**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.