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 $N$.
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.