Modular degrees of Elliptic curves: On a conjecture of Watkins - Couverture souple

Krishnamoorthy, Srilakshmi

 
9783659349416: Modular degrees of Elliptic curves: On a conjecture of Watkins
Couverture souple
ISBN 10 : 3659349410 ISBN 13 : 9783659349416
Editeur : LAP LAMBERT Academic Publishing, 2013

Synopsis

Modular degree is an interesting invariant of elliptic curves. It is computed by variety of methods. After computer calculations, Watkins conjectured that given E over the rational numbers of rank R, 2^R divides (\Phi), where (Phi) : X_0(N) to E is the optimal map (up to isomorphism of E) and degree of (Phi) is the modular degree of E. In fact he observed that 2^{R+K} divides the degree of the modular degree and 2^K depends on {W}, where {W}is the group of Atkin-Lehner involutions, the cardinality of {W}=2^{omega(N)}, N is the conductor of the elliptic curve and omega(N) counts the number of distinct prime factors of N. The goal of this thesis is to study this conjecture. We have proved that 2^{R+K} divides the degree of (Phi) would follow from an isomorphism of complete intersection of a universal deformation ring and a Hecke ring, where 2^K is the cardinality of W^{\prime}, the cardinality of a certain subgroup of the group of Atkin-Lehner involutions. I attempt to verify 2^{R+K} divides the degree of ({\Phi}) for certain Ellipitic Curves E by using a computer algebra package Magma. I have verified when N is squarefree. Computations are in chapter 5.

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Présentation de l'éditeur

Modular degree is an interesting invariant of elliptic curves. It is computed by variety of methods. After computer calculations, Watkins conjectured that given E over the rational numbers of rank R, 2^R divides (\Phi), where (Phi) : X_0(N) to E is the optimal map (up to isomorphism of E) and degree of (Phi) is the modular degree of E. In fact he observed that 2^{R+K} divides the degree of the modular degree and 2^K depends on {W}, where {W}is the group of Atkin-Lehner involutions, the cardinality of {W}=2^{omega(N)}, N is the conductor of the elliptic curve and omega(N) counts the number of distinct prime factors of N. The goal of this thesis is to study this conjecture. We have proved that 2^{R+K} divides the degree of (Phi) would follow from an isomorphism of complete intersection of a universal deformation ring and a Hecke ring, where 2^K is the cardinality of W^{\prime}, the cardinality of a certain subgroup of the group of Atkin-Lehner involutions. I attempt to verify 2^{R+K} divides the degree of ({\Phi}) for certain Ellipitic Curves E by using a computer algebra package Magma. I have verified when N is squarefree. Computations are in chapter 5.

Biographie de l'auteur

2011 - 2013, Post Doctoral Fellow, The Institute of Mathematical Sciences, India.2007 - 2010, Ph.D. at Sheffield University, U.K.Thesis Advisor: Dr. Neil Dummigan. 2006 - 2007, M.Phil at Sheffield University, U.K.Awards: EPSRC – BP Dorothy Hodgkin Postgraduate Award to support study towards a Ph.D., U.K.Gold medalist, M.Sc Mathematics, S.R.C.

Les informations fournies dans la section « A propos du livre » peuvent faire référence à une autre édition de ce titre.

Éditeur
LAP LAMBERT Academic Publishing
Date d'édition
2013
Langue
anglais
ISBN 10 
3659349410
ISBN 13 
9783659349416
Reliure
Broché
Nombre de pages
104
Coordonnées du fabricant
non disponible
Personne responsable
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