yang, Defining equations of modular curves, Adv. Math. 204 (2006), no. 2, 481-508. MR 2249621 (2007e:11068),

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Additional Information

Hao Chen
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98115

Received by editor(s): January 17, 2015
Received by editor(s) in revised form: March 1, 2015, and March 18, 2015
Published electronically: December 9, 2015
Article copyright: © Copyright 2015 Hao Chen

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