On a conjecture of Watkins

Neil Dummigan[1]

  • [1] University of Sheffield Department of Pure Mathematics Hicks Building Hounsfield Road Sheffield, S3 7RH, U.K.

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 2, page 345-355
  • ISSN: 1246-7405

Abstract

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Watkins has conjectured that if R is the rank of the group of rational points of an elliptic curve E over the rationals, then 2 R divides the modular parametrisation degree. We show, for a certain class of E , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain 2 -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove such statements, is necessarily inapplicable to our situation. It seems then that some new method is required if this approach to Watkins’ conjecture is to work.

How to cite

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Dummigan, Neil. "On a conjecture of Watkins." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 345-355. <http://eudml.org/doc/249631>.

@article{Dummigan2006,
abstract = {Watkins has conjectured that if $R$ is the rank of the group of rational points of an elliptic curve $E$ over the rationals, then $2^R$ divides the modular parametrisation degree. We show, for a certain class of $E$, chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain $2$-adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove such statements, is necessarily inapplicable to our situation. It seems then that some new method is required if this approach to Watkins’ conjecture is to work.},
affiliation = {University of Sheffield Department of Pure Mathematics Hicks Building Hounsfield Road Sheffield, S3 7RH, U.K.},
author = {Dummigan, Neil},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {345-355},
publisher = {Université Bordeaux 1},
title = {On a conjecture of Watkins},
url = {http://eudml.org/doc/249631},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Dummigan, Neil
TI - On a conjecture of Watkins
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 345
EP - 355
AB - Watkins has conjectured that if $R$ is the rank of the group of rational points of an elliptic curve $E$ over the rationals, then $2^R$ divides the modular parametrisation degree. We show, for a certain class of $E$, chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain $2$-adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove such statements, is necessarily inapplicable to our situation. It seems then that some new method is required if this approach to Watkins’ conjecture is to work.
LA - eng
UR - http://eudml.org/doc/249631
ER -

References

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  8. E. de Shalit, Hecke rings and universal deformation rings. Modular Forms and Fermat’s Last Theorem, (G. Cornell, J. H. Silverman, G. Stevens, eds.), 421–445, Springer-Verlag, New York, 1997. Zbl1044.11578MR1638487
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  10. M. Dickinson, On the modularity of certain 2 -adic Galois representations. Duke Math. J. 109 (2001), 319–382. Zbl1015.11020MR1845182
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  12. C. Khare, R. Ramakrishna, Finiteness of Selmer groups and deformation rings. Invent. Math. 154 (2003), 179–198. Zbl1104.11026MR2004459
  13. B. Mazur, An introduction to the deformation theory of Galois representations. Modular Forms and Fermat’s Last Theorem, (G. Cornell, J. H. Silverman, G. Stevens, eds.), 243–311, Springer-Verlag, New York, 1997. Zbl0901.11015MR1638481
  14. R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur. Ann. Math. 156 (2002), 115–154. Zbl1076.11035MR1935843
  15. R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141 (1995), 553–572. Zbl0823.11030MR1333036
  16. M. Watkins, Computing the modular degree of an elliptic curve. Experiment. Math. 11 (2002), 487–502. Zbl1162.11349MR1969641
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