Modular degree is an interesting invariant of elliptic curves. It is computed by variety of methods. After computer calculations, Watkins conjectured that given E/IQ of rank R, 2R I deg(<I», where <I> : Xo(N) ---+ E is the optimal map (up to isomorphism of E) and deg(<I» is the modular degree of E. In fact he observed that 2R+K divides the degree of the modular degree and 2K depends on #W, where W is the group of Atkin-Lehner involutions, #W=2w(N), N is the conductor of the elliptic curve and w(N) counts the number of distinct prime factors of N. The goal of this thesis is to study this conjecture. We have proved that 2R+K I deg( <I»would follow from an isomorphism of complete intersection of a universal deformation ring and a Hecke ring, where 2K = #W', the cardinality of a certain subgroup of the group of Atkin-Lehner involutions. I attempt to verify 2R+K I deg(<I» for certain Ellipitic Curves E, when N is 250 by using a computer algebra package Magma. I have verified when N is squarefree. Tables are in Section 5.1.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:531170 |
| Date | January 2011 |
| Creators | Krishnamoorthy, Srilakshmi |
| Publisher | University of Sheffield |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.whiterose.ac.uk/14546/ |
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