Modular degrees of Elliptic curves | On a conjecture of Watkins

Srilakshmi Krishnamoorthy

ISBN 10: 3659349410 / ISBN 13: 9783659349416

Edité par LAP Lambert Academic Publishing, 2013

Modular degrees of Elliptic curves | On a conjecture of Watkins | Srilakshmi Krishnamoorthy | Taschenbuch | Englisch | LAP Lambert Academic Publishing | EAN 9783659349416 | Verantwortliche Person für die EU: preigu GmbH & Co. KG, Lengericher Landstr. 19, 49078 Osnabrück, mail[at]preigu[dot]de | Anbieter: preigu. N° de réf. du vendeur 106052807

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.

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.

Détails bibliographiques

Titre : Modular degrees of Elliptic curves | On a ...
Éditeur : LAP Lambert Academic Publishing
Date d'édition : 2013
Reliure : Taschenbuch
Etat : Neu