Colloquium abstracts

Srilakshmi Krishnamoorthy
TIFR
January 13, 2011

Modular degrees of elliptic curves:  Modular degree is an interesting invariant of elliptic curves. It is computed by a variety of methods. After computer calculations, Watkins conjectured that given E/Q of rank R, 2^R divides deg(\Phi), where \Phi : X_0(N) \to E is the optimal map (up to isomorphism of E) and deg(\Phi) is the modular degree of E. In fact, he observed that 2^{R+K} should divide the modular degree with 2^K depending on W, where W is the group of Atkin-Lehner involutions, \mid W \mid = 2^{\omega(N)}, N is the conductor of the elliptic curve and \omega(N) counts the number of distinct prime factors of N. We have proved that 2^{R+K} divides deg(\Phi) would follow from an isomorphism of complete intersection rings of a universal deformation ring and a Hecke ring, where 2^K = \mid W^{\prime}\mid , the cardinality of a certain subgroup of the group of Atkin-Lehner involutions.